Math is important for building and engineering. People need better math skills. Khan academy is the best online source I have seen. It teaches concepts to quickly get you to higher math. It does not endlessly quiz on problems to build skill in rudimentary areas. Use khan academy to learn the basics and master them quickly so you can advance to the next chapter.

There are four basic types of math. Arithmetic, algebra, geometry, and calculus. From there it just gets more specialized. Arithmetic is adding, subtracting, multiplying, and dividing. This is like grade school math 4th- 6th grade. This is the type of math seem commonly in life. This is the level most people can perform at well and easily understand. After this people have weak math skills since they do not learn it or need it for their life or job.

Algebra is the study of equations where you use variables to represent unknown numbers, usually x and y. In algebra you are trying to get both halves of the equation to be equal to each other. This differs from arithmetic. Arithmetic wants to find an answer while algebra wants to compare two different quantities to each other. For example, "4x= 10" is asking how many fours does it take to equal ten. It is comparing a known quantity to an unknown quantity. We rewrite this as an arithmetic problem to solve in the form, "10/4= x", with x being "?" or unknown. After using division to solve we see that 10/4= 2.5 so x must equal 2.5. This is algebra. Usually prealgebra is taught in 7th grade and algebra is taught in 8th-9th grade.

Geometry comes after algebra during the 10th grade. Geometry is the study of shapes starting with two dimensional and then moving to three dimensional. It uses specific definitions to label lines and parts of shapes in a way that makes referring to them standardized across all industries. For example;

Calculus comes after algebra. It is the study of measuring the rate of change of a point on a line graph. There are two types of calculus: integral and differential. Integral measures the inside of curved spaces. Differential measures the outside of curved spaces. In automotive engineering integral calculus measures spaces such as the inside of engine cams to find displacement. Differential calculus is used for finding gear ratios and determining turning radius of a wheel to a curve. There are other math classes between algebra but multivariate calculus and linear algebra is the highest math and used for rocket science and computer programming. We will be using this for our engineering.

K-6th grade is arithmetic. 7th grade is prealgebra. 8th-9th is algebra. 10th is geometry. 11th is trigonometry. 12th is statistics and pre calculus. Freshman year (13th grade) at college is calculus ab. Second year (14th grade) is discrete mathematics. Junior year (15th grade) is calculus bc. Senior year (16th grade) is linear algebra. After that is advanced classes for scientists, physicists, teachers, and so on. At that point one is truly elite and spends much of there time doing research and writing papers.

There are two main divisions in the academic world. They are theoretical science and applied science. We will focus on applied science. Applied science is work or task orientated. It is using math to build things in the real world. You could consider it vocational training. This includes city engineers, electrical engineers, structural engineers, building architects, computer architects, design architects who specialize in technical drawings and blueprint making, inventors, product manufacturers and so on. Theoretical science majors go on to do secret work for the government. They are involved with designing and building future technology and concepts. They try to find answers to hard questions that are not easy to prove, this often involves space or other non pertinent industries where they are allowed to theorize how various laws of science work to deepen understanding of nature elements like atoms, DNA, evolution, space travel, parallel and multi universes. They do not make or sell goods or services and therefore are dependent on government and private donations to fund their work. Often these are harder subjects to pursue because you are trying to answer questions that have no precedence. Applied science is easier and more rewarding financially because of the prospect of being fully funded through capitalistic commercialization of produced goods and services. The hardest course at Harvard is theoretical physics.

We use arithmetic in our everyday lives for simple counting. We use algebra to equate reoccurring problems in business to sets of data called functions. Functions are used in economics to track just about everything including CPI consumer price index). Khan academy has some excellent material on economics as well. Personally I am more tasked with creating the economy than following it. Geometry has vast applications in medicine, biology, and other health related fields. It probably has applications in soft computer arts such as deep learning and AI (artificial intelligence).

At a basic level geometry is most practical for learning manufacturing and how to program and automate machines and use then to cut material to length and assemble it to build consumer goods. CNC operators use x y z to manipulate cutting machines, in metal working such as auto robotics use similar schemes to cut and weld parts. Using computer assisted robotics is safer and lowers accidents while raising productivity. Blender uses x y z to create 3d animated models. Autocad uses x y z to design just about anything from houses to fashion clothing.

Trigonometry is an extension of geometry and it valued by the government for public and private sector applications. Trigonometry is the study or right angles. It uses sine, cosine, and tangent to calculate the degree of an angle to measure the resulting length of it's sides. Trigonometry is is not very hard but has some usable tools we can exploit to connect our algebra to calculus. The Pythagoras theorem states length of the hypotenuse of a right angle triangle can be calculated using the formula + = . A and b are the legs and c is the hypotenuse. Using this we can take virtually any space or shape and draw imaginary right angle triangles on it to find lengths of the hypotenuse and then use other math forms to deduct solutions. Trigonometry is something to do with navigation and weapons, it is used by the military. Line of sight, ballistics, trajectory of projectiles like missiles/bullets, GPS for using triangulation and transponder locating are all examples of heavy trigonometry use. You want to use trig ratios when you are a sniper on a rooftop and have to calculate the angle and distance to target, then use that information to adjust for wind and elevation to deal with bullet drop and drift. Trig can be used for some building and air traffic control but that is not the scope here. Trig to track weather systems and certain esoteric uses can be employed, using the convention of vectors, multitudes, and magnitudes. Svg graphics uses vectorization to produce extremely smooth lines. Because of the skills taught in it trigonometry is a prerequisite to calculus. We learn enough of it not to solve problems in trig but to understand and explain our calculus better.

Refine our early math enough to dominate algebra. From early math we learn the number line (remember a line has no beginning or end so a number line is -∞ ∞+) with x as our only axis. From algebra we add another direction, ↕, using y as our vertical axis turning our number line into a line graph (this is where all those business charts come from). Geometry gives us yet another direction, ↖ ↘, making our line graph three dimensional (this is used in manufacturing on plasma burn tables, water jet tables, Leigh™ router tables, also 3d printers). Trigonometry gives us trackable data in the form of a 2d shape drawn on the line graph that moves from one position to another. It uses to tell us the multitude of the data. Magnitude would be expressed as an inequity(technically an "inequality" refers to 1>0 while inequity is a tradable commodity such as stock (and its subsequent growth)). Calculus gives our line changeable rate data. Instead of a simple straight line, our line can now bent like to show stock market stability or for volatility. Calculus has a lot of things going on. It uses curved lines where as trig uses dead straight ones. Calculus is used in extreme design to produce impossible looking svelte windswept shapes for curvy car chassis and airplane wing and body design. Curves are hard and would be impossible without calculus. Without calculus to back it up all those incredible looking vehicles would not be functional, instead they would be pretty looking prototypes that are not practical for actual deployment.

(*note-This indented text is for remedial use only. Feel free to skip ahead. I use eccentric notation that will not be familiar to you. Through this course and others you may see symbols and color coding that is recognizable and some that is not. Often times you may see an eclectic mix of knowns and unknowns. It is with meaning. Sometimes it is deliberate and others times it is luck. For xyz I used the convention found in Blender program to ease transition into 3d modeling. It occurred to me after writing this that they in turn probably used their color schemes from other texts and accepted modern norms. I found myself coloring more than I intended to create connections, mnemonics and subliminals to further enhance content though it is not needed.*

Study basic math to have solid understanding of PEMDAS, functions of math laws and how to manipulate equations. Complete algebra and then quickly move through geometry and trigonometry so you start dealing with calculus and engineering math. Once we understand higher math we need to make it functional. We need to add another discipline so that we can dominate that industry. We can add computer programming or we can add manufacturing. Ultimately we are either building computers, building cars, or building architectural works. Out of that we can build anything else.

Year one- Algebra
Year two-Geometry/Trigonometry
Year three- Calculus
Year four- Combinotorics

We can add other useful science to math such as chemistry and physics. Physics is important for understanding electricity. Chemistry is important for understanding industrial science and medicine. These are the main courses to pursue for a math intensive career. Medicine also relies highly on biology as an offshoot of chemistry. Also economics and finance is important but not very math critical. The math used in economics is not very impressive. Economics focuses more on business leaderships and networking skills rather than counting and calculating numbers, only simple formulas are used in business like algebraic functions and interest (prt). Khan academy has excellent videos on physics and chemistry as well as a starting point. Their Economics is good but you should check out other sites like and to learn the terminology of finance and see how finance is actually conducted. Investopedia is highly recommended for teaching yourself how to invest and read financial markets. Reading markets enables you to invest in your own companies more, rely on other companies less, and know when to poach other companies using poison pawn attacks.

So we should also learn a basic amount of:

I am trying to condense math into four areas. Algebra, geometry, calculus, and linear algebra. The point is to quickly teach math to train students to work either in my factories as machine operators and assembly men or in my research and development departments as engineers and scientists. The scope of manufacturing is design using xyz to perform layout of parts, cutting of parts, and finally assembly of parts. When limited to a simple scope all four areas of math become easy to learn and grasp because the student is actually using them in work based environments. Engineering and scientific work is more complex but again is made easier by the fact the student is performing actual work instead of dealing with theoretical work. When they are responsible for designing actual systems they become intimately familiar with them leading to greater retention of knowledge. It is a fast track method that has worked historically in business. Regular school teaches theory that you may never use. Since they do not train for specific industries they do not know how you will use the math and therefore over test and quiz on many different hypothetical scenarios instead of limiting themselves to a single focus. I have specific problems that need to be addressed and that is what I train. Students will at least be able to apply mathematical concepts in my work related fields. From this they are free to explore math on their own and make connections with other applications. This is better than traditional school that wastes time trying to teach all math with an exhaustive list of options. That increases the time it takes to learn specifics of math that you will actually need. Once you learn the math you need you can surmise the rest. The biggest hurdle to math proficiency is accessibility and purpose. If math is several years of mindless study then it is inaccessible due to expense of time and lacks meaningful purpose. No teacher will tell you why you have to learn math. They will just say it will matter one day figure it out. That is demotivational. My course is shorter and purpose structured. I do not teach math games or riddles, I do not troll or try to trick students to see if they can figure it out. I layout examples for them to play with the equations and see if they can understand the meaning, manipulate the structure of the equation, and then let them build whatever connections they wish. I train my employees on the job and guarantee their success as my business model depends on it. Since I am doing this I am incorporating the cost of training them myself. Teachers are already paid by their school so do not have an incentive to ensure student success and even with graduating students can not guarantee employment. Not only am I increasing economic outlook and job prosperity I am also raising grammatical, mathematical, and technological literacy of society. I am a humanitarian based philosophy.

The following link is a google document containing the information below it but with pictures.

Malone Math Google Docs

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This book will cover all math from high school algebra to fourth year linear algebra. It will use mixed examples through out blending advanced concepts with foundational skills. It will start off will algebra then mix in calculus in the beginning with just enough geometry and trigonometry to cover all important topics necessary for passing assessment into a first year course at the university level. It also will cover the first two years of a masters degree.

Since this is an exhaustive undertaking emphasis will be on building foundation through clear understanding of vocabulary and mathematical terms. All math with be cross linked as much as possible so that time can be spared and more ground covered. There will be thorough definition, illustrations, and process fores shown for every operation in math. With an understanding of how to do math the idea is you will not actually have to due much. Will try to cover 4 years of high school math in one year and four years of college math in one year from a strong pure sense while also adding abstraction. Adding abstraction shows readers the order in which math progresses giving them early exposure. The more exposure they have the more likely they will seek advanced math or be able to create their own equations and formulas sooner.

Formal logic and writing proofs will be covered. Early definitions will be given plainly so that children are not overwhelmed. I will try to breeze through each topic as fast as I can while leaving no question as to how I arrived at the logic I did. To bridge different math the early explanations are short so it does not interrupt the flow of instruction. I am putting many pieces of information that has been surmized from my own experience plus the internet and other books. If you feel you need more practice in a certain area buy a dedicated algebra and calculus book used from a thrift store for less than $20 each.

In reality you do not need to spend time extraneously plodding over the same questions over and over. You need a concise view of all the mathematics and their relationship to each other. You will be able to create equations that explain your real life situations and be able to solve them will ever increasing sophistication. In addition you should gain confidence to explore other topics not covered in this book once you have completed it. See what math others are doing and you the internet to join their conversations and work on projects with them. Experiment with their formulas and write your own.

There are two parts to math and they are the pure and the applied. Applied is the area that defines job skills and real life conditions. Pure is abstract in the sense that it only serves as a teaching model to give instruction on the legal operations that can be done within the confines of the agreed upon laws of math. Pure defines what math is and is not and applied is how you use it to enrich your life. Math as a tool is used to solve real world problems and if it is not doing that then it has failed its purpose and real for even being. Concepts will be introduced in a very nontraditional format but still achieve the same goal of teaching.

With that being said consider the points on the following two pages.

Points of consideration

  1. I wrote this book for everyone regardless of skill or age. Only basic arithmetic skills like

    (+, -,, ÷) are required. Most people have already learned these skills in 4th -6 th grade so I do not want to rehash basics that far back. If kids are reading this book it will make clear subjects and terms that they are not likely to see until much later or might have missed. A kid reading this book will improve not only their math but all subjects they face in school as well as their personal life because of the strong logic that it will build in them. Even if the material is too advanced they benefit from having a well referenced book. It will prepare them for college and beyond.

  2. This book is split into three parts. It is at once a formal textbook, a series of notes from an upperclassman, and a conversation with a peer. As it begins slow skip parts that you do not need to deal with. As it is conversational raise questions that you have and we can devise a solution together. There is no such thing as a perfect book and I worked hard to write this at consider cost of my own time and money. I wrote this book because my passion for this project and helping others outweighed any other goals in life that had I pursued I would have been better of financially and socially. I put priority into this project above my own needs so that others would not have to go through the trouble of trial and error doing google or you tube searches to get bits or pieces of information they were looking for. This book truly does replace others attempts and it does not come with a subscription or or other reoccurring costs. Once you own this book it is yours to keep forever and use not only for yourself but for your children and friends as well.

  1. Definitions will get more rigorous but start casually for ease of use. They will not be vague however. Certain parts that are basic may have examples what seem to been overly lengthy taking up a page or more. I only do that so that all my constraints are met to keep in theme of the book. I can not assume what people know or their educational exposure outside this book. This is to be a self contained reference except for the glaring omission of problems to solve. When a problem is given it is dealt with in the most complete manner possible so multiple teaching points are given.

    Other books do not do this. There were points where I was tempted to follow more professional texts that stated the process, gave an example, and then stated process once more and was done. I take a more examined look that goes past giving clues and deconstructs a problem as much as I can until there are no unknowns left about it. I only go onto the next instruction when I am satisfied that I have nothing more to explore or prove. This actually cuts time required to learn because nothing is left hidden or unexplained that will lead to confusion or lack of understanding later. This tends to happen when changing course to course and book to book. There are gaps.

