Math

From Js

1 Negative One
2 Arithmetic
3 Probability
4 Algebra
5 Linear Algebra
6 Calculus
7 Analysis
8 Complex Analysis
9 Abstract Algebra
10 Number Theory
11 Glossary of Symbols
- 11.1 Set and Logic Symbols

This is everything I know about maths; it is not everything that is in maths. Just what I know.
Have fun.
Geoff St. Pierre

Negative One

If this is your first time reading this (book) you should ignore A L L of the hyperlinks (blue text) and just read it. Upon subsequent readings you may find it useful and interesting to explore the hyperlinks. Basically, a straight read through is my take on it and the hyperlinks give other peoples take, which sometimes coincides with mine and sometimes not.

Arithmetic

The act of performing operations on numbers. The most common operations are addition ( $\ +$ ), subtraction ( $\ -$ ), division ( $\ /$ ) and multiplication ( $\ \cdot$ ). Other operations include exponentiation, rooting, and logarithms; but, we will get to those in due time.

Addition

$\ 1 + 1 = 2$

To define the number one is a rather difficult task, but we all have a good intuitive sense of what "oneness" is. Oneness is the property of having or thinking of a singular quantity, i.e. you have one dollar, one bushel of potatoes, or one light year. From here we can recursively define the natural numbers by assigning a new name for each new number of unities that we have:

$\begin{matrix} 1 & : & unity & : & one \\ 2 & : & 1 + 1 & : & two \\ 3 & : & 1 + 1 + 1 & :& three \\ \vdots & & \vdots & & \vdots \\ n & : & 1 + 1 + ... + 1 & : & n\ ones \\ \end{matrix}$

Now that we have named the numbers we can define addition as the process of counting how many unities we have, i.e.

$\ 5 + 3 = (1 + 1 + 1 + 1 + 1) + (1 + 1 +1) = 8$

Subtraction

Subtraction can likewise be defined as counting initial quantity of unities and removing some amount, i.e.:

$\ 5 - 3 = (1+1+1+1+1) - (1+1+1) = 2$

implies 5 unities remove three unities, leaving a result of two unities.

Multiplication

Multiplication is a shorthand for addition, i.e.

$\ 5\cdot3 = 15$

What this means is to add up three five times; or add up five trhee times.

$\ 5\cdot3 = 3 + 3 + 3 + 3 + 3 = 15$

Division

$\ \frac{6}{3} = 2$

The above division problem asks the question if six is 1+1+1+1+1+1, and three is 1+1+1, then how many sets of three can we break six into? The answer is of course 2, i.e.

$\ 6 = 1+1+1+1+1+1 = (1+1+1)+(1+1+1) = 3 + 3$ ; two sets of three.

Note: Division is the first operation where a problem arises. In all the previously defined operations (addition, subtraction, and multiplication) we could perform the operation on any pair of numbers we chose. However, with division we cannot divide by zero. Much will be said about this fact throughout the course of this book, and even through your studies in all of mathematics.

Exponentiation

An exponent is a number or expression that is typeset in superscript to the right of another number or expression, i.e.

$\ 2^{3} = 8$

In the above expression 2 is the base and 3 is the exponent and 8 is the result.

Just as multiplication is a shorthand for addition; exponentiation is a shorthand for multiplication. We all know that 2 cubed (or two to the third power) is 8, just as the above equation indicates. Now let us break it down to first principles: what does it mean to write $\ 2^{3}$ , well it means to take two your base and use it as a factor three times your exponent i.e.

$\ 2^{3} = 2\cdot2\cdot2$

which is of course equivalent to 8.

Note: an exponent is intended only for the symbol/expression immediately proceeding it on the left, i.e.

$\ (2+1)^3 \ne 2^{3}+1^{3}$

it means:

$\ (2+1)\cdot(2+1)\cdot(2+1)$

this brings us to the order of operations.

Order of Operations

Of course in all the operations defined above; the result is unique (with the exception of zero divide). This is good, very good, and is essentially what gives mathematics its power.

Now, if we decide to start combining operations; we still would like a unique result. That is how do we evaluate an expression like:

$\ 7\cdot3^{4}$

if we proceed left to right and tackle each operation as it appears we hit multiplication as the first operation, if we did this now; the result is:

$\ {\color{red}21^{4}}$

but, this is not what we intend, because the exponent should only be applied to the 3 (the expression immediately proceeding it on the left). That is to say this will lead to an errant result (or non unique). Therefore, to be in line with the meaning of the symbols, in this case an exponent, we must evaluate the exponent first, in order to preserve its intended meaning, i.e. we must give precedence to exponents over the operation of multiplication.

Thusly, we produce a hierarchy of precedence to the operations as follows:

 (i)   Exponentiation
 (ii)  Multiplication
 (iii) Division
 (iv)  Addition
 (v)   Subtraction

In some situations we may wish to exercise our intentions as humans and supercede the above precedence. To do this we introduce grouping symbols: (), []. In other words if we truly intend to have $\ 21^{4}$ , then we must write:

$\ (7\cdot3)^{4}$

Now our full hierarchy is in place and we can rest easy:

 P E M D A S
 (i)   Grouping symbols (), []
 (ii)  Exponentiation
 (iii) Multiplication
 (iv)  Division
 (v)   Addition
 (vi)  Subtraction

Some people use the mnemonic device PEMDAS to remember this hierarchy.

