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.


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 From albegra math notes and beyond

MATH VOCABULARY AND CONCEPTS 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 6. Differential equations 7. Number theory 8. Combinatorics GLOSSARY 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 ⅓ . 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”. 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 πr^2 to find area of the circle. You only have to find area of the whole circle not two individual halves. lw+ πr^2 60·120+ π30·30 7200+π900 7200+2827.433388 10027.433388 7. 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: 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. Solution: A solution is a number that solves our equation. Solving the open sentence: When we replace variables with numbers the sentence is said to be solved. 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 arbitrry angles: Of real numbers: Special right angles: Law of cosines: Law of sines: Area: Heron's formula: Special values: degrees radians Sin θ Cos θ Cot θ Sec θ Csc θ 0° 0 0 1 -------- 1 – – – – – – 30° π/6 ½ √3/2 √3 2√3/3 2 45° Π/4 √2/2 1 1 √2 √2 60° Π/3 √3/2 √3 √3/3 2 2√3/3 90° Π/2 1 – – – – – 0 – – – – – – 1 Alt Code Symbol Description Alt 48 - 57 0 - 9 zero to nine Alt Codes for Basic Operators Alt Code Symbol Description Alt 43 + Plus Sign Alt 45 - Minus Sign Alt 0215 × Multiplication Sign Alt 0247 ÷ Obelus / Division ign Alt Codes for Pers Alt Code Symbol Description Alt 37 % Percentage Sign Alt 0137 ‰ Per mille (per thousand) Alt Codes for Bracketing Alt Code Symbol Description Alt 40 ( Open Bracket Alt 41 ) Close Bracked Alt Codes for Degree of Accuracy Alt Code Symbol Description Alt 241 ± Plus or Minus Alt Codes for Fractions Alt Code Symbol Description Alt 47 / Fraction seperator Alt 0188 ¼ Quarter Alt 0189 ½ Half Alt 0190 ¾ Three quarters Alt 46 . Decimal Point Alt Codes for Equality Alt Code Symbol Description Alt 240 ≡ Exactly Identical Alt 61 = Equals Alt 247 ≈ Approximately equal Alt Codes for Inequality Alt Code Symbol Description Alt 60 < Less Than Alt 62 > Greater Than Alt 242 ≥ Greater than or equal Alt 243 ≤ Less than or equal Alt Codes for Powers Alt Code Symbol Description Alt 251 √ Square Root Alt 252 ⁿ Power n Alt 0185 ¹ To the power of 1 Alt 0178 ² squared Alt 0179 ³ cubed Angles and Trigonometric Alt Codes Alt Code Symbol Description Alt 227 π Pi Alt 248 ° Degree sign See also Greek Alphabet Alt Codes General Mathematical Symbols Alt Code Symbol Description Alt 35 # Number Alt 236 ∞ Infinity Alt 230 µ Micro Alt 228 Σ Sum Alt 239 ∩ Suggest definition Integration / Integral Sign Alt Code Symbol Description Alt 244 ⌠ Top half Alt 245 ⌡ Bottom Half ALT Codes for Programming Alt Code Symbol Description Alt 0166 ¦ Unix Pipeline Alt 40 ( Open Bracket Alt 41 ) Close Bracked Alt 94 ^ To the power of Alt 60 < Less Than Alt 62 > Greater Than Alt 61 = Equals Alt 42 * Multiply Alt 47 / Divide or Slash Alt 92 \ Back Slash Alt 35 # Hash Alt 40 ( Open Bracket Alt 41 ) Close Bracked Alt 64 @ At Symbol Alt 91 [ Open Square Bracket Alt 93 ] Close Square Bracket Alt 123 { Open curley bracket Alt 125 } Close curley bracket Alt 42 * Wildcard and Multiply Character Displayed Alt Code Alpha α Alt 225 Beta β Alt 225 Gamma Γ Alt 226 Delta δ Alt 235 Epsilon ε Alt 238 Theta Θ Alt 233 Pi π Alt 227 Mu µ Alt 230 Uppercase Sigma Σ Alt 228 Lowercase Sigma σ Alt 229 Tau τ Alt 231 Uppercase Phi Φ Alt 232 Lowercase Phi φ Alt 237 Omega Ω Alt 234 Alt Code Symbol Description Alt 224 α Alpha Alt 225 ß Beta Alt 226 Γ Gamma Alt 235 δ Delta Alt 238 ε Epsilon Alt 233 Θ Theta Alt 227 π Pi Alt 230 µ Mu Alt 228 Σ Uppercase Sigma Alt 229 σ Lowercase sigma Alt 231 τ Tau Alt 232 Φ Uppercase Phi Alt 237 φ Lowercase Phi Alt 234 Ω Omega 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)= 0∫∞ e-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 UPPERCASE ETA: Η 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 Exponetial 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 3.14 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 angle Lowercase tau: τ Torgue, 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 ohm Lowercase omega: ω Most alt codes are unreliable and produce unintended pictograms. Special characters map works but is slow to find and implement. Making a save it on notepad is slow to but a little better. Αα (alpha) {\displaystyle \alpha } \alpha represents: the first angle in a triangle, opposite the side A the ratio of collector current to emitter current in a bipolar junction transistor (BJT) in electronics the statistical significance of a result the false