  1. Many books exist that teach the basics. I go beyond them by taking advanced concepts and making them easy to understand and then applying them in a real situation where applicable. By doing this I transition from textbook lecturer to enthusiastic individual willing to share his notes on a given subject or problem. This does away from the standard teacher student relationship by giving readers what an authoritive book won't, the clear approaches to process without quizing them or challenging them to find it.

  2. The order of this book will be very simple algebra applied with higher level concepts like functions and Greek notation. I am not making easy problems harder, I am introducing higher concepts earlier using almost to easy problems. This let's readers see the full mechanics under the hood so they can move to any course work any where and know what to do. In this sense this is less a textbook but more a tutorial or how to guide.

  1. After algebra I will show calculus and then linear algebra. After foundations and terminology is covered I will math will be used in a more direct sense. Students will be able to write their own formulas for over 189 real life problems.

  1. I give exhaustive explanations for every math word and sign that exists so there is no gaps in knowledge. This benefits kids and adults alike. If you can read English you can use this book. I write and speak at a university level without leaving readers behind. If you do not know a word feel to look it up and not only get better math skills but reading and writing as well.

  1. Outside of process there are also fun concepts introduced to make math interesting. While technically not required for success at math the history, language and anecdotes makes for a greater experience.

  1. Finally, the most advanced subjects are beyond the scope of this book. This is mostly for foundation building so that readers will be able to read the sequel to this book I plan on producing. These go beyond pure math into masters and doctorate degrees in engineering fields like electrical and programming as well as gravitational and quantum mechanics. These sciences are founded in math but also require specialized study of chemistry and other physics. I can not prime readers for that in one volume. I will touch on what I can but will need to split documents into separate volume for portability and ease of use. I do not try to put all into one book and only combine the information I do for fear of it being separated and do not want that to happen.


Basic math:

  1. Arithmetic- The study of adding, subtracting, multiplying, dividing, powering, and radicalizing.

  2. Algebra- The study of balancing equations using variables to solve for an unknown quantity.

  3. Geometry- The study of shapes and their dimensions, number of sides, measure of their angles.

  4. Trigonometry- The study of right angles and their measure. Using laws of trig to find unknown

    angles or distances.

  5. Calculus- The study of rates of change using formulas for slope such as (y2 -y1) ÷ (x2 -x1).

  6. Statistics- The study of odds and probability. This leads into further number theory.

Advanced math:

  1. Linear algebra- Studying higher level linear equations using matrices, determinants, and Gaussian elimination.

  2. Discrete mathematics- Studying logic in the form of formal written proofs, Getting into set theory and set notation to list things in groups of numbers called sets.

  3. vector algebra- Related to linear algebra, focuses on moving through vector spaces using multi dimensional plane geometry.

  4. Tensor calculus

  5. Lambda calculus- Formalizing mathematical concepts in a pure form for later use in emerging fields such as science and computer technology.

  6. Differential equations – Higher levels of calculus concepts dealing with vector spaces.

  7. Number theory – Statistics using tools like Bayesian theorems and computer modeling. It helps to minor in programming using R language and libraries.

  8. Combinatorics – Set theory with application for computer algorithms.

  9. Topology- Algorithms using ordered data in a set theory environment.


Variable: Letter used to symbolize unspecified numbers. Used in algebra. Ex. x-1 =7. x =8.

Algebraic expression- Using letters as variables in equations.

Factors: Numbers used as a base in multiplication.

Product: Result of numbers multiplied.

Raised dot: This symbol · used for multiplication. Has seven other names including interpunct.

Obelus: This symbol ÷ used for division. Also called multiplication sign. Alt 0247.

Fraction bar: The line between two numbers to indicate a fractional sum such as this . Can also be horizontal but that is mostly used for solving equations as a fill in for an obelus since fractions are the quotient of division. Forward slash can be used in place of a horizontal bar if a computer does not have access to it. Alt 0188= ¼ alt 0189= ½ alt 0190= ¾.

Quotient: The resulting number from a division operation. The dividend is operated on by the divisor resulting in a quotient. 100 divided by 4 equals 25. 100 is the dividend, 4 is the divisor, 25 is the quotient.

Ratio: A ratio is sometimes but not always a fraction comparing two amounts. In chemistry and science we call this a solution or a proportion. Ratios tell either the amount of different elements added to each other at what rate or the rate at which something is changing. Ratios are written using a colon as 10:1 properly and spoke,” Ten to one”. This differs from writing them as a fraction since in this case we want to know the two quantities being compared. Using both numbers tells us that one element is 10x greater than the other. A ratio given as a fraction would entice us to simply the fraction or write it as a decimal. The ratio 3:2 becomes 1.5, causing the second element to be lost and only the first one remaining. Context of subject matter determines whether we want to keep information or not however colon formula is often preferential.

Base: The number to be expanded using exponents. 107 has 10 as the base and 7 as the exponent, or power.

Power: Another name for exponent. 22 is “2 squared”, 23 is “2 cubed”, 24 is “2 to the fourth, 2 raised to a power of four, 2 expanded to a power of four”. Raising to a power of 2 is called squaring it, raising to a power of 3 is cubing, raising to 4 and beyond is raising to a power of.

Squared: Any number times itself is said to be squared. 2·2=4 4 is the square of 2.

Cubed: Any number raised to a power of three is cubed. 27 is the cube of 3. A base times itself twice. 33= 27.

Raised to a power of: Taking a base and raising it to a power of 4 or higher. See previous example of “power”. The product of a powering operation is called a power. 4 is the power of 22.

Gross: 122= 144. This is called a “gross”.

Great gross: 123=1728

Volume: The amount of space inside a three dimensional shape serving as a container for some medium such as air or liquid. Used to measure the size of something using mathematical constants. Formula is lwh= volume when l= length, w= width and h= height. Used in geometry and related fields.

Area: The space occupying a two dimensional plane measured by multiplying length by width. Formula is given as lw= area when l= length and w= width. Used in geometry and related fields.

Perimeter: The length of the borders of a two dimensional shape or plane. Formula varies based on the number of sides a two dimensional plane has.

Formula for perimeter:

  1. Triangle= x+y+z. Measure all three sides and add.

  2. Square= 4z. Measure one side and multiply by 4.

  3. Parallelogram = 2(x+y). Measure sides then distribute out and add.

  4. Trapezoid= x+2y+z. Measure sides then add.

  5. Circle= dπ. Multiply diameter by pi or multiply radius by 2 and them multiply by pi. Called circumference instead of perimeter.

  6. Oval track= πh+2l. Use πh+2l to find length of track. ( pi ·height +2·length).

    π·60 +2(120) π·60+ 240 188.496~ +240 428.496 meters

    Use length·width (lw=area) find area of the rectangle and πr2 to find area of the circle. You only have to find area of the whole circle not two individual halves.

    lw+ πr2 60·120+ π30·30




  1. πr2. Measure the widest part of the circle to find the diameter then divide it in half to find the radius. Take the radius and square it then multiply by pi (π).

Operators/Operations: Operators are the symbols used to indicate the operation to be performed. Operations the method of increasing or decreasing a quantity or number. Operation signs are called addition (+), subtraction or minus (-), times or multiply (× alt 0215] or · [alt 0183]), divide (÷ [alt 0247]), power (^ or write as exponents using superscripts), and radical (√ [alt 251]. There are six main operations, each corresponding to it's own unique symbol telling how to proceed with determining a solution to a given problem. More symbols can and do exist for the same operations so there is some redundancy. These other symbols exist as a convenience so that they make reading an equation easier to do. Often the (×) is replaced in algebra with the center dot (·) so that the × sign is not confused with the variable “x”.

Most if not all math symbols (called operator signs or math signs) come from earlier and archaic sources. Many times these symbols have been repurposed for many different uses. They have variant sizing, patterns, and different names. Sometimes there are different names for the different versions. Mostly used for annotating literature these older names and versions are not likely appropriate for math or proofreading anymore. They are obsolete and many people would not understand them or their purpose making them only acceptable for private use and not public consumption. You are not expected to know these or learn them. You are however expected to learn the Greek letters and symbols commonly used in math, physics, and engineering. Using archaic and obsolete symbolism will make notes messier and hard to read so their use is discouraged. In addition to learning math operators and signs it is also useful to learn keyboard shortcuts to put them in computer text software programs. These signs do not appear on modern keyboards so you but to insert them using alt shortcuts. Alt shortcuts only work on keyboards with a 10 key numerical pad so you need to invest in one if you want to be productive. Learning to insert them onto your documents and websites is handy. They can even be used in search bars and comments. To embed them in a webpage you have to use html codes and they are different but same idea as alt shortcuts. Warning: Some courses have different meanings for use!

Statement: A line of text or something said giving some description to the conditions of the facts at hand. We use these facts to form our equations so that we can solve them. Statements are often used in word problems when explaining the equation in written English rather than using algebraic expressions. Note however that algebraic expressions and equations themselves are forms of statements as well. In closing a statement must be true or false but not both.

Open sentences/Solving the open sentence: Mathematical expressions containing variables whether written in script or as an equation that must be solved before you know if the statement it is making is true or false. When we replace variables with numbers the sentence is said to be solved.

Solution: A solution is a number that solves our equation.

Equation: A set or string of terms with operations that requires working to derive a solution. In Algebra equations are balanced so that two sides, a left and right, or separated by an equation sign (=). To keep the equation true you have to perform equal actions on either side or your equation will become unbalanced and therefore false.

Set: Sets group numbers or objects together so that they are listed using parentheses to differentiate them from other data. Using parentheses like this is called set notation and is mainly a study of set theory. Set theory uses an elongated version ( ) of the Greek letter epsilon (ε [alt 238]) to denote members of a set. Members of a set are also called elements. This is used in computer science for object orientated programming (oop), writing strings, arrays, and other places where programming code deals with nested items. Sets also show up in algebra when dealing with simple linear equations and plotting coordinates on a linear graph. Groups of points, called ordered pairs [(0,1)], can be collected and placed inside set notation to display what valves are relevant or appearing for a given function or chart.

For example if we have the points (0,1), (1,2), (2,3),(…), we would nest them like {(0,1); (1,2); (2,3);(...). The ε means that each pair is a “member” of this set.

Sets are also used in number theory and many other places in math. In spreadsheets the Greek letter sigma (Σ [alt 228]) is used to add. We call this summation or the summing of values inside the nested parentheticals. For example if we had Σ={1,2,3,} it would read as “the summation of {1,2,3,}” and that means that we add and get “6”. Whether actually adding or just knowing that the values inside are to be treated as a singular value this is how sigma is used.

Note: Greek letters vary in meaning from branch of one mathematics to another. Uppercase epsilon looks just like the English letter E. To avoid confusion exaggerated versions are used to illustrate the meaning better. This is why is used instead of E or the regular Greek letters. Uppercase epsilon is usually for elements of a set and lower case epsilon (ε) is for denoting a small quantity. There are many uses for the Greek letters and I have included as many of them and their alt codes that I know but sometimes there will be contradictions in terms so you will have to substitute whatever you can and just improvise. If you are using eccentric notation that a reader is likely to be unfamiliar with make that clear before using ambiguous terms. It really only matters that you and your intended audience can understand what notation you are using not anybody else.

Subset: A set with a set. Infinity (∞ [alt 236]) is the set of all numbers. All even numbers are a subset of ∞. All even numbers ending with zero are a subset of that set. All even numbers ending with zero under 100 are a deeper nested subset still.

For example: {+∞ (2,4,6...(10,20,30...100,110,120...(10,20...90)))}.

This shows that within positive infinity (as denoted by the +) a subset exists that contains the set of infinite positive even numbers (as denoted by the ellipsis (…)). Then exists all positive even tens, finally there exists a set of even tens that ends at 90.

Null: Ø. This means that a set contains no numbers. {Ø} means that there are no numbers in this set, not even zero as a placeholder. Literally {} instead of {0}.

Empty set: A set with no elements is called empty or null.

Replacement set: The solution to an open sentence occurs when all variables have been accounted for using numerals. The replacement set for the variables is the solution to the expression. In factoring polynomials you often use replacement sets to find the factors that make up a trinomial.

Negation: Negation means reversing an operation or taking the opposite of something. Subtraction negates or is the negation of addition. By multiplying by a -1 we can negate terms of equations (indeed the entire equation itself) to be the opposite of sigh of what it already is. This is useful simplifying and reducing equations to make them cleaner looking or easier to solve.

Properties: Properties are rules that dictate how order of operations should behave and what is permissible when reordering terms to restructure an equation. Properties show an how the numbers behave and interact with each other on a foundational level as opposed to simply solving for an answer. Understanding and having innate sense of properties shows an advanced mastery of mathematics and goes beyond superficial plugging in of values to get a result. You know how the numbers behave conceptually rather then mechanically.

Additive identity property: For any number a, a+0 = 0+a. This states that the order of terms is irrelevant and moving them around does not change the answer.

Multiplicative identity property: For any number a, a·1 = 1·a. This states that the order of terms is irrelevant and moving them around does not change the answer.

Multiplicative property of zero: For any number a, a·0 = 0·a. This states that the order of terms is irrelevant and moving them around does not change the answer. [In addition any number times zero is zero. Further diving by zero is undefined. It is undefined since dividing by zero does not give us non zero factor.]

Properties of equality: The following properties are true for any a, b, or c.

Reflexive: a = a

Symmetric: if a = b then b = a.

Transitive: if a = b and b = c then a = c.

Substitutive: if a = b then a may be replaced by b.

Distributive property: For any numbers a, b, c:

a (b + c) = ab + ac and (b + c) a = ba + ca

a (b – c) = ab – bc and (b – c) a = ba – ca

Commutative property: Rehashes additive and multiplicative property.

Associative property: For any numbers a, b, c (a + b) + c = a + (b + c) and (ab) c = a (bc). It says that if you change the grouping for + or ÷ the result is the same.

Term: Is a number, variable, product, quotient, or mixed number. Terms are the groups of numbers separated by operands. Each term may be composed of multiple parts but is considered a single value. A trinomial has three terms each separated by a plus or minus sign.

Simplest form: An expression without like terms or parentheses. This considered the final answer when solving equations.

Like terms: Terms that contain the same variables raised to the same power. 5x2 and 4x2 are like terms but 2x3 and 5x are not. 5xy5z and y5xz are like but 5x5yz and 5xy5z2 are not.

Coefficient: The constant numeral factor in a mixed number. In 5x 5 is the coefficient because it is a constant number.