This concludes our initial discussion of operations. If nothing else remember first principles and everything else can be reduced to addition or simple counting.

Fractions and Arithmetic

We can perform all of the above operations on fractions. It just takes a little more oomph!

Adding fractions

$\ \frac{1}{2} + \frac{1}{2}$

Well we can reason this out pretty easily, i.e. if you have half of a dollar and someone gives you another half of a dollar, then you have two halves of a dollar or one whole dollar.

$\ \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1$

The key here is that both of the fractions that we are adding have the same denominator. Let us review some vocabulary:

 denominator:  the number in the bottom of the fraction, it indicates your denomination.
 numerator:  the number in the top of a fraction, it enumerates how many denominations you have.

If you take this view, it is easy to see that in order to add these objects, it is first necessary to put them onto the same footing, or of the same denomination. In other words we must obtain a common denominator.

$\ \frac{1}{2} + \frac{1}{3}$

To put them on the same footing we must find a common multiple between the denominators: 2 and 3. We can always multiply the numbers together to get such a multiple, i.e. $\ 2\cdot3 = 6$ , so 6 is a common multiple. Now we can manipulate each of the fractions to have a denominator of 6.

$\ \frac{3}{3}\cdot\frac{1}{2} + \frac{2}{2}\cdot\frac{1}{3}$

by multiplying by $\ \frac{3}{3}$ we are not changing the value of the fracion just how it looks.

$\ \frac{3}{6} + \frac{2}{6}$

Now, we can just add the numerators.

$\ \frac{5}{6}$

Done.

Probability

The definition of probablity is:

Probability of an Event = # of Desired Outcomes / # of Possible outcomes; Mathematically we write:

$\ P(E) = \frac{possible}{total}$

Algebra

Performing arithmetic on variables. Well that is the action, but usually one has some purpose in mind; like trying to prove something. A variable is a symbol that represents a quantity that is unknown and or changing. The symbol 1 is considered constant. All numbers are constant, but x is a letter that we can use in a mathematical context to be an unknown or changing quantity.

That is to say an equation like:

$\begin{matrix} 14 + 2.7 - \sqrt{4} & = & 14.7 & \mathrm{is\ not\ algebraic} \end{matrix}$

it is true, but not algebraic. It is arithmetic.

However, the equation:

$\begin{matrix} x + 1 & = & 5 & \mathrm{is\ algebraic} \end{matrix}$

That is to say the letter x represents an unknown quantity. One that can be determined by logic, but at its current position in the equation is unknown. If we subtract 1 from each side of the equation respectively, we can get x by itself, and thereby determine its value.

$\begin{matrix} x + 1 - 1 & = & 5 - 1 \\ \\ x & = & 4 \end{matrix}$

so x must posses the value of 4 else the equation be false.

Number Sets

Venn Diagram of Number Sets

It is probable that the advent of numbers came with the need for humans to cooperate and collaborate; things like trade and barter. And in this fashion we started counting. It is easy to list these sets, but harder to convey the human and mathematical pains and distances in years that are actually between them. For instance, the only difference between the natural numbers and whole numbers is zero, but mathematically speaking zero is hugely important, and conceptually a giant leap.

Likewise with the integers mathematicians would often discard negative solutions as absurd. But, in time we have accepted, use, and apply all of these sets and more in modern mathematics.

The Natural Numbers:
$\ \mathbb{N} = \lbrace 1, 2, 3, ... , n, n + 1, ... \rbrace$

The Whole Numbers:
$\ \mathbb{W} = \lbrace 0, 1, 2, 3, ... , n, n + 1, ... \rbrace$

The Integers:
$\ \mathbb{Z} = \lbrace 0, \pm 1, \pm 2, \pm 3, ... , \pm n, \pm (n + 1), ... \rbrace$

The Rational Numbers:
$\ \mathbb{Q} = \lbrace \ \frac{a}{b} \ \vline \ a,\ b \in \mathbb{Z},\ b \ne 0 \rbrace$ , $2.1 \in Q$ since $2.1 = \frac{21}{10}$

The Irrational Numbers:
$\ \mathbb{I} = \lbrace a\ \vline\ a \notin Q\ \and\ a \in \mathbb{R} \rbrace$ , examples: $\sqrt{2}, \sqrt{3}, \sqrt{10}, \pi, e$

The Transcendental Numbers:
$\ \mathbb{T} = \lbrace$ Transcendental Numbers $\ \rbrace$ , examples: $\ \pi, e$

The Real Numbers:
$\ \mathbb{R} = \lbrace$ all of the above number sets $\ \rbrace$

The Complex Numbers:
$\ \mathbb{C} = \lbrace a+bi\ \vline \ a,b \in \mathbb{R}, i = \sqrt{-1} \rbrace$

There are other objects like quaternions, but these are the most common objects studied in High School Mathematics.