positive rate in statistics ("Type I" error) the fine structure constant in physics the angle of attack of an aircraft an alpha particle (He2+) angular acceleration in physics the linear thermal expansion coefficient the thermal diffusivity In organic chemistry the α-carbon is the backbone carbon next to the carbonyl carbon, most often for amino acids right ascension in astronomy the brightest star in a constellation Iron ferrite and numerous phases within materials science the return in excess of the compensation for the risk borne in investment the α-conversion in lambda calculus the independence number of a graph Ββ (beta) See also: Beta (disambiguation) Β represents the beta function {\displaystyle \beta } \beta represents: the thermodynamic beta, equal to (kBT)−1, where kB is Boltzmann's constant and T is the absolute temperature. the second angle in a triangle, opposite the side B one root of a quadratic equation, where α represents the other the standardized regression coefficient for predictor or independent variables in linear regression (unstandardized regression coefficients are represented with the lower-case Latin b, but are often called "betas" as well) the ratio of collector current to base current in a bipolar junction transistor (BJT) in electronics (current gain) the false negative rate in statistics ("Type II" error) the beta coefficient, the non-diversifiable risk, of an asset in mathematical finance the sideslip angle of an airplane the first-order effects of variations in Coriolis force with latitude in planetary dynamics a beta particle (e− or e+) sound intensity velocity divided by the speed of light in special relativity the beta brain wave in brain or cognitive sciences ecliptic latitude in astronomy The ratio of plasma pressure to magnetic pressure in plasma physics β-reduction in lambda calculus The ratio of the velocity of an object to the speed of light as used in the Lorentz factor In organic chemistry, β represents the second carbon from a functional group Γγ (gamma) See also: Gamma (disambiguation) Γ represents: the circulation in fluid dynamics the reflection coefficient of a transmission or telecommunication line. the confinement factor of an optical mode in a waveguide the gamma function, a generalization of the factorial the upper incomplete gamma function the modular group, the group of fractional linear transformations the gamma distribution, a continuous probability distribution defined using the gamma function second-order sensitivity to price in mathematical finance the Christoffel symbols of the second kind the neighbourhood of a vertex in a graph the stack alphabet in the formal definition of a pushdown automaton {\displaystyle \gamma } \gamma represents: the circulation strength in fluid dynamics the partial safety factors applied to loads and materials in structural engineering the specific weight of substances the lower incomplete gamma function the third angle in a triangle, opposite the side C the Euler–Mascheroni constant in mathematics gamma rays and the photon the heat capacity ratio in thermodynamics the Lorentz factor in special relativity the damping constant (kg/s) Δδ (delta) See also: Delta (disambiguation) Δ represents: a finite difference a difference operator a symmetric difference the Laplace operator the angle that subtends the arc of a circular curve in surveying the determinant of an inverse matrix the maximum degree of any vertex in a given graph the difference or change in a given variable, e.g. ∆v means a difference or change in velocity sensitivity to price in mathematical finance distance to Earth, measured in astronomical units heat in a chemical formula the discriminant in the quadratic formula which determines the nature of the roots the degrees of freedom in a non-pooled statistical hypothesis test of two population means {\displaystyle \delta } \delta represents: percent error a variation in the calculus of variations the Kronecker delta function the Feigenbaum constant the force of interest in mathematical finance the Dirac delta function the receptor which enkephalins have the highest affinity for in pharmacology[1] the Skorokhod integral in Malliavin calculus, a subfield of stochastic analysis the minimum degree of any vertex in a given graph a partial charge. δ− represents a negative partial charge, and δ+ represents a positive partial charge chemistry (See also: Solvation) the Chemical shift of an atomic nucleus in NMR spectroscopy. For protons, this is relative to tetramethylsilane = 0. stable isotope compositions declination in astronomy the Turner function in computational material science depreciation in macroeconomics noncentrality