Constant: A number that is not a variable. Numbers 0-9 are constants because they are not alphabetic. In 5x the coefficient 5 is the constant and x is the variable. All variables have letters to denote their changing values. All constants except higher level ones have numbers to denote their value is always the same. Some constants like Euler's number “ e ”, the imaginary number “ i ”, and π use letters to represent their value but this is because they are very long and only used as a convenience. They are considered constants because their value is known and it does not change.

Pi: The mathematical constant approximated by 22/7 to denote the symbol π. Pi is used to calculate circumference and area of circles using the formulas dπ for circumference and πr2 for area. There are different approximations for pi but the symbol always remains the same.

Euler's Number:

Imaginary number i:

Fundamental identities:

Addition formulas

Subtraction formulas:

Formulas for negatives:

Cofunction formulas:

Double angle formulas:

Half angle formulas:

Product to sum formulas:

Sum to product formulas:

Tangent and cotangent Identities

Formula: Equation that states a rule for the relationship between certain quantities.

Triangle: ½ bh= area.

Sum of angles+ A+B+C= 180°

Equilateral triangle: h=3s/2 area= 3s/4

Right angle hypotenuse: a2+ b 2 = c2.

Trigonometric functions:

Of right angles:

Of arbitrary angles:

Of real numbers:

Special right angles:

Law of cosines:

Law of sines:


Heron's formula:

Special values:



Sin θ

Cos θ

Cot θ

Sec θ

Csc θ






– – – – – –

























– – – – –


– – – – – –


Greek Alphabet

Alpha Α α

Digamma Ϝ ϝ

Kappa Κ κ ϰ

Omicron Ο ο

Upsilon Ο ο

Beta Β β

Zeta Ζ ζ

Lambda Λ λ

Pi Π π ϖ

Phi Φ ϕ φ

Gamma Γ γ

Eta Η η

Mu Μ μ

Rho Ρ ρ ϱ

Chi Χ χ

Delta Δ δ

Theta Θ θ ϑ

Nu Ν ν

Sigma Σ σ ς

Psi Ψ ψ

Epsilon Ε ϵ ε

Iota Ι ι

Xi Ξ ξ

Tau Τ τ

Omega Ω ω

Uppercase Alpha: Α

Abstraction in lambda calculus

Lowercase Alpha: α

Probability and finance

Uppercase Beta: Β

beta function, also called the Euler integral

Lowercase Beta: β

Gödel's incompleteness theorems

Uppercase Gamma: Γ

Gamma function (improper integration of the derivative of another function). Γ = Γ(p+1)/p

Γ(p)= 0e-x xp-1 dx p>0

Lowercase gamma: γ

Euler–Mascheroni constant, Lorentz factor

Uppercase Delta: Δ

Used in calculus to denote change in y over x to give slope.

Lowercase delta: δ

Used in calculus to find limit of a function. Calculus of variations

Uppercase Epsilon: E

Not used

lowercase epsilon stretched out: ∈ ε

Used in group theory to mean member of a group. Levi-Civita symbol

Uppercase Digamma: Ϝ

digamma function is defined as the logarithmic derivative of the gamma function

lowercase digamma: ϝ

Not used

Uppercase Zeta: Ζ

Not used

lowercase zeta: ζ

Riemann zeta function


H-theorem in Statistical mechanics

Lowercase eta: η

Minkowski metric, index of refraction, η-conversion, eta meson, regression analysis

Uppercase Theta: θ

Big O notation, Option time value,

Lowercase theta: ϑ

Angle, Bragg's angle of diffraction, potential temperature, MTBF , Compton scattering, Jacobi's theta function, Chebyshev function

Uppercase iota: Ι

Not used

Lowercase iota: ι

Orbital inclination, inclusion map, index generator function (apl)

Uppercase kappa: Κ

ISO 302

Lowercase kappa: κ ϰ

Kappa curve, Hooke's law, Riemann manifold, Von Karman constant, Von Mangoldt function,

Uppercase lambda: Λ

Cosmological constant, lambda baryons,

Lowercase lambda: λ

Exponential decay constant, Poisson distribution, abstraction in lambda calculus

Uppercase mu: Μ

Not used

Lowercase mu: μ

Mobius function,

Uppercase nu: Ν

Not used

Lowercase nu: ν

Hertz,Stoichiometric coefficient, Poisson's ratio, TRUE ANOMALY,

Uppercase xi: Ξ

Lowercase xi: ξ

Uppercase omicron: Ο

Lowercase:omicron: ο

Uppercase pi: Π π ϖ

Lowercase pi: π

3.14159 the coefficient for finding circumference

Uppercase rho: Ρ

Lowercase rho: ρ ϱ

Uppercase Sigma: Σ

Used in group theory to mean summation or sum. Also covariance matrix.

Lowercase: sigma: σ ς

Stefan–Boltzmann constant

Uppercase tau: Τ


Lowercase tau: τ

Torque, shear stress, tortuosity, tau lepton,

Uppercase upsilon: Y

Upsilon meson

Lowercase upsilon: ο

Not used

Uppercase phi: Φ

Magnetic flux

Lowercase phi: ϕ φ

Golden ratio, Work function,

Uppercase chi: Χ

Not used

Lowercase χ

chi-square distribution (statistics)

Uppercase Psi: Ψ

Water potential

Lowercase Psi: Psi ψ

Wave and particle physics

Uppercase omega: Ω


Lowercase omega: ω






∃⁂⁕ ⁆ ⁁

 π




¥£¢^ — –

± ¢ ˣ ˢ ˟ ˠ ˡ θ





Alt codes

Alt 1 ☺

Alt 33 !

Alt 65 A

Alt 97 a

Alt 129 ü

Alt 161 í

Alt 193 ┴

225 ß

Alt 2 ☻

Alt 34 "

Alt 66 B

Alt 98 b

Alt 130 é

Alt 162 ó

Alt 194 ┬

226 Γ

Alt 3 ♥

Alt 35 #

Alt 67 C

Alt 99 c

Alt 131 â

Alt 163 ú

Alt 195 ├

227 π

Alt 4 ♦

Alt 36 $

Alt 68 D

Alt 100 d

Alt 132 ä

Alt 164 ñ

Alt 196 ─

228 Σ

Alt 5 ♣

Alt 37 %

Alt 69 E

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Alt 133 à

Alt 165 Ñ

Alt 197 ┼

229 σ

Alt 6 ♠

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Alt 70 F

Alt 102 f

Alt 134 å

Alt 166 ª

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230 µ

Alt 7 •

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231 τ

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232 Φ

Alt 9 ○

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233 Θ

Alt 10 ◙

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Alt 74 J

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234 Ω

Alt 11 ♂

Alt 43 +

Alt 75 K

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Alt 139 ï

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235 δ

Alt 12 ♀

Alt 44 ,

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236 ∞

Alt 13 ♪

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237 φ

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238 ε

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239 ∩

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240 ≡

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241 ±

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242 ≥

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243 ≤

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244 ⌠

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245 ⌡

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246 ÷

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Alt 24 ↑

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248 °

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249 ∙

Alt 26 →

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250 ·

Alt 27 ←

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251 √

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220 ▄

252 ⁿ

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223 ▀


Alt 32

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224 α

…— – ‒  

Chapter 1 Algebra Math Problems

Cement Staircase Problem

We need to build a solid cement staircase. The staircase is 3 feet wide, 6 feet tall, and each step has a depth (run) of 10 inches and height (rise) of 6 inches . We want to know how much cement to buy and how much it will cost. This is a project estimation problem but will also help us picture how the staircase is constructed.

To visualize our staircase could draw a map:

  1. First let's draw a square that is the extreme outer limits of our three dimensional shape.

    We know that it is 3'w x 6'h but we do not know the total length (depth) because we do not know how many steps there are. We only know that each step has to be 10” deep (thread length). To figure total depth (and number of steps) take height of staircase and divide by height (rise) of steps.

    Length = (height of case ÷ height of step) · (thread). Why?

    Because this is a cube that's why. We know that all steps are the same size so that when we draw a square to represent them they will fit evenly into our container. Since all steps are the same each step must be a equal fraction of the length and height of the cube. Each step is equal to the length divided by number of steps. Since we only know the height and not length we must

    divide the height of cube by the height of step to get number of steps. Then to find length multiply thread by number of steps.

    Formula for linear length: length = unit length · number of units.

    (72” ÷ 6”) · (10”) = 120” We converted 6' into inches by multiplying by 12 (12”= 1')

    The answer is 120” or 10' long (120”/12”= 10')

  1. We now know the dimensions of the outer extremes but we do not know the number of steps.

To find the number of steps we could write the problem as an equality. Let s stand for steps and l stand for length. Equalities compare two different units of measurement. In this case we are comparing length to steps or steps per length. “Per” means division. Used in formulas it represents a fraction or percentage. “To” means ratio and simply tells the change of rate.

120inches (length) ÷ 10inches (length) = 12” ? (120L ÷ 10L= 12s)

We can not simply divide 120” by 10” because our answer would be 12”. We know this does not make sense because the unit we are looking for is steps not inches (length). As such we must use variables to denote what we have and what we are trying to solve.

120L ÷ 10L= s writing as an equality

120L = s rewriting as a fraction


120 = s the L's cancel out (L÷ L=1, any number times 1 equals itself)

10 when dividing we would have (10)(1) = 10

12= s dividing the constants.

Our answer is 12= s or steps = 12. There are 12 steps. In other words: “120inches (length) ÷ 10inches (length) = 12steps” is written “(120L ÷ 10L= 12s)”.

  1. Okay so we know more about our stairs and steps let's represent that visually. In the square exists real space and negative space. The real space depicted in black is physically occupied by the staircase and the negative space shown in red is empty. You can think of the red as being air.

Our box looks like a grid and we could plot points on it. In fact we could write a linear function for our box if we wanted depending on how we mapped our values. Right now we can see how the real space occupies more area than the negative space. This will hold true for any box we make where the box is formed around our real object making it taller than the negative space. The real space has a height of “12” and the negative space has a height of “11”. It would appear that the two spaces are almost mirrored except that “real” has an extra row of 12. Let's count. We counted red to be 66 blocks so since black is red + 12 it must be 78. This looks like an arithmetic sequence. Arithmetic sequences are written using recursive formulas:

Series number formula: Sn = (first # + last #) ÷ 2 · # of terms

(1+12) ÷ 2 · 12

78 If worked!

(1+11) ÷ 2 · 11

66 This arithmetic series is the average of the first and last number divided by the number of terms, also called the series number.

  1. We know we have a staircase fitting the dimensions of 10' x 6' x 3'. This gives us 180 cuft3. There are several ways to solve this. I will just Go down the list.

We could calculate the the size of a step and then multiply it by the number of blocks it takes up in our drawing, this case being 78.

36” · 10” · 6” = 2160” cu.in3. we multiply all dimensions to find cubic inches of a single step

2160” in3÷ 123 we divide cubic inches by a “great gross” to get cubic feet

2160” in3 ÷ 1760 = 1.25' cu ft. the cubic feet of a step (1.25ft3)

78 · 1.25ft3 = 97.5ft3 This is the volume of our staircase.

  1. An 80lb. Bag of cement is on sale for $3 and fill 0.6 cubic feet. To find how many bags we will need we divide 97.5ft3 by 0.6ft3. Then we multiply bags of cement by price per bag to get total cost of cement. Since units are the same they will cancel out so we do not have to write ft3 on 97.5 and 0.6.

97.5 ÷ 0.6 = 162.5 bags needed is 162.5 (163 in real life).

162 · $3 = $487.5 it costs $487.5

This is the most straight forward answer. We need 162.5 bags and it costs $487.5 In real life round up because you can not buy a half bag plus add sales tax. 163 bags at $3 with 10.2% tax costs $538.88.

163·$3 = price writing out equation

$489 · 1.102 = $538.878 calculating price after tax

= $538.878 final cost

Another way to find the cubic volume is to visualize the staircase as a ramp with half blocks as steps. You can think of the steps as prisms attached to the ramp another larger prism.

First example this superb chart:

After step #2 we figured out number of steps and and the length so we know there are 12 steps and 10' is the length. Putting that into our box we get a clear representation of the staircase. Note that the points are not necessary to plot but are interesting because it shows a definite pattern.

For every odd point y/x= 0.6. This is the function of the graph. We can prove this by taking the height and dividing by the entire length. 6'/ 10' = 6/10 or a slope of 3/5 when reduced. Since the steps are congruent for every rise of 6” we get a run of 10” and visa versa.

Making a chart and draw a line from (0,0) as your starting origin to (120,72). This shows the distance from 0” long, 0” high to 120” long, 72” high. This is our first ordered pairs where x = run and y = height.

Having drawn the black line to form the hypotenuse of a large prism we can see that our area is simply a ramp with inverted prisms for steps. Finding and adding the volumes of the prisms will give us our total volume for the whole. To find the volume of the ramp we need to treat it like a solid cube and multiply the furthest points of 10'x6'x3' and divide by two.

Formula for volume of a prism: Prism ft3 = LWH/2

10' · 6' · 3' = 180ft3 volume of the ramp times two

180ft3 ÷ 2 = 90ft3 volume of ramp

We have found the volume of the prism but we still need to find volume of stairs to find total volume of the staircase.

The stairs measure 36”x10”x6”. This is 2160ft3. Divided by 123 to convert to inches to feet we get 1.25ft3. But removing them from the large prism has changed their shape

Since we have removed the stairs to get the large prism shape it leaves 12 half stairs instead of whole ones. Just like the area for the large prism is found by dividing a cube of the same dimensions in half, we have to fit the 12 halves together to make 6 whole stairs. This

The black line forms the hypotenuse of the large prism. The prism is 10'x3'x?'. We know it is 3' wide and 10' long but we do not know the length of the hypotenuse. To find the length of the hypotenuse use the Pythagorean theorem a2 + b2 = c2 where a and b are legs of the triangle and c is the hypotenuse.

102 + 62 = c2 plugging in our values for a and b

100 + 36 = c2 squaring our base and height legs of triangle

136 = c2 adding legs to find hypotenuse squared (c2)

136 = c showing that the radical negates the square

136 = 11.661903789690600941748305755091‬ finding the radical of 136

c = 11.661903789690600941748305755091‬ our answer

Yikes! This number makes no sense and using it in further operations would only induce error. If we use it we either have to round off or use complex math and still only get and approximation. Using it to find the volume of the prism we would get

Function defined plainly

y = x+1 then ƒ (x) = x+1 y =ƒ (x)

Ƒ or ƒ is not the English “ f ” used as a variable. f[x] ƒ (x) . The “ ƒ ” is a stylized lowercase Latin f that is a shorthand notation for “function”. Do NOT multiply the the left side of the equation.