Exercises for Number Sets

 Determine the smallest number set for each of the following:

 1.       2.       3.     4.       5.  


 6.       7.       8.      9.    10.

Dots

Dots; more formally referred to as points are zero dimensional objects.

General:
$(x,\ y)$

Specific:
$(2,\ 5)$
$(-1.5,\ 22.7)$

Dots are usually placed into a coordinate plane; x-axis typically parallel to the earth's surface at your current position, and a y-axis perpendicular to the earth's surface at your current position.

It takes two dots to define a line.

Lines

y = x

Lines are straight, extend indefinitely in two directions, one dimensional objects. Lines have a degree of one.

There are two "special" types of lines: horizontal and vertical.
Horizontal:
$\ y = a$ , where a is some fixed constant, i.e.: $\ y = 2$ is a fixed horizontal line that is always two units above the x-axis.

Vertical:
$\ x = a$ , i.e. $\ x = 2$ is a fixed vertical line that is always two units to the right of the y-axis.

General Forms:
$\ y = mx + b$ ; slope intercept form of a line.
$\ y - y_1 = m(x - x_1)$ ; point slope form.
$\ Ax + By = C$ ; Standard form of a line.

Slope Intercept Form

$\ y = mx + b$ is the slope intercept form of a line.
m represents the slope of a line and is computed:
$\ m = \frac{y_2-y_1}{x_2-x_1}$ where $\ (x_1, y_1),\ (x_2, y_2)$ are points on the line.
This is often referred to as: $\ \frac{rise}{run}$ or $\ \frac{\Delta y}{\Delta x}$ ; the change in the y-coordinate divided by the change in the x-coordinate.

For example if you wanted to compute the slope of the line containing the two points $\ (2, 3),\ (5, 8)$ it would be:
$\ m = \frac{8 - 3}{5-2} = \frac{5}{3}$ .

Solving Linear Equations

Solving an equation is the act of isolating a variable and finding its value or an expression that represents it (the variable). The most basic form of this act is solving for "x", i.e.:

Example 1

$\begin{matrix} 2x & = & 6 \\ \frac{2x}{2} & = & \frac{6}{2} \\ x & = & 3 \end{matrix}$

x is not by itself it has a coefficient of 2.
To get x by itself we must divide by 2.
now x is "by itself", and is solved for x.

Example 2

$\begin{matrix} 2x + 1 & = & 6 \\ 2x & = & 5 \\ x & = & 2.5 \end{matrix}$

If you learn nothing else about mathematics; you should realize that an equation is a representation of equivalent statements and if you change one side of the equation you must accordingly adjust the other side to preserve the balance. If you divide one side by two you must also divide the other side by two.

Exercises Solving Linear Equations

Solve for x:

1. $\ x + 1 = 5$

2. $\ 2x - 3 + 4x = x - 7$

3. $\ x + x = 4$

4. $\ 9 - x = 4x$

7. $\ 4x - 9 = 16x + 4$

8. $\ 2(x + 3) = x - 9$

9. $\ 5(x+1) = 4(x - 3)$

10. $\ 1.2x = 9$

Functions

A function can be thought of like a machine. It takes some input does some stuff to it and creates an output. Like a toaster it accepts input of bread, heats it on both sides, and outputs toast. In the former example the toaster is the function, the bread is the domain, and the toast is the range.

In mathematics functions have four things:

 (i)   A name: like  $f$ 
 (ii)  A rule: like  $2 x$ 
 (iii) A Domain
 (iv)  A Range

Mathematically we write: $F: D \rightarrow R$ this is read the function F maps elements of the set D to elements of the set R. Let us look at some examples and explore the concepts of domain and range.

Domain

 Domain: The set of allowable inputs to a function.

Dr. Harrison Milzzz points out that in common speak this usually means acceptable x-values. This is fine for functions of x, which is what we often have in math, but sometimes we might have:

$\ s(t) = 16t^t + 8t$

In which case we would be talking about acceptable values for t. In general allowable inputs to a function.

Often the domain consists of numbers, i.e. for the function:
$\ f(x) = 2x$
The domain is: $\ D_f = \mathbb{R}$

for the function:
$\ g(x) = \frac{1}{x}$
The domain is: $\ D_g = \mathbb{R} \setminus \lbrace 0 \rbrace$ ; since an input of 0 would cause a divide by zero error, zero is not an allowed input.

Range

$g(x) = \frac{1}{x}$

 Range:  The set of outputs from a function.

Perhaps the easiest way to get the range is by looking at the graph of a function and observing what y-values are possible (a horizontal line test if you will).

If you run an imaginary horizontal line over the graph on the right you see it bumps into the function $\frac{1}{x}$ for all values of y except at the horizontal line y = 0. Mathematically we write:

$\ R_g = \mathbb{R} \setminus \lbrace 0 \rbrace$

This is read the Range of the function g is all real numbers except for zero.

Excercises: Functions, Domain and Range

 Given , , and 

 Evaluate:

 1.             2.             3.             4. 


 Find the domain and range of the following functions:

 5.              6.             7.  


 Graph:

 8.