measure in statistics[2] Not to be confused with ∂ which is based on the Latin letter d but often called a "script delta." Εε (epsilon) See also: Epsilon (disambiguation) {\displaystyle \epsilon } \epsilon represents: a small positive quantity; see limit a random error in regression analysis the absolute value of an error[3] in set theory, the limit ordinal of the sequence {\displaystyle \omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\dots } \omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\dots in computer science, the empty string the Levi-Civita symbol in electromagnetics, dielectric permittivity emissivity strain in continuum mechanics permittivity the Earth's axial tilt in astronomy elasticity in economics expected value in probability theory and statistics electromotive force in chemistry, the molar extinction coefficient of a chromophore. set membership symbol ∈ is based on ε Ϝϝ (digamma) See also: Digamma Ϝ is sometimes used to represent the digamma function, though the Latin letter F (which is nearly identical) is usually substituted. A hypothetical particle Ϝ speculated to be implicated in the 750 GeV diphoton excess, now known to be simply a statistical anomaly Ζζ (zeta) See also: Zeta (disambiguation) {\displaystyle \zeta } \zeta represents: the Riemann zeta function and other zeta functions in mathematics the coefficient of viscous friction in polymer dynamics the damping ratio relative vertical vorticity in fluid dynamics Ηη (eta) See also: Eta (disambiguation) Η represents: the Eta function of Ludwig Boltzmann's H-theorem ("Eta" theorem), in statistical mechanics Information theoretic (Shannon) entropy {\displaystyle \eta } \eta represents: the intrinsic wave impedance of a medium (e.g. the impedance of free space) the partial regression coefficient in statistics elasticities in economics the absolute vertical vorticity (relative vertical vorticity + Coriolis effect) in fluid dynamics an index of refraction the eta meson viscosity energy conversion efficiency efficiency (physics) the Minkowski metric tensor in relativity noise in communication system models η-conversion in lambda calculus Cost-push supply side shocks in the Phillips curve equation (economics)[citation needed] A right angle, i.e., π/2, as a follow-up to the tau/pi argument[4] Θθ (theta) See also: Theta (disambiguation) Θ (uppercase) represents: an asymptotically tight bound related to big O notation. Debye temperature in solid state physics sensitivity to the passage of time in mathematical finance in set theory, a certain ordinal number in econometrics and statistics, a space of parameters from which estimates are drawn {\displaystyle \theta } \theta (lowercase) represents: a plane angle in geometry the angle to the x axis in the xy-plane in spherical or cylindrical coordinates (mathematics) the angle to the z axis in spherical coordinates (physics) Bragg's angle of diffraction the potential temperature in thermodynamics the mean time between failure in reliability engineering soil water contents in soil science in mathematical statistics, an unknown parameter theta functions the angle of a scattered photon during a Compton scattering interaction the angular displacement of a particle rotating about an axis. ϑ ("script theta"), the cursive form of theta, often used in handwriting, represents the first Chebyshev function in number theory Ιι (iota) See also: Iota (disambiguation) {\displaystyle \iota } \iota represents: an inclusion map in set theory the index generator function in APL (in the form ⍳) the orbital inclination in celestial mechanics. Κκ (kappa) See also: Kappa (disambiguation) Κ represents: the Kappa number, indicating lignin content in pulp {\displaystyle \kappa } \kappa represents: the Von Kármán constant, describing the velocity profile of turbulent flow the kappa curve, a two-dimensional algebraic curve the condition number of a matrix in numerical analysis the connectivity of a graph in graph theory curvature dielectric constant {\displaystyle (\varepsilon /\varepsilon _{0})} (\varepsilon /\varepsilon _{0}) thermal conductivity (usually a lowercase Latin k) thermal diffusivity a spring constant (usually a lowercase Latin k) the heat capacity ratio in thermodynamics (usually γ) the receptor which dynorphins have the highest affinity for in pharmacology[1] Λλ (lambda) See also: Lambda (disambiguation) Λ represents: the von Mangoldt function in number theory the set of logical axioms in the axiomatic method of logical deduction in first-order logic the cosmological constant the lambda baryon a diagonal matrix of eigenvalues in linear algebra the permeance of a material in electromagnetism a lattice {\displaystyle \lambda } \lambda represents: one wavelength of electromagnetic radiation the decay constant in radioactivity function expressions in the lambda calculus a general eigenvalue in linear algebra the expected number of occurrences in a Poisson distribution in probability the arrival rate in queueing theory the average lifetime or rate parameter in an exponential distribution (commonly used across statistics, physics, and engineering) the failure rate in reliability engineering the fundamental length of a fabrication process in VLSI design the mean or average value (probability and statistics) the latent heat of fusion the lagrange multiplier in mathematical optimization, known as the shadow price in economics the Lebesgue measure denotes the volume or measure of a Lebesgue measurable set longitude in geodesy linear density ecliptic longitude in astronomy the Liouville function in number theory the Carmichael function in number theory a unit of measure of volume equal to one microlitre (1 μL) or one cubic millimetre (1 mm³) the empty string in formal grammar a formal system in mathematical logic the thermal conductivity. Μμ (mu) See also: Mu (disambiguation) {\displaystyle \mu } \mu represents: the Möbius function in number theory the ring representation of a representation module the population mean or expected value in probability and statistics a measure in measure theory micro-, an SI prefix denoting 10−6 (one millionth) the coefficient of friction in physics the service rate in queueing theory the dynamic viscosity in physics magnetic permeability in electromagnetics a muon reduced mass chemical potential in condensed matter physics the ion mobility in plasma physics the Standard gravitational parameter in celestial mechanics the refractive index of a medium with respect to another medium or vacuum. Νν (nu) See also: Nu (disambiguation) {\displaystyle \nu } \nu represents: frequency in physics in hertz (Hz) degrees of freedom in statistics Poisson's ratio in material science a neutrino kinematic viscosity of liquids stoichiometric coefficient in chemistry dimension of nullspace in mathematics true anomaly in celestial mechanics the matching number of a graph Ξξ (xi) See also: Xi (disambiguation) Ξ represents: the original Riemann Xi function, i.e. Riemann's lower case ξ, as denoted by Edmund Landau and currently the grand canonical ensemble found in statistical mechanics the xi baryon {\displaystyle \xi } \xi represents: the original Riemann Xi function the modified definition of Riemann xi function, as denoted by Edmund Landau and currently a random variable the extent of a chemical reaction coherence length the damping ratio universal set Οο (omicron) See also: Omicron (disambiguation) Ο represents: big O notation (may be represented by an uppercase Latin O) {\displaystyle \mathrm {o} } {\displaystyle \mathrm {o} } represents: small o notation (may be represented by a lowercase Latin o) Ππ (pi) See also: Pi (disambiguation) Π represents: the product operator in mathematics a plane the unary projection operation in relational algebra osmotic pressure {\displaystyle \pi } \pi represents: Archimedes' constant, the ratio of a circle's circumference to its diameter the prime-counting function profit in microeconomics and game theory inflation in macroeconomics, expressed as a constant with respect to time the state distribution of a Markov chain in reinforcement learning, a policy function defining how a software agent behaves for each possible state of its environment a type of covalent bond in chemistry (pi bond) a pion (pi meson) in particle physics in statistics, the population proportion nucleotide diversity in molecular genetics in electronics, a special type of small signal model is referred to as a hybrid-pi model in relational algebra for databases, represents projection ϖ (a graphic variant, see pomega) represents: angular frequency of a wave, in fluid dynamics (angular frequency is usually represented by {\displaystyle \omega } \omega but this may be confused with vorticity in a fluid dynamics context) longitude of pericenter, in astronomy[5] comoving distance, in cosmology[6] Ρρ (rho) See also: Rho (disambiguation) Ρ represents: one of the Gegenbauer functions in analytic number theory (may be replaced by the capital form of the Latin letter P). {\displaystyle \rho } \rho represents: one of the Gegenbauer functions in analytic number theory. the Dickman-de Bruijn function the radius in a polar, cylindrical, or spherical coordinate system the correlation coefficient in statistics the sensitivity to interest rate in mathematical finance density (mass or charge per unit volume; may be