Function of [x] is stating that the output of [x] is given by the equation on the right side of the equals sign. Output is defined as the y value of an ordered pair.

The “ x ” is in brackets to indicate that it is a number in a series and not in parentheses as used in multiplication. As parentheses indicate multiplication in algebra you will see calculators use brackets for f [x] most times.

Although you may see it written as f (x) or ƒ(x) this is sloppy. It is inconsistent and used only as a convenience if a keyboard or individual does not possess tools to write “ƒ” in proper notation. It can be confusing if you do not recognize f as a statement rather than a variable. It is also confusing for beginners hurting understanding moving forward when more eccentric notation and Greek letters are introduced.

y= x+1

ƒ [x] = x+1



func is the output

func of x = output

func of x = y

output = y

y of [x] = func of x

Saying the y-value of x is interchangeable with saying the function of x.

There are differences between numbers and statements and it is vital to understand them. When using notation one of the most important things is to fully understand the grammar and sentence as spoken in English.

It is okay to use a regular f if you are in a hurry or not presenting your work to others but if you are sharing your work publicly it is better to write function as a stylized f and epsilon ( Σ ) as a Greek e. I would also make the argument that all special notation should be differentiated from other variables and English letters so that your equations are clearer and not ambiguous.

Some formulas written as functions:

f(x) = x + k

f(x) = (slope) · x + k where m is slope, and k is some constant for the y intercept


y= x + k and ƒ(x) = x + k y equals ƒ(x). K is the variable for a constant number

If y = x+1 then,

ƒ (x) = x+1

x=x and ƒ(x)= y

ƒ of x means function of x. Function is written on calculators as func. You chose a number for x such a 0-9. Substitute the number you chose for x in your equation. You are only working on the right side of the equation. When you find a value for the right side of the equation your work looks like:

ƒ[x]= answer

y= answer

This describes a Cartesian pair. Cartesian pairs “map” a ratio, also called slope, between x & y values.

y= x+1 This is our function. We need to pick a value for x. We choose 0.

y= 0+1 Substitution

y= 1 Solved

This means that our Cartesian pair is (0,1). We have infinite pairs. Since it is hard to find slope from zero let's use 1. Using zero is undefinable since dividing by zero is undefined. (Honestly slope needs two points to find but x has a hidden coefficient of 1 making our slope 1 since 1y/1x = 1m).

y= x+1 This is our function. We need to pick a value for x. We choose 1.

y= 1+1 Substitution

y= 2 Solved

This means that our pair is (1,2). Using a different x will produce a different y. Choosing for x is called “picking an input”. We input our number for x. Solving is called “receiving an output”.

Function notation: f(x) = (x + 1) ·10

is the same thing as

Intercept notation: y =10x + 10

The first one is function notation and the second one is y-intercept notation. Line graphs use these notations called formulas to find chart points. A chart point looks like (x,y). The x is the number of units moved to the right on the x axis and y is the number of units moved up the y axis.

f(x) = (x+1) · 10 Using function notation we choose 5 for x.

f(5) = (5 + 1) · 10 Inputting 5 for x. (Substitution)

f(5) = (6) · 10 adding

f(5) = 60 solving for f(5)

There are no more steps to do so this is solved. Function of 5 equals 60. Our pair is (5,60).

That means when x is 5 y is 60.

As such f(5) = y

and y= 60

Remember the function of x is y not x is y. f(x) = y but x  y. I do not want to see people trying to do this: f(5) = y(5) = 5y. This makes no sense, is wrong, and can not be solved for a given pair.

Using y-intercept notation will give the same pair because algebraically it is the same as function notation. You use function notation when you want to find further pairs of (x,y), you use y-intercept notation when you want to find the value of y when x is zero or when want to find the slope.

The y value is called the y-intercept if x is zero. Using zero for x will cancel out the x term leaving the constant term as the y-intercept. The coefficient of x is the slope. Slope is the rate of change. You can use slope and rate of change interchangeably.

y =10x + 10 Using y-intercept notation

y= 10 · (5) + 10 Using 5 for x

y= 50 + 10 multiplying

y= 60 solving

Because y = 60 when x is 5 our pair is (5,60). This means that both formulas are equivalent.

In this sense we can consider a formula to be some quantity of x to equal a quantity of y. There for a formula is a ratio of x to y. (5,60) is 5x for every 60y or 60y/5x. When we reduce we get a slope of 12/y. This called “ delta of 12y/5x”. In function notation formula: ƒ(x) =  12y/5x. Delta is  for slope.

I tend to write pairs in parentheses. I may also write f(x) with parentheses as well. Because ordered pairs come in set notation it is understandable the desire to use associate functions using parentheses. This is common practice but improper. The use of parentheses for f(x) ties the idea of (x,y) together, which we are doing, but it is too closely related to the multiplication of two variables.

The use of brackets for f[x] strongly enforces the concept of input/output, (i/o), and directly instructs that this is an operation not to be multiplied but a place holder for the number of the nth term of x we are using to derive a y. Brackets are the correct choice but on boards it is faster and more familiar to use curved brackets in the form of parentheses.

Due to this bad habit f(x) is used in print and other placed when it really should not. () 's means multiply and []'s means something goes in here to denote the substitution of x on the opposing side of the argument.

A point is the nth term in a series. Terms of x are called the domain. Therefore the term is x.

F(x) is expressing y in terms of x. Term means number. We are referring to x-value a certain x is plotted on a chart. Function of x is the function of the term.

The domain is all possible x-values that are plotted.

The term of a function is the x-value of an ordered pair.

f(x) = ⅓ x3 + ½ x2 -14x+25


Adding monomials to get a binomial: A binomial takes the form of x + k such as (x + 1). Therefore you have to add two monomials that do not share the same variables to get a monomial. A polynomial is a linear equation. It follows the formula: ax2 + bx + c = k.

To make a polynomial you must square a binomial. Binomial means “two monomials”. A monomial is a factor of a bi-, tri-, or polynomial.

(1x) is a monomial. So mono means one, bi means two, tri means three, and poly means many or more than three.

The square of (x+1) is (x+1)2. This means we are multiplying (x+1) · (x+1). Anytime we square a number we multiply it by itself.

(x+1)2 a monomial raised to a power of 2

(x+1) · (x+1) written out in factor notation

x2 +2x +1 The result after multiplying the factors

The number x2 + 2x + 1 has three terms: x2, 2x, 1.

x2 is a variable because it is a letter with no coefficient.

2x is a mixed variable because it has letters and numbers.

1 is the constant because it is a numeral with no letters.

2x monomial (Greek for one number)

x2 + 2x binomial, also it is a polynomial since it has more than one term

x2 + 2x + 1 trinomial has three monomials

6x3 + 5x2 -1x + 6 This has four monomials so it is a four term polynomial

The degree of a polynomial is the power of the largest exponent. x2 + 2x + 1 has a degree of 2 since x2 is the largest termed exponent. All others have an exponent or power of 1. When a number has a power of one it equals itself. 2x1 is the same thing as saying 2x or 2x/1= 2x. Remember 2x/1= 2x.

This is the law of powers:

30 = 1 All numbers have a hidden coefficient of 1. 1 followed by nothing.

30 = This is a special case requiring proof.

31 = 3 Any number raised by 1 equals itself. 1 followed by multiplying 3.

32 = 9 A number squared is itself times itself. 3·3 = 9. There are two base factors.

33 = 27 A number cubed is three factors of itself multiplied.

30 = 1 with no operation following it. We do not have a “ · ” to multiply with zero. Doing so would give false answer of 0.

31 = 1 · 3 Using a power higher than 1 forces an interpunct (·) into play so that now we times bases.

32 = 1 · 3 · 3 We multiply the 1 by groups of the base. The exponent tells how many bases to times.

33 = 1 · 3 · 3 ·3 Three cubed is 3 groups of three.

34 = 1 · 3 · 3 · 3 · 3 We multiply 1 by the base nth times

nth means number. We are saying a number in a series. 34 equals bnth. Plug in b=3 and nth= 4.

Formula for powers:

bnth = 1· (b · b, “nth times”) where b is the base and nth is the power. Write b nth times and multiply to find the power of b. Power can mean either the exponent or value of b raised to the power of nth.

Ex: 34 = 81. 4 is “the power 3 is raised to” or “the exponent of 3 is 4”, 81 is “the power of 34”.

Using power to describe both the result and exponent is confusing; nonetheless:

(23 · 3) = 3·2·2·2= 24 because (3,6,12,24...) are multiples of 3.

33 = 3·3·3·3= 81 because (3,9,27,81...) are POWERS of 3.

Power is the total value of the expression not just the power it is raised to. This is ambiguous but how it is. When multiples of a number are the result of a power operation they are called powers of the base.

Sometimes we say things like “100 is the square of 10” but we also say “100 is the second power of 10”. We say 9 is a power of 3, 27 is a power of 3, 81 is a power of 3. When powering say power instead of product...

This is the law of negative powers:

The law of powers states that the negative sign in front of the power negates the multiplication by inducing division of 1 using the the power as the divisor. To find a negative power, write the power as a denominator with one as the numerator. 34 = -4 and 3-4 = 4 as they are reciprocal.

Writing power as the denominator with 1 as the numerator:

3-1 =

3-2 =

3-3 = 29

Since we know the power of the base we simply write it as a fraction under 1. So if you can find the power of a number then you will know it's negative power. Why and how does this work?

Negative powers are the negation of positive powers.

Negation means reversing an operation. We are NOT multiplying by a negative number. Since powering a number means multiplying by groups, then it's negation must mean dividing by groups. This is why no power can be a negative number unless it is imaginary. ( i2 = -1).

Law of negation:

In multiplication division is the negation and visa versa :

2· 3 = 6

6 ÷ 3 = 2 or division undoes multiplication

6 · ⅓ = 2 multiplying by the reciprocal divides (undoing multiplication)

In addition, subtraction is the negation and vice versa:

27+3= 30

30- 3 = 27 or subtraction undoes addition

30 + (-3) = 27 negating by adding a negative sum

Special cases

10+1= 11

-(10+1) = -11 negating a positive sum by multiplying by -1

We distributed the terms out to negate a positive sum by using the law of the hidden coefficient of 1. Usually this is used in reverse to add negative sums like money: “I am owed $11”.

Usually we simplify math by negating negative sums so we can add the difference. This gives the absolute value. Absolute value is always positive. Then use as a positive or negative as situation warrants.

$0 - $10 - $1 = $? becomes

-($0 - $10 - $1) writing in () to isolate negation sign on the outside

$0 + $10 + $1(-) flipping signs by multiplying each term by -1

$11(-) summation of terms

- $11 negating the positive back to a negative so sign is not lost

It is important when negating problems using absolute value for ease calculation you put the minus sign in parentheses so that you do not forget the original sign of the equation. If you want to use a positive value you stop at $11(-) but you must have the sign there. If you need to convert the answer back it's original form having the sign properly isolated so you can keep track of your negations is critical.

1-10 a negative sum

-(1-10) = (-10) + (-1) distributing the sum out flips the signs

+1 +10 = +10 + 1

Negation is useful for removing negative operators so that equations can be simplified. When all like terms are combined you change the sign of the answer to show the actual value.

The negation of negative powers:

Negating a power means to turn a power into it's reciprocal. A reciprocal is what ever multiplied to it makes it equal one. The reciprocal of any whole number is it's fractional opposite. 5 shares reciprocity with because 5 · 0.2 = 1. Written fractionally it is 5/1 · 1/5 = 5/5. Reducing by dividing by 5 makes it 1/1. both 5/5 and 1/1 equal 1 since any number divided by itself equals 1.

If we are undoing the negative of a power we have to use its reciprocal:

3-4 a negative power

1 · · · · expanded out using reciprocal notation

1 · 1/81 multiplying

1/81 the reciprocal of 81/1

81 · 1/81 = 1 so they are reciprocal. Also note that (81/1 · 1/81) = 1. This is true for any number.

The law of reciprocals:

For any non zero number a, a/1 · 1/a = 1 and -a, -a/1 · -1/a = 1.

The law of negative factors:

Negative sums having an even number of terms multiplied equals a positive sum always. For any non zero numbers -a · -b = ab. Negative sums having an odd number of terms multiplied equals a negative sum always. For any non zero numbers -a · b = -ab.

If you have three terms all with negative signs the two negative signs cancel out making the sum positive but the third sign negates it and your sum is negative.

If you have four terms with all negative signs they all cancel out making the sum positive. When two numbers have a negative sign it cancels out leaving the product positive. Each pair of negative terms cancel out their (-) so if you have an even number of negative terms the sum will always be positive. If you have an odd number of terms the sum will be negative because the last term is negating it.

Two negatives equal a positive:

You can only multiply two numbers at a time. With 5 numbers to be multiplied You start at the left of the expression and multiply the first two terms creating a new product with four left over. You multiply that with the third term creating a newer product leaving fourth and fifth term. You multiply your product with the fourth term and the product of that with the last term. This is the full meaning of the above.

-3 · -3 · -3 · -3 · -3 An expression without an equals sign

9 · -3 · -3 · -3 multiplying terms first and second

-27 · -3 ·-3 multiplying first product with third term

81 · -3 multiplying second product with fourth term

-243 final product of remaining terms

We performed four multiplication operations using five terms which created four products. Our products were (9, -27, 81, -161). The first two terms canceled out the signs. Each additional operation changed the sign from positive to negative. Since we had an odd number of terms the first four canceled out the signs and the last term negated it to a negative.

-30 = 1 (1)

-31 = -3 (1 · -3)

-32 = 9 (1· -3 · -3)

-33 = -27 (1· -3 · -3· -3)

-34 = 81 (1· -3· -3 · -3· -3)

-35 = -243 (1· -3 · -3· -3 · -3· -3)

A negative raised to a negative:

-3-1 = -⅓ (1· -⅓)

-3-2 = (1· -⅓ · -⅓)

-3-3 = -29 (1· -⅓ · -⅓ · -⅓)

Write the power under a fraction bar and use 1 as the numerator. Numerator is the top number and denominator is the bottom number. The answer of a division operation is called the quotient. A fraction is a division problem. Also (dividend ÷ divisor =quotient) if strictly speaking division and not fractions.

In “ -3-3 = -29 it's factors are (1, -⅓, -⅓, -⅓). Taking a fraction of a fraction will make it smaller. When we are raising a number to a negative power that is what we are doing. We are dividing by groups of the base.