Inequalities

Simply put an inequality is a relation with an inequality relational operator instead of an equals sign. For example:

$\ x + y = 1$ is an equation.
$\ x + y < 1$ is an inequality.

One Dimensional

One dimensional inequalities or 1-D are inequalities that contain exactly one variable.

Click here for audio:

Here is the audio of the lecture to remind you how to do the homework problems. If you need to listen to it while you review your notes.

$\ \circ$ ~ open dot to exclude from solution.
$\ \bullet$ ~ closed dot to include as part of solution.

Two Dimensional

Two dimensional inequalities or 2-D are inequalities that contain exactly two variables. Here is an example of a two dimensional inequality.

Click here for audio:

Here is the audio of the lecture to remind you how to do the homework problems. If you need to listen to it while you review your notes.

Systems

A system of equations is a set of equations, i.e.

x = 2, y = 7

$\begin{Bmatrix} x = 2 \\ y = 7 \end{Bmatrix}$

is a system containing two equations. The solution to a system are the values of that make both equations true. In this case: $(2,7)$ is the solution to this system.

Geometrically speaking the solution to a system is any place that the graphs of each of the individual equations intersect.

This system is already solved so to speak; it explicitly defines x and y to specific values. This is not always the case take:

$\begin{Bmatrix} x + y = 1 \\ x - y = 1 \end{Bmatrix}$

This system does not explicitly define a solution, but rather gives us boundaries on what can happen to x and y, namely they must be on the two lines specified by the equations in the system.

There are three possible cases when we speak of linear systems in two dimensions:

Substitution

Elimination

Exercises for Systems

Quadratics

y = x 2

Quadratics are polynomial functions of degree two. There are two primary forms of a quadratic:

Standard Form

Standard Form: $\ y = ax^2 + bx + c, a \ne 0$

Often with quadratics and with functions we are interested in finding the roots. Roots are where the function crosses the x-axis. So, they are also called zeros of a function. The terms x-intercept, roots, and zeros can be used interchangeably.

Vertex Form

Vertex Form: $\ y = a(x-h)^2+k$

For vertex form h is the x-coordinate of the vertex and k is the y-coordinate. The vertex of the parabola is $\ (h, k)$

It takes two points to define a line; it takes three points to define a quadtratic.

Exercises: Vertex/Standard

 Find the degree of the following polynomials, and determine if they are linear, quadratic, or neither.
 1.                       2.     


 Sketch the graphs of the following quadtratics.
 3.                 4.  


 5. Do standard form and vertex form ever coincide?  
 Write a paragraph discussing this questions, be sure to include some examples in support of your argument.

Factoring

Factoring is taking a polynomial that is expressed as sums and differences of terms and expressing it as a product. We are used to doing the reverse, i.e. take: $\ 2(x - 6)$ and we are ready and willing to take the $\ 2$ and distribute it across the quantity $\ (x-6)$ . The reverse taking the $\ 2$ out of each of the terms of $\ 2x - 12$ is factoring.

Greatest Common Factors

The first step in factoring a polynomial is checking for a Greatest Common Factor (GCF), does there exist a common factor in each term of the polynomial?

Example One:

$\begin{array}{cc} 16x^{2} - 4 & \mathrm{has\ a\ GCF\ of\ 4.} \\ & \\ 4(4x^{2} - 1) & \mathrm{Factor\ out\ the\ 4.} \\ \end{array}$

Example Two:

$\begin{array}{cc} 27x^{2} + 81x - 9x^{3} & \mathrm{has\ a\ GCF\ of\ 9x.} \\ & \\ 9x(3x + 9 - x^{2}) & \mathrm{Factor\ out\ 9x\ from\ each\ term.} \\ \end{array}$

Factoring Trinomials

Once taking care of the GCF of a trinomial sometimes we factor it further to a product of binomials, i.e.:

Example One:

$\ x^{2} - 12x + 35$

$\ (x - 5)(x - 7)$

That is we "unFOIL" the polynomial. Notice that "FOIL"ing (x - 5)(x - 7) returns the original trinomial. That is to say they are equivalent expressions; just the first trinomial
$x 2 - 12 x + 35$ is expressed as sums and differences of terms, and (x - 5)(x - 7) is expressed as a product.

In general to factor a trinomial (after factoring out the GCF) we must ask ourselves the following quesions:

 (i)  What two numbers multiply together to give the constant term?
 (ii) Of the above factors which (if any) add to give the coefficient of the middle term?

Consider: $x 2 - 12 x + 35$ again. The third term: 35 has factor pairs of (1, 35), (-1, -35), (5, 7) and (-5, -7); it is pretty easy to see that the only pair that adds to -12 (the coefficient of the middle term) is (-5, -7).

Example Two:

$\begin{matrix} 4x^{2} - 16x - 48 & GCF: 4 \\ 4(x^{2} - 4x - 12) & \mathrm{ now\ factor\ the\ trinomial\ again } \\ 4(x + 2)(x - 6) \end{matrix}$

Visit the Factor Worksheet for some practice with factoring.