replaced by the capital form of the Latin letter D) resistivity the shape and reshape operators in APL (in the form ⍴) the utilization in queueing theory the rank of a matrix the rename operator in relational algebra Σσς (sigma) See also: Sigma (disambiguation) Σ represents: the summation operator the covariance matrix the set of terminal symbols in a formal grammar {\displaystyle \sigma } \sigma represents: Stefan–Boltzmann constant in blackbody radiation the divisor function in number theory the real part of the complex variable s = σ + i t in analytic number theory the sign of a permutation in the theory of finite groups the population standard deviation, a measure of spread in probability and statistics a type of covalent bond in chemistry (sigma bond) the selection operator in relational algebra stress in mechanics electrical conductivity area density nuclear cross section surface charge density for microparticles Ττ (tau) See also: Tau (disambiguation) {\displaystyle \tau } \tau represents: torque, the net rotational force in mechanics the elementary tau lepton in particle physics a mean lifetime, of an exponential decay or spontaneous emission process the time constant of any device, such as an RC circuit proper time in relativity one turn: the constant ratio of a circle's circumference to its radius, with value 2π (6.283...).[7] Kendall tau rank correlation coefficient, a measure of rank correlation in statistics Ramanujan's tau function in number theory a measure of opacity, or how much sunlight cannot penetrate the atmosphere the intertwining operator in representation theory shear stress in continuum mechanics an internal system step in transition systems a type variable in type theories, such as the simply typed lambda calculus path tortuosity in reservoir engineering in topology, a given topology the tau in biochemistry, a protein associated to microtubules the golden ratio 1.618... (although φ (phi) is more common) the number of divisors of highly composite numbers (sequence A000005 in the OEIS) in proton NMR spectroscopy, τ was formerly used for physical shift Υυ (upsilon) See also: Upsilon (disambiguation) Υ represents: the upsilon meson Φφ (phi) See also: Phi (disambiguation) Φ represents: the work function in physics; the energy required by a photon to remove an electron from the surface of a metal magnetic flux the cumulative distribution function of the normal distribution in statistics phenyl functional group the reciprocal of the golden ratio (represented by φ, below), also represented as 1/φ the value of the integration of information in a system (based on integrated information theory) Geopotential Note: A symbol for the empty set, {\displaystyle \varnothing } \varnothing , resembles Φ but is not Φ. {\displaystyle \phi } \phi represents: the golden ratio 1.618... in mathematics, art, and architecture Euler's totient function in number theory a holomorphic map on an analytic space the argument of a complex number in mathematics the value of a plane angle in physics and mathematics the angle to the z axis in spherical coordinates (mathematics) the angle to the x axis in the xy-plane in spherical or cylindrical coordinates (physics) latitude in geodesy a scalar field radiant flux electric potential the probability density function of the normal distribution in statistics a feature of a syntactic node giving that node characteristics such as gender, number and person in syntax the diameter of a vessel (engineering) capacity reduction factor of materials in structural engineering Χχ (chi) See also: Chi (disambiguation) {\displaystyle \chi } \chi represents: the chi distribution in statistics ( {\displaystyle \chi ^{2}} \chi ^{2} is the more frequently encountered chi-squared distribution) the chromatic number of a graph in graph theory the Euler characteristic in algebraic topology electronegativity in the periodic table the Rabi frequency the spinor of a fundamental particle the Fourier transform of a linear response function a character in mathematics; especially a Dirichlet character in number theory the Sigma vectors in the unscented transform used in the unscented Kalman filter sometimes the mole fraction a characteristic or indicator function in mathematics the Magnetic susceptibility of a material in physics Ψψ (psi) See also: Psi (disambiguation) Ψ represents: water potential a quaternary combinator in combinatory logic {\displaystyle \psi } \psi represents: the wave function in the Schrödinger equation of quantum mechanics the stream function in fluid dynamics yaw angle in vehicle dynamics the angle between the x-axis and the tangent to the curve in the intrinsic coordinates system the