-3-1 = -⅓ (-0.33333) a third

-3-2 = (-0.111111) a third of a third

-3-3 = -29 (-0.0370370) a third of a third of a third

Adding powers

32 + 33 32+3 Adding the exponent in the superscript area multiplies instead of adds

32 + 33 = 9 + 27 The terms are separated by the add sign making them discrete quantities

Raising to a power, expanding, the expansion of are all ways to say the final value of the exponential term. Expansion can mean writing out in expansion notation like (3 · 3 · 3 · 3 · 3) or solving like 243.

Subtracting powers

321 - 33 = 321-3 Add the exponent in the superscript area


318 combine the exponents

10,460,353,203 expand to solve

Exponential means that the bigger the exponent you larger the interval between powers become. A version of exponential notation is used to write very large numbers called scientific notation. This bakes the number and reduces it to the ones unit with a limited number of decimals and then writes ten multiply sign in front of it with powers of ten being used to represent the length of the sum.

Multiplying powers

33 · 33 standard form

33 +3 grouping

36 multiplying the exponents as factors to get a productive

729 simplifying


(3 ·3 · 3 · 3 · 3· 3) write out all your bases so you can easily times them.

(9 · 9 · 9) or (27·27) multiply by grouping pairs and squaring

(81 · 9) or 729

Do not Multiply different bases by adding exponents:

4a2 · 4b2 can not be simplified any further adding exponents since a  b. Since the base is different you would be grouping together different groups of groups i.e.

4a2 · 4b2 = 4(32) · 4(22)

4(9) · 4(4)

36 ·16= 576 20736

16 (ab4) is wrong (does not equal 4a24b2 )

4a24b2 is right but coefficients need to be multiplied

16a2b2 is right (does equal 4a24b2 )

3n2 · 7n2 can simplified further since they have the same number of variables raised to the same power. If bases are different you would be grouping together different groups of groups i.e. triangles3

3n2 · 7n2 We find that n = 5

3(52) · 7(52) substitution

3(25) · 7(25) expanded

75 · 175


3n2 · 7n2





3n2 · 7n2 = 21n4

Find like terms. 3n2 and 7n2 are like because they share the same base of n2.

3n2 · 7n2

(3·7) · (n2 · n2) multiply your constants ( the coefficient 3 and 7)

(21) · (n2 · n2) multiply your variables (the bases n2 and n2)

(21) · (n2+ 2 ) add to multiply exponents

(21) · (n4)


Radicals: The inverse operation of a power is called a radical.

To negate a power you root it to find it's square. The radical sign looks looks like “ √ ”. We are dividing in groups to reverse the multiplying in group we did when we raised a number to a power.

Like long division the large number, called a power, is put inside a box. A line called a vinculum is used to show what numbers are inside the box to be rooted. The process of finding a root is called rooting and a root is some factor of the power that we are trying to find.

4 We need to find the root of 4.

1 · 4 We look for factors of 4 that equal 4 when squared

1 · 2 · 2 22 = 4 so it is our root

2√4 The square root of 4 is 2.

49 We need to find the root of 49

1 · 49 looking for factors...

1 · 7 · 7 these are the only factors of 49... 72 = 49 so...

7√49 The square root of 49 is 7.

The number inside the box being rooted is called the radicand. We make a tree under the radicand to find it's factors. When we find the number that when properly powered gives us the root we are looking for we write it infront of the radical sign.

When a radical with does not have any numbers outside it we only look for the square root. When a number has a 3 or 4 outside it we look for the cube or fourth root respectively.

327 We need to find the cubic root of 27

1 · 27 looking for factors...

1 · 3 · 3 · 3 = 27 these are the only factors of 27... 33 = 27

3 327 The cube of 27 is 3

4625 The four means find the fourth root

1 · 625

1 · ? · ? finding factors...

2 does not divide evenly. We look at the ones unit and see that it is a five so 2 can can not be a factor of it. 2 only works as a factor for even numbers and this one is odd. Hmm... we try the next number up which is 3...

625 ÷ 3 = 200 + (25 ÷ 3). Mentally using long division we can see that 3 goes into 6 twice removing

the hundreds unit leaving us with 25. 25 does not divide evenly by 3. When we look for a factor and the number we are using does not divide evenly we go to the next number up. We start with the lowest factor of 1 and the radicand itself and then go up sequentially until we find a number that divides evenly.

625 ÷ 3 = 208⅓ 3 will not work

625 ÷ 4 = 156.25 4 will not work

625 ÷ 5 = 125 5 will work so we put that into of tree of factors


1 · 625

1 · 5 · 125 we must now factor the largest number on the right

1 · 5 · 5 · 25 we know that 52 = 25 so lets see what is happening

1 · 5 · 5 · 5 · 5 we have fully factored 625 and found two perfect squares

5 4625 5 is the fourth root of 625.

So what is meant by two perfect cubes or a perfect cube?


1 · 625

1 · (25) · (25) notice that 25 is the power of 52. This is a perfect square.

1 · (5 · 5) · (5 · 5) finding perfect squares will make factoring easier.

These examples were just to show concept and process. You can find the root to any number if you make a large enough tree. Often people just use a calculator as they do not have paper handy. Also any number can be raised to a power and it will be a whole number that is positive but finding the root of a power usually leaves an irrational neverending fraction. Between 1 and 2 there are infinite irrational numbers so it is not weird for then to occur it is weird for them not too. Look at number line below.

There are neverending numbers, both rational and irrational between and two numbers. Ancient Greeks theorized about numbers and asked the question how long it would take to reach something if each step you took was half as big as the last. Half the distance from 0 to 1 is ½ . To halve the distance you divide by 2. Doing so makes each number smaller than the last. Your current interval is half as big as your last and twice as big as your next. At this rate you will never reach 0 from 1. If this was a negative exponent your denominator would be keep going until you reached 1/∞ ( 1 over infinity). To make the imperial fractions irrational adding a 1 the ones unit in the denominator would do it.

Harder radicals:

By 2's:

1024 need to find what x equals x2 = 1024

1 · 1024

1 · 2 · 512 because this is an even radicand we can always factor out

1 · 2 · 2 · 256 as long as our factor is even we can keep factoring out 2

1 · 2 · 2 · 2 · 128 factoring by 2 keeps the process simple...

1 · 2 · 2 · 2 · 2 · 64

1 · 2 · 2 · 2 · 2 · 2 · 32 We can keep doing this but it is getting excessive

1 · 2 · 2 · 2 · 2 · 2 · 2 · 16... we need to still factor 16

when you have this many common factors there is likely to be a square. Instead of factoring 16 all the way down like:

1 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 8

1 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 4

1 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 ·2

Then solving by finding pairs of common factors and squaring like:

1 · (2 · 2) · (2 · 2) · (2 · 2) · (2 · 2) · (2 ·2) = 4·4·4·4·4 = 16 · 16 · 4 then

16 · 16 · 4 We have two squares and a 4. Factor the 4.

16 · 16 · 2 · 2 Rearrange the 2's to pair them with the squares.

(16 · 2) · (16 · 2) multiply

32 · 32 Since 32 · 32 = 322 , 32 is the root of 1024

32√1024 32√1024

We could do that but there was a shortcut that we missed. Either we should have groups the 2's when the biggest factor was 64 or grouped them as we went.

1024 need to find what x equals x2 = 1024

1 · 2 · 512

1 · = 4 · 256

1 · = 8 · 128

1 · = 16 · 64

(32 · 32) times'ing the 2's gives a pair of 32 . We have two perfect squares.

Both methods work. The first one took sixteen steps and the second one took five. Use a combination of both methods to get your answer. Start by taking out factors of two until you see a pattern and can solve by instinct. The more you practice patterns become familiar and you will recognize more pairs of factors of high level powers.

Making trees of factors to group or sum is called recursion. Recursion is used heavily in computer programming and other high level math. It is based on shapes and patterns. Each time you group pairs from rows or columns you are performing a recursive action. Sierpinski's triangle is a type of recursion and we will study it later. The staircase problem seen earlier with the series formula was also a recursion action to group the 12 elements into 6 pairs for summation and then further multiplying.

31331 It has a small 3 so we must find the cube

1 · 1331 We know this is a pair of factors

1 · ? · ? We need scratch paper to figure out the factor

To find large cubes use paper to factor using division. If you do hundreds of rooting operations you will start to remember them just like learning the times table. Usually we try to find the largest pair of factors first but this one it hard for beginners.

Start with 2 and try to divide 1331. If it works keep factoring, if it doesn't use 3 then 4 then 5 till you can start getting usable pairs. Write the dividend 1331 in the division box and the divisor outside of it to use long division. It should look like this:

2 1331

Divide from left to right. Does 1 divide evenly by 2? No we get ½. If a unit is to small to be divided evenly move one unit to the right to add a digit. Divide try to divide the two digit number. If it is still too small continue moving to the right until you have a number that is big enough to divide.

0 write 0 above the 1 because 2 is too big for it.

2 1331

-0 then bring down the product of 2 times 0

1 subtract 1 -0 =1 .


2 1331




2 1331






2 1331








2 1331









Guide to long division for kiddos

Use this if you have to tutor a younger learner:

It works like this:

2 1 2 is bigger than one so we can not divide. 2 goes into 1 zero times.

0 write a zero to say you divided 1 zero times.

2 1

0 multiply the 2 with the 0 and write your product under the 1.

2 1

-0 our product was zero.


2 1

-0 Subtract zero from 1 and write result under the vinculum.


2 1


1 1- 0 = 1. We subtract from the top down. A vinculum is the bar under the zeros.


2 1



Lets try something else

3√1331 using a mental factoring method

4 in the hundreds

4 in the tens

3 in the ones

Add up all the bits and you get 400 + 40 + 3= 443. we will have a remainder so this number will not work. We should stop here but will keep going for illustrative purposes.

From memory we know the value of the product and the remainder.

Take our factors of 3 and 443 and multiply it. It is 1329 meaning we have a remainder of 2.

I do not like abstracting the answer this way but is useful for studying data structure like trees, strings, arrays and so forth.


9 Added up the red is our product 1329 with a remainder of 2






3 We can see how it makes a tree structure.

When doing mentally everyone has there own methods, find the one that works best for you. Done correctly it is an invaluable time saver. This may help autistic or dyslexic learners due to the highly creative patterns this displays. We should explore tree'd structures as it relates to algorithms much later.

So this is a large radicand we will need some way of quickly solving these. When accuracy matters always use a calculator followed by long division if you do not have one.

Long division will take up too much space to try all the numbers the numbers so I will truncate quickly for brevity...

3//1313= 13 ÷ 3 = 4+ r.1 so 12+1 ÷ 3 = 4 + r. 1 so we are left with 11 ÷ 4. This will not divide evenly so we have to try a number different than three.

It helps to make a list of possible candidates for a rooted to the third problem or large problems and then quickly eliminate them through mental division.













13 after this I am getting my calculator

  1. 1 is always a factor but to small so we skip this one

  2. 2 does not divide into odd numbers so we skip this one

  3. 3 divides into 13 with 4 leaving a remainder of one, bringing down the 3 we have 13 again so use 4 again with another remainder of 1. Bringing down the last number is 1 making 11. 3 can not divide evenly so we skip 3 and go on to the next higher number on our list.

  4. 4 divides 13 with 3 leaving 1 making 13 again so we use 3 again getting a remainder as 1, bringing down the one again a getting 11 again. We can not divide 11 by 4 so we go onto the next number. Note we got similar responses with 3 and 4 because they were the factors used to make 12 to subtract from 13 to get a remainder of 1 to put in front of 3 to make 13. This is a pattern. Look for patterns.

  5. 5 can not divide because all powers AND multiples of 5 end in zero or five. Go on to the next number.

  6. 6 goes in 13 with 2 and leaves 1, making 13 again so we use 2 to get 1 again and then 11 after bring down the 1. 6 can not go into 11 so go onto the next numbers.

  7. 7 goes in once leading to 5, then 63. 7 goes in 63 with 9 leading to 0 . Bringing down the one we get a remainder of 1. It can not divide 1 so we move on. We started with 1331 so this is going to just start the process over and become redundant making a repeating decimal. 1331÷

  8. 8 goes in once leading to 5, then 53. 8 goes in 53 with 6 leaving 5. Bringing down the one we get 51. It can not divide 51 because using 6 is too small (84) and 7 is too big (56).

  9. 9 goes in once leading to 4, then 43. 9 goes 43 leaving 7. Bringing down the one we get 71. 9 can not divide 71 evenly because the two candidates are 7 and 8 and they equal 63 and 72 respectively.

  10. 10 can not be a root because all multiples of ten even in zero. Skip and go to the next number.

  11. 11 goes in leading to 2, the 23. 11 into 23 leaves 1, bringing down the one we have 11 again and 11 goes into 11 once. We are done. This is our third root. Check to prove it.

11*11= 121.... 121 *11= (121 + 1210) = 1331. There were patterns here that apply to other rooting problems. We need to find all them so we look intuitively at problems instead of mechanically.

We tried eleven numbers in our list. The first two were automatically eliminated. We eliminated 3 when it had 11 as a dividend. We eliminated 4 when it had 11 as a dividend. We eliminated 5 automatically. We eliminated 6 when it had 11 as a dividend. We eliminated 7 when it had 1 leading to 1/7. we eliminated 8 after getting 51 and 9 when getting 71. We eliminated 10 automatically. We finally selected 11 when it had a dividend of 11 because it goes into 11 once. We eliminated most of the numbers we had before we even got started. The ones we didn't (3,4, 6) gave a remainder of 11. in dividing 13 3 will use 4 to make 12 and 4 with use 3 to make 12, as they are the coefficients when factors of 12. That is why they both had the same remainder of 133, and the same dividend of 11 from 1331. 6 had dividend of 11 from 1331 and that is because 6 is a multiple of 6. Multiples of numbers often leave the same dividends from radicands. 7, 8, and 9 had to be done but this was not to hard. 9 was easy because its times tables are intuitive. After 9 the only number to divide with was 11 and that was our answer.

Analysis: eliminated numbers (1, 2 , 5 ,10)

numbers with 11 as dividend ( 3, 4, 6, 11)

numbers eliminated after full division without remainder of 11 (7, 8, 9)

Conclusion: 11 came up as a remainder too often. Could it be that when trying numbers and getting a consistent dividend towards the end that it is somehow linked with the root we are after? There must be patterns that can help us solve the problem by looking at the last digit of the radicand or a sun dividend we seem to be hitting or approaching.

Further this list of numbers was easy to do mentally as they were number 10. Most people memorized there times tables through shear practice. Practicing larger number will increase your mental math ability so you do not have to rely on a calculator to multiply, divide, square, or take roots.

When rooting: 1 is always eliminated as it is the coefficient of the radicand. 2 is always eliminated for odd numbers, we look at the last digit of the radicand and if it is odd we skip using 2. 5 gets eliminated if radicand does not end in 5 or 0. Ten gets eliminated if radicand does not end in 0.