Completing the Square

One of my proudest moments as a young math student was successfully proving the quadratic formula. It took me four days and there were many mistakes, but the satisfaction I felt and still remember; I cannot properly explain.

This picture is an attempt at the expression of how beautiful the journey was and is. It is about the journey and the experience more than any singular event.

Some quadratics do not factor over the integers and so a new technique is needed to solve such equations. One such technique is Completing the Square.

Take the equation:

$\ y = x^{2} + x - 1$

The factors of -1 are {-1, 1}. Add 1 - 1 is 0. In other words this quadratic does not factor over the integers. So we force it to be a perfect square as follows.

$\begin{matrix} 0 & = & x^{2} + x - 1 & \\ & & \\ 1 & = & x^{2} + x & \mathrm{add\ 1\ to\ both\ sides.}\\ & & \\ \frac{1}{4} + 1 & = & x^{2} + x + \frac{1}{4} & \mathrm{take\ the\ coefficient\ of\ the\ linear\ term,\ divide\ by\ two\ and\ square\ it}.\\ & & \\ \frac{5}{4} & = & (x + \frac{1}{2})^{2} & \mathrm{by\ using\ the\ } \frac{1}{4} \mathrm{we\ have\ made\ the\ right\ side\ a\ perfect\ square.}\\ & & \\ \pm \sqrt{\frac{5}{4}} & = & \sqrt{(x + \frac{1}{2})^{2}} & \mathrm{take\ the\ square\ root\ of\ both\ sides.}\\ & & \\ \pm \frac{\sqrt{5}}{2} & = & x + \frac{1}{2} & \mathrm{simplified,\ now\ solve\ for\ x\ using\ linear\ techniques.}\\ & & \\ \pm \frac{\sqrt{5}}{2} - \frac{1}{2} & = & x & \mathrm{subract\ } \frac{1}{2} \mathrm{\ from\ both\ sides.}\\ \end{matrix}$

In the previous example $\ a = 1$ if $\ a \ne 1$ , then you must force it by dividing the entire equation by $\ a$ and then proceed with the same process as demonstrated above. In practice if $\ a \ne 1$ then it is often the preferred to use the quadratic formula to solve the quadratic at hand.

Quadtratic Formula

The quadratic formula is: $\ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Degree of a Polynomial

The degree of a polynomial is the highest exponent on a variable. For example: $\ y = x^2$ has degree of 2. Often I write this as $\ deg(y) = 2$ Where $d e g$ is a function that returns the degree of a function.

Example:

$\ f(x) = 2^{32}x - x^4$

$\ deg(f) = 4$ , 32 is the highest exponent, but 4 is the highest exponent on a variable.

Constant Functions:

$\ q(x) = 22.5$ ; $\ deg(q) = 1$ ; for a constant function the degree is 1; the left side q has exponent 1.

Multivariable Functions: $w(x,\ y,\ z) = x + xy - xyz$ $\ deg(w) = 3$ when you have variables multiplied in terms of a function, you add the exponents of that term to evaluate the degree of each term, and the term with the highest number is the degree; in this case 3.

Exercises for Degree

 Evaluate the degree of each function:

 1.      2.      3.      
 

 4.  
 

 5.      6.

Polynomial Worksheet

Fractional Exponents (Roots)

$\ 20^{2} = 400$

We know the meaning of an exponent, but how about:

$\ 8^{\frac{1}{3}}$

Well this simply asks the question: What is the cube root of 8?

In other words $8^{\frac{1}{3}} = \sqrt[3]{8}$

So,

$\ 8^{\frac{1}{3}} = 2$

2 is the number you would cube in order to obtain 8.

In general:

$\ b^{\frac{p}{r}} = \sqrt[r]{b^{p}}$

b: is the base; p: is the power; r: is the root.

So, $\ 4^{\frac{3}{2}} = 8$ .

Exercises

 Evaluate:

 1.

Inverse Functions

Inverse functions are functions that "undo" each other.

Identity Elements

An identity element is an element that does not change other elements through a given operation.

Addition

The identity element for addition is $\ 0$ since it does not change the other numbers through addition.

$\begin{matrix} 0 + 1 & = & 1 \\ 0 + 2 & = & 2 \\ \vdots & & \\ 0 + a & = & a \end{matrix}$

$\forall a \in \mathbb{R}$

Two numbers are said to be additive inverses of each other if they add to the identity element for addition, i.e. the additive inverse of 7.4 is -7.4, since:

$7.4 + - 7.4 = 0$

Multiplication

The identity element for multiplication is $\ 1$ since one does not change other numbers under the operation of multiplication.

$\begin{matrix} 1\cdot 1 & =& 1 \\ 1\cdot 2 & = & 2 \\ \vdots & & \\ 1 \cdot a & = & a \end{matrix}$

$\forall a \in \mathbb{R}$

Two numbers are said to be additive inverses of each other if they multiply to the multiplicative identity, i.e. $\ 14$ and $\frac{1}{14}$ are multiplicative inverses of each other since:

$14 \cdot \frac{1}{14} = 1$

Identity Function

The identity function is $\ f(x) = x$ since this is the one function that does not change domain elements when input into the function, i.e.