reciprocal Fibonacci constant the second Chebyshev function in number theory the polygamma function in mathematics load combination factor in structural engineering the supergolden ratio[8] Ωω (omega) See also: Omega (disambiguation) Ω represents: the SI unit measure of electrical resistance, the ohm angular velocity / radian frequency (rev/min) the right ascension of the ascending node (RAAN) or Longitude of the ascending node in astronomy and orbital mechanics the rotation rate of an object, particularly a planet, in dynamics the omega constant 0.5671432904097838729999686622... an asymptotic lower bound related to big O notation in probability theory and statistical mechanics, the support a solid angle the omega baryon the arithmetic function counting a number's prime factors the density parameter in cosmology {\displaystyle \omega } \omega represents: angular velocity / radian frequency (rad/sec) the argument of periapsis in astronomy and orbital mechanics a complex cube root of unity — the other is ω² — (used to describe various ways of calculating the discrete Fourier transform) the differentiability class (i.e. {\displaystyle C^{\omega }} C^{\omega }) for functions that are infinitely differentiable because they are complex analytic the first infinite ordinal the omega meson the set of natural numbers in set theory (although {\displaystyle \mathbb {N} } \mathbb {N} or N is more common in other areas of mathematics) an asymptotically dominant quantity related to big O notation in probability theory, a possible outcome of an experiment in economics, the total wealth of an agent in general equilibrium theory vertical velocity in pressure-based coordinate systems (commonly used in atmospheric dynamics) the arithmetic function counting a number's distinct prime factors a differential form (esp. on an analytic space) the symbol ϖ, a graphic variant of π, is sometimes construed as omega with a bar over it; see π the last carbon atom of a chain of carbon atoms is sometimes called the ω (omega) position, reflecting that ω is the last letter of the Greek alphabet. This nomenclature can be useful in describing unsaturated fatty acids. 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. (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') Formula for linear length: length = unit length · number of units. 2. 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 10L 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)”. 3. 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. 4. 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. 5. 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 p = $538.878 final cost 6. 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 x=x ƒ(x)=y 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 also; 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. A famous math person absent mindedly confused this with range or the constant inbetween intervals hand misdefined his terms. As a result most if not all math books carry this error and the term is abused to the chagrin of math and language itself. f(x) = ⅓ x3 + ½ x2 -14x+25 Polynomials () 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 = ⅓-4 and 3-4 = ⅓4 as they are reciprocal. Writing power as the denominator with 1 as the numerator: 3-1 = ⅓ 3-2 = ⅑ 3-3 = ⅟29 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 ⅕ 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 · ⅓ · ⅓ · ⅓ · ⅓ 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 = ⅑ (1· -⅓ · -⅓) -3-3 = -⅟29 (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 = -⅟29 ” 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 = ⅑ (-0.111111) a third of a third -3-3 = -⅟29 (-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 321-3 318 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 36 (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(12) · 4(22) 4(1) · 4(4) 4 ·16= 64 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 the base is different you would be grouping together different groups of groups i.e. 3·(n2)  7· (n2) . 3n2 · 7n2 We find that n = 5 3(52) · 7(52) substitution 3(25) · 7(25) expanded 75 · 175 13125 3n2 · 7n2 21(n4) 21(54) 21(625) 13125 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) 21n4 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 multiplying monomials to get a binomial: 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. Try again (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. Foil stands for FIRST, OUTSIDE, INSIDE, LAST. (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  (repeats the pattern from blue area) i−3 = i i−2 = −1 i−1 = −i i0 = 1 i1 = i i2 = −1 i3 = −i i4 = 1 i5 = i i6 = −1 in = in(mod 4)