The 9nth law: For all coefficients of 9, coefficient = 10n – n.

9 · 0 = 0 This is because 9 is recursive.

9 · 1 = 09

9 · 2 = 18

9 · 3 = 27

9 · 4 = 36

9 · 5 = 45

9 · 6 = 54

9 · 7 = 63

9 · 8 = 72

9 · 9 = 81

9 ·10 = 90

9 ·11 = 99

9 ·12 = 108

9 ·13 = 117

9 · ∞ = 0 (incalculable, infinity is two zeros or one zero twisted in a knot over itself note 9 · 0 over 9 · 1. 0/09 = ∞. This is true is delta of y/x approaching unbounded infinity in calculus. Also note 0+0+9=9 .)

We were taking what is called the principle root of of a radicand. Numbers have a positive root and a negative one. Usually we talk about the positive one because a power is any number times itself and any negative times a negative is a positive, i.e. -32 = 9. In engineering we take problem where usually negatives as roots help like subtracting an area or using negative space. There are numbers call imaginary numbers and the imaginary number i . It works like this:

Imaginary I :

Powers of I as negative root of -1

I−3= i

i−2= −1

i−1= −i

i0= 1


i2 = −1

i3= −i

i4= 1


i6 −1

in=in(mod 4)

Don't worry about this for now

3-125 Take the negative 3rd root

-1 · (5 · 5 · 5) this is what imaginary i is saying

-5 · -5 · -5 is the negative 3rd root.

-5 √-125 remove the index “ 3 ” to show final answer


4 · 10 factoring out largest factors fist

2 · 2 ·5 noticing a perfect square

2√5 taking all perfect squares outweighed

2√5 no more factors or perfect squares so this is considered prime


4 · 27 pulling out biggest factors

4 · 3 · 9 noticing 4 and 9 are perfect squares

(2 · 2) · (3 · 3) 3 factoring the squares

2 · 3 √3 we pull out perfect squares in their root form

6 √3 times the squares in front of radical and write the remainder in it

We attempt to pull biggest factors out of radicand to speed up factoring and get the square on a sooner go. In this case taking out the biggest factors let us immediately know 4 and 9 were squares. Since we had more than one square and they were different we had to factor them fully down to their bases of 2 and 3. We take the bases out of the radical (the sign and vinculum indicating 2 and 3's state as a being under the sign) and put them in front or out of the sign.

Having them out of the sign indicates that they are the closest squares of the radicand we could find.


How come we were able to put 3·2 into 108 ? 102 = 100, 10 · 11 = 110 sooo the square of 108 must be a number between 10 and 10 times 11. 10 times 11 is not even a square, then next square after 102 is 112 and it equals 121! Therefore the square of must be a fractional number and not a whole number. In fact it is 10.3923048454. To fully understand the mechanics first examine -144 :

First method

144 factoring by 2

2, 72 dividing our dividend by 2

2,2, 36

2,2,2, 18

2,2,2,2, 9 9 is not divisible

2,2,2,2, 3,3 factoring 9 and noticing there are squares in our radicand

6, 6, 2, 2, reversing to find squares

12, 12 found 12 as a perfect square since 12 2

Second method


2, 72 writing 2's in a column to save space and easily see

2, 36

2, 18

2, 9

2222, 33 we have four 2's and two 3's but how to remove them?

(4,4) (3,3) finding and grouping lowest possible squares

4 · 3 √144

12 144 The product of the squares 3 and 4 is the square of 144

Third method


12 · 12 Pulling out biggest squares first

12 √144 This is fastest and the recommended approach

The third method is fastest and why it is the recommended approach. The other two work but imply a lack of knowing bases and their powers. You should know that 12 is a dozen and a dozen dozen is a “gross”. Further you should know that a dozen dozen is 122 or 144. We use this to teach Pythagorean theorem a2 + b2 =144 (122) to lead into trig formulas and converting cubic inches into cubic feet by dividing by 1728 (123).

In the first method we used a method called reversing to find the largest squares. This method takes the longest but is taught in case you have a big number like 65,536. It shows a crucial point that powers of two can be broke down into smallest form them built up to simply find the largest squares. We you have pull the largest squares out of a radicand the radical is considered solved.

How come this works? Does this work for some or all radicals?

It works for all problems because it looks for the largest squares. Essentially when you are rooting you are trying to find what number times itself the number of times indicated by the indice (“index”). To find the highest square and write it in final form you must put it outside the radical sign. This is called negation.

While inside the radical it is under the radical sign. This means that the radicand is being factored producing it's factors underneath it. When we see the factors making pairs of squares they are the byproducts of the process. To solve the radical we must take the squares out in highest number. A square taken out of a radical by definition must lose it's matching coefficient. In 4 we have our matching pair 2 · 2. Written under the radical they are the byproducts of √4 .

The full demonstration is :

22 makes a power 4. To unmake it we find it's radical 2. We write 2 by itself because literally that is the answer.

22 = 4


2 · 2

2 √4 read as, “Two is the square root of four”. We only listed one factor of 2 not both of them. The radical sign implies square roots but there are other roots. The number indicating the exponent/root to be sought are called indices plural and indice or index singular. Indexes list information, here it is listing the number of times the base is to be timesed or in other words it is stating the power to with the exponent was listed at. The wording lone invokes a time when factories and power plants where all the rage making financial markets rich hence the S&P (Standards and Poor) 500 and the Dow Jones report are called indexes.

In the radical √108 we know that the answer is 10 < √108 < 11 because 102 is 100 and 112 121. Then 100 < 108 < 121. Our answer lies within the interval [10, … ,11]. Because there are no whole numbers in this domain we will have an answer that does not divide evenly. It will be irrational, likely producing a long non terminating decimal. That means we do not have a perfect square to pull out and will be left with a remainder. When doing this our response looks funny and this is deliberate. First we do not want for spend forever calculating some insignificant some for eternity. Second we want people to know that there is something wrong with our answer and if the need a more precise calculus then they need to expand on what we have given. We list our answer as the closest possible square but write it as a square of the remainder so that no way can it actually be a true square of the number in the radical.

In we gave our answer as 6√3 read as, “ six is the square root of three”. Is six the square root of three? Heavens no! Six squared equals thirty-six! The actual square root of 3 is ≈ 1.732008076. So why are we saying this?


9 · 12

9 · 3 · 4

Twelve is not square so we must keep factoring. Leave the 9 in until we have all our byproducts.

Nine and four are square but three is not so it is a remainder. Remember taking squares is dividing by groups.


9 · 12

9 · 3 · 4

3 · 3 · 3 · 2 · 2

This is a set of perfect squares. You remove 2, put it outside the radical, and erase or discard the byproduct. By definition byproducts are literally industrial waste. The 2 is the square of the 2 we removed so we erase it. We do the same thing for 3 by moving it outside the radical and erasing it's coefficient. This leaves only the non square three as a remainder. We leave remainders in the radical so it will stay there as part of our final answer.


3 · 3 · 3 · 2 · 2 The smallest set of factors. There is nothing more to factor.

108 is erased because producing factors unproduces radicands

3 · 3 · 3 · 2 · 2 we need to move what we can out and leave our remainders

3 · 3 · 3 · 2 · 2 Each square is going to be moved to outside the radical sign

so we cross out the coefficient it needs to be square. This is

the negation of the square.

3 · 2 √ Our squares are outside the radical. The non square three stays behind.


6 √ 3 z 3 is put in the remainder as it is the only thing left. Outside squares are


6 √ 3 This statement expresses that a prime number was rooted leaving squares multiplied to 6 with a remainder of 3. Trial and error can undo this and give us back our original number.

Written this way someone who needs to know that √108 equal 10.3923048454 can reverse engineer the process to find values to the farthest decimal they need to satisfy their equating needs.

That is most of the answer.

Bring a numbering out of the radical makes numbers smaller because the radical acts as a fraction bar to imply division. Putting a number out of a radical implies needing to multiply it to be able to get the original radicand again.

Leaving a remainder that is less then the square outside the radical implies that the radicand we now have is a byproduct of a larger process. Once the outside number will not square to the inside radicand we know the radicand is not the larger power but the smaller byproduct.

This will be hard for beginners to understand at its first induction but is important to know for actuarial sciences like curing cancer and other medical research. The body and it's systems particularly the blood and cells create a lot of waste in the form of free radicals and as a byproduct of the body's chemical processes. This is complex and can not go into detail at this point. Let's just say statistics and probability are related to health science and that is the highest level medicine, a PhD working on research to find breakthroughs. Takes 12 years plus college. A bachelors of 4 years, then a masters of 4 years, and finally a PhD of 4 years. They have more school then a general doctor working in a hospital unless then are a researcher at a leading research hospital like John Hopkins and others.

If you don't get it you will. Find the squares and everything else is a byproduct.

The cycle is triangular like a algorithm: From a number to be powered, to a power, to a root. From the root to a root squared to the power again and then back to the root.

Caveats exist. Any whole number can be powered and get a whole just like multiplication any two whole numbers will produce as whole number as its product. When you divide any whole number you start having weird things happen like getting a remainder. That remainder is written as a fraction attached to the quotient. A remainder is what remains. “Remains” is slang for waste. A corpse or stomach contents after eating (poop). So when you take the root of a number and get byproducts they are the fractional remains of the quotient. Fractions forming from regular division lead to irregularities and that is why people do not like dividing, because you get long decimals that do not make sense. This decimal is written in a way that looks weird to people and upon seeing the answer they will avoid that number or expression unless they absolutely have to deal with that value for further equations. Likewise writing the quotient of a radicand looks weird to let people know: this is unusual, do you really want to deal with this number?

We can assume the expression is stating an irrational nonrepeating decimal. We can approximate it's value but do not write it in it's entirety. Usually we can calculate a decimal to it's fourth digit and that should be accurate enough for most needs. If it is not then it is up to who ever is receiving this number to calculate it out to the digit they demand. As people have different requirements we do not impose our criteria on them by approximating a value for them but instead rephrase the expression, “the root of something” to, “the square and remainder of that something”.

Tricks for finding very large square roots

To find square of numbers like √2304 we make a list of the first 9 perfect squares:

12 = 1

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81 We need to memorize these numbers.


2304 We need to rewrite this using a pipe symbol.

23|04 like this

|04 we look at the first two digits and see what square would end in the last #

From our list 22 and 82 would produce a “4”. We will have to chose one of those based on our next step.

23| we look at the left half of or number and see what square equals it

42 is 16 and 52 is 25. We can not go over 23 so 52 is eliminated. We use 42 and write as 4. Using 4 we multiply it with the next higher up number 5, 4·5= 20. Now we compare 20 to 23. If 20 is higher then we use 22, if it is lower then we use 82.

20 < 23 It is lower so we will use the higher number of 8 instead of 2.

48 √2304 write the 4 in the tens unit and 4 in the ones unit.

12769 we need to rewrite this

127|69 rewritten in pipe notation

|69 find the squares that end in “9”

(3,7) we find 3 and 7. I will call this pair the candidates for the ones unit.

127| find the square that is under 127 and over 127

(11,12) we find 11 and 12. (121,144)

121, 127, 144 the number we want, the target, the coefficient

11 does not go over 12 so it is the number we want. I will write this down:


11| writing down 11 under our radicand in pipe notation.

I need to find whether to use 3 or 7 as our ones unit. To do this I will multiply the 11 with the coefficient of the next number higher than it 12. 12 is also the number we used to find 144 so we call the next number up the coefficient.

11·12= 132 doing the math to get our productive

132 > 127 finding 132, our coefficient is larger than our target 127.

If coefficient is greater than the target use the smaller of your pair, if it is less than target use the larger of your pair.


23|04 rewriting in pipe notation


|04 looking for a candidate pair for the ones

|04 = (2,8) finding (2,8) as our candidate pair for the ones


23| looking for the candidate pair for the tens

23| = (4,5) finding (4,5) as the candidate pair for the tens

(4)·(5)= 20 multiplying the tens candidate pair to get a comparison number of 20

20 < 23 comparing 20 to 23 and finding it smaller

If 20 is “ < ” target number use 8 to make up the difference.

If 20 is “ > ” target number use 2 to avoid going over.

Using 2 will produce a number smaller than 2304, using 8 will make up the difference so that our square equals 2304.

In a candidate pair the small number avoids going over target and big number makes up difference to reach target. In another situation where we need to reach our target we use the larger of the candidate pair.

20 < ” means than 42 is far from 2304 so we have to use 48 to equal 2304.

20 > ” means that 48 is over 2304 and we have to use 42 to avoid going over 2304.


_ _ _

_ _ 4 find the ones


_ _ 6 because 42 and 62 equal 16 and 36 both ending in 6

5 _ 4 find the hundreds


5 _ 6 because 52 equals 25 which is closest to 29 without going over. 62 is 36, which is over 29.

|811 a

5 4 4 find the tens


5 4 6

Problem #2

We are building a yard around a foundation for a home. To protect the foundation we need to drain water away from it using a rise and run of ¼” per 1'. From the foundation to the edge the yard we need a ramp of soil exactly 16' long. The distance from the foundation to end of yard is a 16' long grade.

How much soil do we have to remove to create a 16' long grade, how deep do we have to dig and what is the horizontal distance of the foundation to the end of the yard?

Rise and run problems involve slope. The formula for slope is (y2 – y1) ÷ (x2 – x1). This is given using graph points. The slope is written y /x . Since we already have been given the slope as a ratio of ¼”:1'

we should rewrite it so that it is in the same units. Use the smaller scale so in this case inches.

To convert the 1' in ( ¼”:1' ) to inches multiply feet by 12 to get it's equivalency in inches.


¼”:12” 1 · 12 = 12” .

There are 12 inches in a foot is what we are saying. The full formula is ( 1f/f · 12”/1 ) where the apostrophe for foot has been replaced with the letter f. We are canceling out feet by dividing 1f by f. This gives us a result of 1 for our value. We then multiply it by 12” to get 1 · 12” = 12”. The 1 as a denominator is superfluous and can be omitted shortening the formula to simply ( 1f/f ·:12” ).continued below #1

¼”:12” our ratio of y:x

1” : 48” times out the fraction by multiplying by the reciprocal of 4/1

1”/48” rewriting as slope notation for clarity

We have our slope of – 1”/48” and a grade of 16' descending to the edge. We can use this to find more information.

The red line is the current ground level. We need to excavate enough soil to create our grade. Our grade is to be 16' diagonally with the ground. It also needs to have a 1'/48' slope.

The green line is the grade line. The blue lines are the depth of the lowest point of the grade. The white triangle is the area of dirt that needs to be removed to create the grade.