$\begin{matrix} f(1) & = & 1 \\ f(2) & = & 2 \\ f(3) & = & 3 \\ f(-22) & = & -22 \\ f(-313.678) & = & -313.678 \end{matrix}$

In other words what you put in is what you get out.

Two functions are inverses of each other if under the operation of composition we arrive at the identity function; this is exactly analogous to the idea of additive inverses and multiplicative inverses.

Take the functions: $\ f(x) = x + 1$ and $\ g(x) = x - 1$ .

Are $\ f$ and $\ g$ inverses of each other? Well they are if they compose to be the identity function: $\ y = x$ , Let us check.

$\begin{matrix} f(g(x)) & = & f(x - 1) \\ & = & (x-1) + 1 \\ & = & x \\ & & \\ & & \\ g(f(x)) & = & g(x+1) \\ & = & (x+1) - 1 \\ & = & x \end{matrix}$

Since $\ f(g(x)) = x$ and $\ g(f(x)) = x$ we say that $\ f$ and $\ g$ are inverses of each other. We write:

$\ f(x) = x + 1$
$\ f^{-1}(x) = x - 1$

Cubics

y = x 3

A cubic is a polynomial of degree 3.

For example: $\ y = 44x^{3} - 17$ is a cubic; and $\ y = 44x^{3} - 5x^{7}$ is not.

The general form of a cubic is:
$\ y = ax^{3}+bx^{2}+cx+d$

Quartics

y = x 4 - x + 2 x 3

A quartic is a polynomial of degree 4.

General form of a quartic:
$\ y = ax^{4}+bx^{3}+cx^{2}+dx+e$

Quintics

y = 0.5(x - 2)(x - 1)(x)(x + 1)(x + 2)

A quintic is a polynomial of degree 5.

The General Form:
$\ y = a + bx + cx^2 + dx^3 + ex^4 + fx^5,\ f \ne 0$

Insolvability of the Quintic

Basically we can find a formula for the roots of a constant, linear, and quadratic functions using methods of radicals. Cubics have a formula, but I heard that it is long; so long that it is not useful. Quartics I have not tried nor heard anything about. But for quintics no such formula exists; see Abel and Ruffini theorem.

Polynomials

A polynomial has the form:

$\ P[x] = \lbrace \ \sum_{i=0}^{n}{a_ix^i}\ |\ a_i \in R,\ i \in W \ \rbrace$

For example:
$\ p_1(x) = 1$ ; is a constant polynomial $\ deg(p_1) = 1$
$\ p_2(x) = x + 1$ ; is a linear polynomial.
$\ p_3(x) = x^2$ ; is a quadratic polynomial.

and

$\ \mu (x) = 2x^{-1}$ ; is not a polynomial at all since $-1 \notin W$

FTA - The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states:

Every polynomial of degree n has n roots over the complex numbers $\ \mathbb{C}$ .

For example; $h (x) = x 3$ has three roots over the complex numbers. $y (x) = x 2 + 1$ has two roots over the complex numbers. Notice that the three roots of h are real namely x = 0; multiplicity three, and the roots of y are complex numbers $\ x = \pm \sqrt{-1}$ .

Rational Functions

A rational function is a ratio of polynomial functions. For example: $\ \frac{x}{x+1}$ is a rational function becuase x is a polynomial and x + 1 is a polynomial and in ratio we call it a rational function.

 Definition of a rational function:

  ; The set of polynomial functions of x.
  ;  The set of rational functions of x.

Perhaps the simplest rational functions are ratios of constants like:

$\ q(x) = \frac{2}{3}$ , but this is no different then a constant function, a horizontal line. Similarly all polynomials qualify as rational functions, i.e. $\ h(x) = x + 1$ can be thought of as $\ \frac{x+1}{1}$ , and therefore qualifies as a rational function, in other words: $\ P[x] \subset Q[x]$ just as $\ \mathbb{Z} \subset \mathbb{Q}$ .

Graph of $\ \frac{1}{x}$

The simplest rational function that is also not in P[x] is $\ \frac{1}{x}$ . Let us take a closer look at this function.

$\ r(x) = \frac{1}{x}$ ; the domain of this function is $\ D_{r} = \ \mathbb{R} \setminus \lbrace\ 0 \ \rbrace$

perhaps the first thing you notice is that the graph comes in two pieces. This is the first time we have seen anything like this; the break is called a discontinuity and is caused by the zero divide error. The fact that zero is not a member of $\ D_{r}$ .

Vertical Asymptotes

A vertical asymptote is when the behaviour of a function explodes to $\infty$ or $-\infty$ . Like in the function $\frac{1}{x}$ , x = 0 is a vertical asymptote.

$r(x)=\frac{3}{x+4}$

Take the function:
$r(x) = \frac{3}{x+4}$
has a vertical asymptote at x = -4.

In general a vertical asymptote of a rational function occurs for a number that makes the value of the denominator equal to zero. In this case the denominator of $r (x)$ is $x + 4$ ; take this expression and set it equal to zero and solve for $x$ in order to obtain the vertical asymptotes.

$x + 4 = 0$
$x = - 4$
i.e. $D_r = \mathbb{R} \setminus \lbrace-4\rbrace$

$y=\frac{15.8}{x^2+1}$

Some rational functions do not have vertical asymptotes, take the function:
$y = \frac{15.8}{x^2+1}$
notice:
$x 2 + 1 = 0$
yields only complex solutions: $x = \pm \sqrt{-1}$
thus no vertical asymptotes in the Cartesian coordinate plane.

Singularities

Singularities occur when a division by zero error can be removed by simplification of the function. These are sometimes called removable discontinuities because they can be removed by the simplification of the function.

$q(x) = \frac{x}{x(x-1)}$

For example:
$\ q(x) = \frac{x}{x(x-1)}$ has two numbers that make the denominator equal to zero, namely x = 0 and x = 1.
But we can cancel the common factors of x from the top and bottom of the function yielding a new simplified function:
$\ q_s(x) = \frac{1}{x-1}$ ; this function only has a divide by zero error at x = 1, i.e. the problem of x = 0 was eliminated.

So we say that:
$\ x = 0$ is a singularity (removable discontinuity).

and

$\ x = 1$ is a vertical asymptote.

Discontinuities

Horizontal Asymptotes

Exponential Functions

y = e x

An exponential function in its very simplest terms is a function in which the variable is in the exponent. In general:

 Definition of an Exponential Function:

Notice the x is used as an exponent.

Terminology:
$\ a$ is the initial condition of the exponential function.
$\ b$ is the base of the exponential function.

Exponential Growth

Consider the functions $\ y = x$ , $\ y = x^2$ , and $\ y = 2^{x}$ ; each function increases as you move to the right, but if you examine the graphs you can see that in the long run $\ y = 2^{x}$ overtakes the other functions and grows much more quickly.

In the short term between x = 0 and x = 4, there is a struggle, but ultimately $\ y = 2^x$ explodes and grows much more quickly than the other two functions.

When a people speak of exponential growth they mean to demonstrate this explosion of growth. A situation that grows very quickly.

Exponential Decay

Logarithmic Functions

y = l o g (x)

A logarithmic function is the inverse of an exponential function. The logarithm asks the question what exponent would you use on the base to obtain the desired result?

i.e. log(100) ; asks what exponent would you use on 10 (the base) to obtain a result of 100? The answer is 2.

Examples:

$log 2 8 = 3$ , i.e. 3 is the exponent to use on 2 in order to obtain 8.

$log0.1 = - 1$

Trigonometric Functions

The trigonometric functions are defined as the ratio of lengths of sides of a triangle. Take the triangle:

An angle in standard position.

This triangle has sides of length: x, y, and r. The length x is horizontal and is on the x-axis, the length y is a vertical distance, and r is a radius from the point (x, y) to the origin (0, 0). The angle $\ \theta$ is the angle with vertex at (0, 0), between the rays positive x-axis and r.

Using this picture we define the trigonometric functions as follows:

$\ sin(\theta) = \frac{y}{r}$ ; sine is defined as the ratio of the length y to the length r.

$\ cos(\theta) = \frac{x}{r}$ ; cosine is defined as the length x to the length r.

$\ tan(\theta) = \frac{y}{x}$ ; tangent is defined as the length y to the length x.

Hyperbolic Functions

Linear Algebra

Linear algebra is the study of linear systems, operations, and equivalence.

Scalars

A scalar is any number or element representing a singular thing or quantity. Speed is a scalar 55mph, but velocity is a vector.

Vectors

Vectors are lists of numbers. Scalars are singular quantities or numbers. For instance in physics speed is a scalar quantity and velocity is a vector. Speed may be something like 35 mph one number indicating a rate; whereas velocity indicates both a speed and a direction. That is there are to components to velocity a number for speed and for direction of travel.

Coordinates in the xy-plane can be considered vectors: $\ (x, y)$ is an example of a vector. We can also write $\ \begin{pmatrix} x \\ y \end{pmatrix}$ ; the former being a row vector and the latter a column vector. Vectors are usually named with lower case letters with an arrow overhead: $\vec{v} = \left( 3, 45^{o} \right)$ , for say a velocity vector. $\vec{v}$ is said to be a two dimensional vector since it contains two components. We write:

$dim(\vec{v}) = 2$

All of the vectors listed thus far have been two dimensional, but in general a vector can be $n$ dimensional.

$\vec{v_n} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{pmatrix}$

An $n$ dimensional vector where $a i$ are the components of the vector. That is to say $dim(\vec{v_n}) = n$ .

Addition of Vectors

Addition of vectors is performed by simply adding corresponding components, i.e.
if $\vec{q} = \begin{pmatrix}1 & 2 & -6 \end{pmatrix}$ and $\vec{r} = \begin{pmatrix}0 & -4.6 & 2.3 \end{pmatrix}$
then:

$\vec{q} + \vec{r} = \begin{pmatrix} 1 & -2.6 & -3.7 \end{pmatrix}$

Scalar Multiplication

Scalar multiplication is the act of taking any member of $\ \mathbb{R}$ , and distributing it across a vector:

Examples:

$\ 2(6, 8) = (12, 16)$

$\ -3\begin{pmatrix}3 \\ -1 \\ 7.1 \end{pmatrix} = \begin{pmatrix} -9 \\ 3 \\ -21.3 \end{pmatrix}$

$a \begin{pmatrix} 2 & -1 \\ 0 & 14a \end{pmatrix} = \begin{pmatrix} 2a & -a \\ 0 & 14a^2 \end{pmatrix}$

In other words just distribute the scalar number to each element in the vector or matrix.