By first drawing the grade line we were able to draw the lines second as the edges our grade. Then drawing parallel lines for the surface and base of our grade next. This creates a rectangle with a line dissecting it in half starting at the height of our grade and ending the end of it.

Our perspective is from stand on the red line as solid soil. The upper left corner is the side with the foundation. We know how long to make our grade but we do not know how to make it because we do not know where it ends and we do not know how deep to dig. Finding the base will tell us where to end and subsequently where to mark on the topsoil to make our mark and start digging. Finding the depth will tell us how deep to dig.

We can consider the area to be removed as a white triangle and the grade as a hypotenuse of a triangle with the left blue line as the height and black line as the base. The brown represents dirt and the green line the grass that will grow on it. Once we find our missing sides we will measure the distance from the top of the grade to the end and make our first dig in the dirt at the right corner and work our way back to the left as we go leaving the white space as air.

So how do we find the missing sides with only the hypotenuse? A2 + b2 = c2 will give us the length of a hypotenuse but we already know that. Rearranging the formula we know that c – b = a and c – a = b so 16 -a = b and 16 – b = a. Maybe we can use that later but first let's try something else.

We know that a circle has 360° degrees and that a square does also. Remember that a right angle is 90° and a square has 4 right angles so 4 · 90° = 360°.

A square that is 1 unit long dissected from opposite corners will form a line that has a slope of 1, since 1/1 = 1, and it will be a 45° angle, also called a miter. Mitering a square divides it evenly in half. Notice the dual 45° on either side of the miter line. 90°/ 2 = 45°. Math is symbolistic. The wording is not arbitrary. There is literally (literature) a forward slash in the blue square.

A 45° miter has a rise and run of 1/1 making y = x. This is called an isosceles triangle for equal measure. Iso- means equal and sceles means scale. Exercises called isometric means equal measure.

We have just proved that 1/1 · 90/2 = 45°. This means that if we know the slope of a line we can use that to calculate it's angle from a base. Further we know that the slope is also the hypotenuse of any line although we are not formally proving that here. Slope equals hypotenuse in a right triangle.

This means that we have also uncovered the formula for converting slope to degrees. Y/x · 45° = in degrees. A symbol ∟ means 90° angle and is called the “right angle” sign and is the “acute angle” sign for degrees less than 90° although you can use it for any degree angle. There is also an obtuse sign,, for degrees larger than 90° and is shown with a 135° angle for the hypotenuse leg.

Basically slope is any fraction that shows y divided by x where x is the base and y is the height. Any given fraction can be put into our formula and give in degrees the angle associated with sine. Sine is one of the six function of trigonometry, for all intents and purposes it is angle a or the first angle. I should probably list all my angles but I want a little more information first.

Each of these triangles has a base of 144px. They are all right angles so angle c is 90° on all of them. A right angle is always 90°. All triangles have three sides and three angles whose angles sum is always 180°.

The tallest one is half a square so measures 144px by 144px. Since y = x the angle of a is 45° making angle b 45° as well. C – a = b so 90° – 45° = 45°. Both angles a and b are the same at 45° so it is called isosceles.

The 30 - 60 – 90 triangle is 144px by 72px. It's is twice as long as it's height, or you could say it's height is half it's base. It's slope is 72/144 making y = ½x. C – b = a meaning 90° – 60° = 30° and 90° – 30° = 60°. This is a special triangle that is very useful for solving problems.

The 11.25° triangle is 144px by 36px. It is the most acute triangle of three best it's angle is the smallest.

The slope tells me the measure of angle a. In higher math these angles are referred to as alpha, beta, and gamma using lowercase Greek letters. They are written as or α, ß, γ.

Finding angles is called evaluating a triangle. To do so it is easiest to solve for angle a using the triangle orientated as shown as this correlates to line graph algebra.

To evaluate the triangle with only slope we attempt to find the degree of angle a and call it our alpha.

Y/x · 45° = θ of α. This is read as, “ theta of alpha”. Theta is the ninth letter of the Greek alphabet, it indicates an angle. This is often used for trigonometric functions and discussing the tilt of planets to there axes (plural of axis). If you see θ it with probably be preceding an angle like θ45 meaning “ angle of 45° ” or θa meaning “angle of a”.

We can only devise the angle of a at this point since we really only know slope. To find the length of a missing side or angle using trigonometry we need 2 pieces of information other than the right angles measure.

Let us find alpha of the acute triangle before we try on our landscaping problem.

Slope · 45° = alpha°

36/144 · 45° = θα original pixel dimensions given as slope

9/36 · 45° = θα dividing by 4

¼ · 45° = θα dividing by 9

¼ of 45° = 11.25° of means division by multiplying using a fractional reciprocal

11.25° = α our answer

Alpha of triangle three is 11.25°. We can check this many ways, come to this conclusion many ways, and state this many ways. The more eccentric and fancier we get the less clear we sound to others.

I am using eccentric notation to show some of the various forms that occur when reading others notes. Some people use cryptic or sloppy notation so if you are collaborating with others or reverse engineering using someone else's notes do not be surprised if it is not written in standard form.

36/144 · 45° = α

1/4 · 45° = α simply performing long division or taking the largest factor out of both which is 36

45° ÷ 4 = 11.25° using the reciprocal of ¼ to divide

Above is the standard form with no unnecessary steps.

Since the the height of the acutest triangle is ½ of ½ of of the miter triangle it's alpha degree is ½ of ½ of 45°. This is the logic we base our formula in that a slope ratio 1/1 equals 45, ½ equals 22.5, ¼ equals 11.25.

Roughly put a slope of 1 = 45°, any division of 1 must apply the same division to 45°.

slope 1 = 45°

slope ½ = (½)45°

slope ¼ = (¼)45°

Any slope times 45 over 1 will give the exact degree of alpha. (Y/x)45° = α°

Recap: Our slope is 1'/48'.

= (1'/48')45°. the above equals etc.

45° ÷ 48 = 0.9375° divide miter by x if scale units are the same i.e. foot/foot , inch/inch

Our alpha is 0.9375°. See following picture:

This means alpha or angle a is 0.9375. This is excellent information and we solved it simply by converting our rise and run to all inches and then dividing 45 with 48 to get 15/16 or 0.09375°. If we subtract this from 90° we will get the degree of angle b, 89.625°.

90° – 0.9375° = 89.625° c – a = b This is a Pythagorean triple because of the right angle

alpha = 0.9375° the acute angle of hypotenuse ÷ alpha's adjacent side

beta = 89.625° the large acute angle of hypotenuse ÷ beta's adjacent side

gamma = 90° the right angle of tangent ÷ base

WARNING : {The previous sides are relative to the angle in question.

Arcs in the corner denote angle indicators usually followed by the degree given. For 90° right angles a right angle forming a 90° bend appearing to be a square box is made to show that the angle is square. In actually the symbol is really the letter gamma and does not actually touch the height leg but is often drawn that way in 5th grade math as an introduction to shapes. 10th grade geometry formally introduces gamma as a Greek letter. It appears as an upside capital “ L” or an “F” with the lower horizontal segment missing. Gamma is written “ γ=90° in lowercase for values and in uppercase Γ as shorthand while omitting the value 90°. In my picture there is one pixel missing between the gamma top and the blue line so they are not touching.

Alpha = green ÷ black

Gamma = blue ÷ black

Beta = green ÷ left blue

These drawing are not even remotely scale. 89.625° should be almost 90° and therefore much closer to a vertical line making it much closer to the red line. The arc for beta appears much smaller than the arc for alpha even though beta is ≈ 90 times larger than alpha. When using arcs size of arc has no bearing on actual size of angle, arcs are hard to draw so scale can not be achieved.

This is what our landscape looks like more drawn to scale. Notice the slight grade and roll off.

Pic is 900px by 9px. This is the minimum for building code. You can make it steeper.}}}}}}}}}}}}}}}

This is a chart of angles found using our formula (slope times 45°). You can use it for future reference, embellish it, or make your own.

I only did fractions with one as a numerator because that was the simplest for learning and therefore most useful.

Pic was made in paint using a (1000px,1000px) square. Paint uses line graphs to chart where color is stored on the canvas. Zooming 800% a grid appears and you can see each individual pixel. Each one is a point on the graph assigned an ordered pair of (x,y) values.

Zeroing in on the top right corner, it's (1000,0), I then made a circle (2000px,2000px) to fit perfectly aligned in the square. Paint and Javascript canvas both use a unique quadrant scheme. (0,0) is the first pixel in the top left corner. Moving right increases the x value, moving down increase the y value. This is unlike conventional math where the (0,0) is at the center of two 90° axes with standard labeling of the quadrants 1-4 starting at the top right for (+,+) then moving counterclockwise to (-,+), (-,-), (-,+).

Paint programs only have one quadrant and use the upper left corner to zero everything to make editing fast and more precise. This is where images are loaded and where copied items are pasted so that less manual dragging with a mouse it needed for proper alignment.

I drew red lines to contrast with the black square and use white paint to quickly erase the square once my lines where done. The lines all extend past the protractor to y or x values of 1,000 using normal convention so that they are where the border of the square was. The lines are all squared up.

Here is a chart of the slope to degree enlarged. Note the blue are multiples of 1/3 and could be used to find the value of (1',48') by doing the necessary divisions by 2 till 1/3 it has been divided 4 times.

Slope of y/x

Angle in degrees

0/x ; y/0 = ∞

0 ; 90°

1/1 ; y = x


y = 1/2x


y = 1/3x

























































This is a good reference as an intro to trigonometry and studying the unit circle.

You can think of the unit of measure as the base or coefficient. Therefore you must divide by the base to cancel it out. That is point of algebra, using variables to perform analytics this way. We are going from a base of feet to a base of inches.

To avoid confusion, base in a general means a group we are using to count with. In length base is the units we are counting be it feet or inches. When we count in feet we are really counting groups of 12 inches. With inches we are only counting groups of 1 inch.

Base changes notation depending on what measure we are using so it is very important to define base. Otherwise we will mechanically derive answers only after we understand what to do with the numbers we are given and there will be some always be some unclarity when reading formulas. Example:

In distance:

1' = the apostrophe is the notation for a base of linear feet. The 1 is simply the numeric value or numeral of the base. We are saying we have one big unit of 12 small units. We are saying we have 1 number of a base 12 system. Base = how many units we are counting by.

12” = the double apostrophe is the notation for a base of linear inches. (12) = unit while ( “ ) = base.

In powers:

72 = The base is the numeral 7. Here the base is not the sub or superscript but the numeral while the exponent serves a unit purpose. 7 = base while 2 = units. We are multiplying by groups of 7, so 7 is our base we are counting by, while the exponent 2 is the units or number of 7's to be multiplied.

Compare 72 and 73. 7 · 7 versus 7 · 7 · 7. The base of seven does not change but the units of seven does.

Compare 2' and 3'. 2· 12” versus 3 · 12”. The base of 12” does not change only the units of twelve inches.

2' is literally saying 2 = 2 while ( ' ) = [· (12”)]. In English, “ The single apostrophe is shorthand for ' times twelve inches ' ”.

In binary and hex:

Binary is a base two system, hence it's name bi-, meaning two, and nary meaning (not any) number. When using binary notation we are using only two digits 0 and 1. To do conversions will be explained at length in the next chapter but for cursory review it is notated as units2. So a binary number 22 and 32 both have subscript two as a base indicating it's system, also called a radix. Radix is the positional notation. Being base two means that after two numbers are counted, 0 and 1, the position of digits is increased on the lefthand side. For every number composed of all 1's we witness an increase in digits digits by one place when added an additional unit of 1. Observe:

02 = 0

12 = 1

22 = 10

32 = 11

42 = 100

52 = 101

62 = 111

72 = 1000

82 = 1001

92 = 1011

102 = 1111

112 = 10000

The base of subscript 2 does not change but the input output does. This requires explanation later. Note that base can differ of whether it is a symbol, subscript, superscript or other notation used. Hex uses base 16, decimal uses base 10, sexagisimal uses base 60. These are written as 116 , 110 , and 160. These are all quite different uses with specialized purposes, each requiring their own section of study.

The point is math is ambiguous and different conventions arise leading to conflicts of understanding. I wanted to make sure that the concept of base was clear so there was not any ambiguity. For a final point consider: 22 and 2'. The base of the units is underlined. I think it would be reasonable for someone to question why I underlined the digits in one and not the other. Thus understanding the language of math is important when dealing with terms for in this it seems base is not always in the digits place.

0 = 0000

1 = 0001

2 = 0010

3 = 0011

4 = 0100

5 = 0101

6 = 0110

7 = 0111

8 = 1000

9 = 1001

10 = 1010

11 = 1011

12 = 1100

13 = 1101

14 = 1110

15 = 1111

16 = 10000

ABBRIEVATED 0b for base binary

ASCII - Binary Character Table






































































































































































































































































































































































The terms of a polynomial are the individual numbers with or without a variable. A mixed term is a number with a coefficient and a variable.

Constant = 5 or another defined immutable quantity

Variable = x or another letter or symbol

Term = a monomial separated from another by an operation sign like plus or minus

Coefficient = The constant factor in a monomial

Constant = 1

variable = x

mixed term= 2x

binomial= 2x +1

trinomial = x2 +x + 5

polynomial with four terms = x3 + x2 +x + 5

polynomial with five terms = x4 + x2 + x2 +x + 5

2 is a monomial because it has one term. 2 + x is a binomial because it has two terms. 2 + x can not be simplified any further so it stands as its own number or value until we can solve for x by substituting a value for it. A binomial it a number of two different sums acting as one since we can not solve for it and get a single value.

multiplying binomials to get a trinomial:

(x + 1) (x -1) Take two monomials

(x + 1) (x -1) first: (x) · (x) = x2

(x + 1) (x -1) second: (x) · (-1)= -1x

(x + 1) (x -1) third: (1) · (x) = 1x

(x + 1) (x -1) fourth: (1) · (-1)= -1

x2 -1x + 1x -1 add the terms together

x2 + 0x -1 the “ x1 ” terms cancel out

x2 -1 we are left with a binomial. Use a different binomial

(x -1) ( x -1)

x2 multiply first terms

-1x multiply both outside terms

-1x multiply both inside terms

1 multiply last terms

This is called the foil method of solving the product of two binomials.


(x -1) ( x -1) multiply these to get x2. These are the first two terms.

(x -1) ( x -1) multiply these to get -1x. These are the second two terms.

(x -1) ( x -1) multiply these to get -1x. These are the third two terms.

(x -1) ( x -1) multiply these to get 2. These are the fourth two terms.

first outside inside last

x2 -1x -1x +2 Add a + sign for the positive constant and line them in order

-1x -1x = -2x Combine like terms. like terms have the same variables and the same degrees on each one

x2 -2x + 2 Is our final answer. This is called quadratic formula or quadratic notation.

(x -1) ( x -1) Practice on easy ones using paper till you can do this mentally.

x2 -2x + 2 x2 makes an umbrella shape on line charts called a parabola or parabolic.

factoring a polynomial by grouping

dividing polynomials using long division

dividing polynomials using synthetic division


a candy bar equals ten pieces that are solid and joined by thin perforations for separating 10p

you give half to someone 10p – 5p = 0

the amount someone has is now 5p

he gives some to a girl but we do not know how much 10p = ap + bp

the bars are trapezoidal prisms of chocolate containing rich gooey caramel

each piece tapers and extends to the next only joined by a thin layer of chocolate meant to make snapping into individual pieces easy. To give someone his 5 you had to snap along this thin section leaving the 5 individual square prisms intact. For him to share pieces he must separate pieces the same way. If he does not share this way he would have to crush a square and cause the goop to fall out ruining it. This would ruin the piece ans make all the chocolate turn to crumbs and go flying like split atoms. This is gross and therefore forbidden.

How make pieces does each person have?

What we know:

You had 10p

you gave 5p

they gave x amount of p

10p= (5p - x) + (5p + x)

The above statement is true but we can not solve with this information, we can however use it to devise tools to solve for it.

We need to reorder this information using logic but we need to be careful because:

10p= 10p -x +x

10p =10p + 0x all we did was cancel x's and get our original statement or

0p = -x +x all we did was cancel the p's and find the absolute value of the integer


Systems of Linear equations


Quadratics Polynomials

Factoring trinomials

Quadratic equations factoring

(a+b)2 = (a+b)(a+b) = a2 +2ab +b2

(a-b)2 = (a-b)(a-b) = a2 – 2ab +b2

Perfect Square Trinomials

(x+4)2 = x2 +8x +64

(x-8)2 = x2 -8x +64

Ax2 + Bx + C

Bx = two times the product of one factor from the first and last term.

Knowing the first and last terms are squares mean it could be a perfect square trinomial.

Perfect square if:

1. 1st and last terms are square

2. o +i = middle term. Factors of

Factor: 6x2 – 18x + 9

Can be factored using (6x-?)(x-?) to get 6x2 but this would not be a perfect square.

Using (3x-?)(3x-?) gives 6x2 and is less work so let's use that.

(3x- ?)(3x- ?) writing our x2 terms

What factors of 9 add to -18 when multiplied by 3?

Factors of 9:

1∙ 9

3· 3

-1∙ -9

-3· -3 [(-1,9);(-9,1);(-3,3);(3,-3) are not considered because they make the last term -9)]

Since the middle term is -18 we have to use a pair of factors that are negative. Adding negative factors produces a negative number. If we used a positive pair we would get a positive result for the middle term and that would be wrong here.

3x(1)+ 3x(9) = 3x + 9x = 12x wrong

3x(3)+ 3x(3) = 9x + 9x = 18x right value but wrong sign

3x(-1)+ 3x(-9) = -3x + -9x = -12x right sign but wrong value. No combo of 3 and 9 will ever equal 18.

3x(-3)+ 3x(-3) = -9x + -9x = -18x right sign AND right value. This is our factors.

Other factors of 9 use mixed signs and would give a negative last term because a negative times a positive is a negative. So we did not include those. Likewise our middle term was negative so we could have skipped the two pair of all positive factors because they would have given a positive middle result when added.

(3x- ?)(3x- ?) plug in -3 as the coefficient of o and i.

(3x- 3)(3x- 3)

6x2 + o + i+ l 3x·3x =6x2

6x2 + -9x + i+ l o = 3x · -3

6x2 + -9x + -9x + l i= 3x · -3

6x2 + -9x + -9x + 9 l = 3· 3

6x2 + -18x + 9 combining like terms. O + I = -18x

6x2 -18x + 9 removing the plus sign.

There are other ways to do this. The way I have written is long and drawn out. You can remove the plus sign in the adding o+i step. The point is that we are not subtracting the middle terms but adding negative quantities. This has the same affect as subtracting and is called subtracting by addition. From this point all equations use subtraction by addition to deal with negative numbers. It helps make clear what the signs ofare of the values and which direction we are going in the number line. Usually this is done mentally as it looks bad to have a plus sign if front of a negative number but this is done to be clear and shown that we are indeed adding instead of subtracting.

In -7 + -7 = -14, we use the plus sign to shown 7+7 = 14 as the absolute value of the integer, the total distance moved on the number line, but we must not forget that this is a negative number! We always add numbers whether they are negative or positive so we do not get confused whether we are adding or subtracting, moving up the number line or down, canceling out or retaining a value. This is the explanation.

In polynomials we add like terms and this can be a long list. When doing this peole get confused whether we adding or subtracting negative or positive numbers. The in multiplication we have sigh changes and this adds to the confuse. It is easy to confuse the values of our integers and move in the wrong direction on the number line either canceling out a value we nee or keeping values we need to eliminate. When writing out brackets leave empty spaces separated by plus signs. The empty spaces will be filled in with your numbers later. Your numbers will either be positive or negative. Negative numbers have a minus sign in front of them positive numbers do not. It helps to write out long sums using this notation to avoid mistakes that occur when computing mentally. You must keep track of your signs!

There are 8 ways to write a given sum operation and by using plus signs we eliminate half of them, this avoids the issue of whether we are adding or subtracting. Straight up we are always adding. So that removes the four subtraction possibilities and really removes the other three so that in any given situation there is only one way we can sum a list of integers.

This is the way to write out numbers. If you try to do it another way it can be done but is more prone to error. If you can do it mentally and get the right answer fine but if you need to write it out only write it this way. Other ways are unreliable.

If I have to add six numbers:

N + n + n + n + n + n

and my numbers are -7,2,-1,3,-8,10:

-7+2+ -1+3+ -8+ 10

I left the plus signs and simply filled in the numbers with their minus signs if they were negative.

At this point you can remove the plus sign in front of a number IF it has a minus sign:

-7+2 -1+3 -8+ 10

-7+2 -1+3 -8+ 10 = -1

In 2 + - 1, the signs “cancel” out making it 2-1. We are adding a negative quantity so we can express this as pure subtraction. We do not need the plus sign anymore. It just help keeps the sign and operation clear in the beginning when we are writing down our numbers. So a plus sign is negated by a minus sign directly after it to the right. “2+-1” = “2-1” because “+-” = “-”.

In 7 - -7 we are subtracting a negative number. This becomes 7 + 7. When you subtract a negative number the signs cancel. We say we negated the operation This is because the minus sign acts as negation. A minus sign will negate both a “+” and a “-”. Negation means reversal, it is reversing direction. “=” reversed is subtraction, moving left on the number line, and “-” reversed is addition moving right on the number line. To some extent confusion is avoided by always separating terms and constants with a plus sign. Unless the operator sign is expressed as “-” we write it as “+”.

f= x2

o= f1x



(x+f1)(x+f2) our factors of a polynomial

x2 f = x2 This is the first term, if follows x·x.

f1x o= f1x This is the second term, it follow f1 ·x

f2x i= f2x This is the third term, f2 ·x

f1f2 l= f1f2 This is the fourth term, f1 ·f2

This is an attempt to formalize two things that are happening with our polynomial that otherwise would have to be intuitively understood rather than concretely. It states that the middle term, some x with a coefficient, is the sum of o+i where o is the product of x and the first factor of our last term and i is the product of x and the second factor of our last term. This combines many rules into one rule to rule them all.

X2 + f1x + f2x +f1f2 is the form for a trinomial. Trinomials are also called quadratic polynomials or just quadratics or polynomials for short. This is important to understand how to factor quadratics to solve them.

There are many rules and notations stating a polynomial. I will run through them and try to combine the different notations. This is sticky but true.

1. A polynomial is the product of two binomials. A binomial is a polynomial containing two numbers that cannot be further simplified. They can not be further simplified because the use of variables prohibits knowing the numerical value of the term. The entire binomial contains two parts. Technically the two parts are separate parts but are often thought of as one term in of itself.

2. A binomial is a two part expression. The simplest one is x+1. Since x is not defined in a numerical sense we can not find the value obtained when x is added by 1. As such x+1 is the simplest form we can write this value, also called a number. x+1 is a linked value that represents one number even though it is written using two numbers, namely x and 1. To know what x+1 equals we would have to assign a value to x and since we do not know what value to assign the expression x+1 stands as is. It is a singular value written in simplest form at this point and there is no way to write it as a single integer (or real number) to represent this sum. In short they can not be added and get some number removing the “+”. What I'm saying is a binomial can not be solved for x.

3. A quadratic is given in the form Ax2 + Bx + C where capital letters are constant numbers and x is some variable term. Some quadratics have 0 for B. When this happens the term is not written, Bx would simply be 0x since it cancels out. Some quadratics do not have any number for A, not even zero, and are written as x2. You can not have 0x2 in a quadratic as it would cancel out the x2 term. Quadratics and called what they are because it comes from Latin “quadrare” which means to make square, which is what we do when we square x. This means Ax2 + Bx + C can look like x2 + Bx + C.

4. x2 + Bx + C Has three terms:




It's factors are (x+ a factor of C)(x+ another factor of C)

It can also be written as (x+b)(x+b) = x2+2b+b2 but is is not guaranteed.

The constants can be expected to differ from each other and be two different numbers. If they are the same number than indeed the C term is a number squared.

Examine: (x+b)(x+b) = x2+2b+b2

x2+2b+b2 = (x·x)+(b+b)+(b·b)

x2 + 2b + b2

The term 2b looks like multiplication but really we are adding. We only are multiplying the last term.

This because when we are multiplying the middle terms we do so by adding the coefficients since they share the same base (variable). What is (b+b)= ? 1b+1b= 2b. We added the constants in front of b.

The first term x2 is formed when multiplying x·x.

The second term Bx comes from finding o and I and adding them together.

O and I are found during the outside and inside phase of foil.

The last term C comes from multiplying our two constant terms together.

The first and last phase of FOIL is straight forward. It is the outside and inside phases that require a closer look and greater skill. The takeaway is that the middle term is always the sum of o+i.

The two things that make the middle term is finding the factors of C, then multiplying the with the factors of Ax2.

Strictly speaking C = (f1)(f2) and Ax2 = Ax·x.

This means that o= Ax(f1) and i = x(f2)

Since the middle term equals o+i then it must be Ax(f1) + x(f2)

5. The notation for f1 and f2 stand for 1st factor and 2nd factor of C.

Take x2 + 7x +10

What are the factors of C (the Constant term with no variables)? They are 2 and 5 (2,5).

Ignoring everything but the “+10” I then compare it to “+7x” and ask what factors of 10 add to make 7.

My choices are (1,10) and (2,5). 1 + 10 7 so it must be (2,5). Note changing order does not matter here because this is commutative and the negative factorings are excluded because the middle term 7x is positive.

   >x2 + 7x + 10    A trinomial that we want to factor

2· 5 Finding the smallest factors that adds to 7 in 7x

2x+5x Writing in x as coefficients for our factors. This is our i+o 7x Adding our inside term and outside term to simplify our middle term x2 + 7x+ 10 writing x2 + in front and +10 to show the finished product (polynomial)

F = x2

O = 2x

I = 5x

L = 10

We need to write this data as the factors of the trinomial in the form (x+M)(x+N). I write this as

(x+f1)(x+f2) where the f's are factor one and factor two of C, in this case (2,5).

(x+  )(x+  ) x already filled in with blank space for our factors of C

(x+2)(x+5) writing (2,5) as our factors. This would be our final answer if asked to factor

Does (x+2)(x+5) = x2 + 7x + 10? We should check.

(x+2)(x+5) x2 The “First” in foil

(x+2)(x+5) x2 + 5x The “Outside” in foil

(x+2)(x+5) x2 + 5x+2x The “Inside” in foil

(x+2)(x+5) x2 + 5x+2x+10 The “Last” in foil

x2 + 7x + 10 adding o+i to simplify middle term

It checks out.

How about x2 -9x +18?

First what factors of +18 add to make – 9?

The middle term is negative which means that at least one of the factors is negative. It also must be the larger factor since when combining o+i the larger factor determines the sign of the middle term.

+18 is positive which means both factors must be negative so when they multiply a positive 18 results. In summary a mixed pair(-,+) would make 18 negative and also add to the wrong middle term. Since one of the signs is wrong we would be moving the wrong direction on the number line away from -9 instead of arriving at it. Knowing these rules and patterns avoids tedious trial and error of trying to guess what to use.

This is rote not critical thinking. Try to preprocess as much as possible an save critical thinking for your thesis and doctorate papers.

We must have 2 factors that are both negative. -f1·-f2 = “+ C”

These two negative factors will add to equal -9. (-f1) + (-f2) = -9

Intuitively I know the answer is (-6,-3) because -6-3= -9 and -6· -3 = 18.

Very quickly let's look at other factorings of 18

18 our product

1·18 18+1 ≠ 9, 18-1≠ 9, -18-1≠ -9, -18+1≠ -9. 19,17, -19, -17 and not -9.

2·9 This is not the pair because x≠ x +1. This is a special rule. 11,7-11,-7 ≠ -9

3·6 3+6≠ -9, 3-6≠ -9, -3+6≠ -9, -3-9 = -9. (-3,-9) satisfies both constraints.

No combination of 18 and 1 will ever equal -9. It is too large and far from -9. 2 and 9 is a special case.

Special case: There is no way (2,9) will ever equal -9 because whatever we do to 9 we are moving away from it instead of arriving at it since it was our origin to begin with. Consider x≠x+1. There is no number we can substitute for x that will equal itself plus one. Therefore x=x+1 is a false statement that can never be true. Likewise 9= 9+2 is false, as is 9= 9-2. From the four operations we get 7,11,-7,-11.

The first rule was our pair had to have one negative factor and it had to be the bigger of the two since the middle term -9x was negative. The sign of the larger factor determines the sign of the middle term. If you have a negative middle term your larger factor is negative. If your constant C term is positive then both your factors are negative.

-9x was negative so we had to have a big negative factor. 18 was positive so we had to have the other factor be negative so that when multiplied the two negatives canceled out leaving a positive number.

Because both factors are negative when added together we are stating at a negative sum and then subtracting from it by adding the other negative sum. We are moving left on the number line. So then we ask what -n-m= -9? There are only three factorings for 18 and two of them are eliminated right away for reasons already stated: 18,1 is too far away in an obvious way. 9,2 is absurd and invokes a special case that stands out as such. This leaves only 3,6 as a solution.

With 3,6 as a solution we all ready know from -n-m= -9 that the pair is -3,-6.