Vector Multiplication

Two primary types of vector multiplication are the dot product and cross product.

Dot Product

Given two vectors $\vec{a} = \begin{pmatrix}1 & 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix}-4 & 3 \end{pmatrix}$ then the dot product is:

$\vec{a} \cdot \vec{b} = 1\cdot(-4) + 2\cdot3 = -4 + 6 = 2$

notice that the dot product of two vectors is a scalar.

Cross Product

Matrices

A matrix is a rectangular array of elements. A matrix can also be thought of as a list of vectors. We usually name matrices with capital letters; like:

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

The dimensions of $\ A$ are 2x2; it has 2 rows and 2 columns. In general a 2x2 matrix has this form:

$M_{2x2} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$
Not all matrices are square; here is a rectangular matrix
$M_{2x3} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}$ .
In general
$M_{m x n} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots & \ddots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$ is a matrix with m rows and n colomns.

Addition of Matrices

Scalar Multiplication

Multiplication of Matrices

Matrix Equations

Calculus

Calculus is usually described as the study of change; for instance acceleration can be considered as a continuous change in velocity; calculus helps us study the relationships of changing quantities like perhaps changing velocity.

Limits

A limit is the an idea of a function approaching a certain value. The notation is as follows:

  is read the limit of  as x approaches 3.

For example take the function: $\ g(x) = 2 - x$ , and the limit:

 

 Here we can simply set x equal to 9 since 9 is a valid domain element for the function g; and the limit is 2 - 9 or -7.

Sometimes we cannot directly substitute a the approaching value into the function, i.e. the element we are approaching is not a valid domain element of the function, e.g.

$\ \lim_{x \rightarrow 7} \frac{x - 7}{x - 7}$ direct substitution of x = 7 here results in the indeterminate form $\ \frac{0}{0}$ ; illegal cannot divide by zero. As you know 7 is not a valid domain element of this function, i.e. the function has no defined value at x = 7, but that does not mean that the limit does not exist. The function may be approaching some value, or it may not be, further investigation is required; here is that investigation: if you look at the function and ignore the limit, you can see that algebraic simplification is possible: $\ \frac{x-7}{x-7}$ simplifies to $\ 1$ so we have an equivalent expression: $\ \lim_{x \rightarrow 7}1$ , and as x approaches 7, 1 does not care, so: $\ \lim_{x \rightarrow 7}1 = 1$ therefore $\ \lim_{x \rightarrow 7}\frac{x-7}{x-7} = 1$

So the limit as x approaches 7 of $\ \frac{x-7}{x-7}$ is 1. Even though the function itself is not defined at x = 7, the limit exists and the function approaches a value of 1 from the left and the right as x approaches the value of 7. You can see this both graphically and numerically.

Graphically:

$\frac{x-7}{x-7}$

as we ride along the function we see that everywhere it has a value of 1, except for the undefined domain element of 7, but by inspection theres is nothing to indicate that the function "wants" to have any other value at this point as we approach x = 7 from both the left and the right.

Numerically:

We can evaluate functional values that are close to x = 7 on the left and right of 7.

$\ f(x) = \frac{x-7}{x-7}$
$f (6.9) = 1$	$f (7.1) = 1$
$f (6.99) = 1$	$f (7.01) = 1$
$f (6.999) = 1$	$f (7.001) = 1$

We can get as close as we want to 7 and always the functional value of $f$ is 1.

Exercises

Evaluate the following limits:
$\begin{matrix} 1. & \lim_{x\rightarrow 2}{\left(x - 7\right)} & & 2. \lim_{x \rightarrow -1}\left(\frac{1}{x}\right) \end{matrix}$

Left Hand Limits

A left hand limit is the act of examining x approaching some value only from the left; the notation is:

$\lim_{x \rightarrow 3^-}f(x)$

Right Hand Limits

A right hand limit is the act of examining x approaching some value only from the right; the notation is:

$\lim_{x \rightarrow 3^+}f(x)$

Existence of a Limit

A limit is said to exist iff both the left hand and right hand limits agree.

Derivatives

Secant Line of a Function

A derivative is an operation on a function that measures how much the function is changing at a given instant or value of x.

 Definition of a Derivative:

Which is an exact analogy to our definition for slope of a line: $\frac{\Delta y}{\Delta x}$ , but applies to general functions and gives the slope of the tangent line to the function at a given point. The points on on the curve $\ f(x)$ are $\ (x, f(x)), (\Delta x, f(x+\Delta x))$ the line through these two points is a secant line to the curve. If we take the limit as $\ \Delta x$ goes to zero, then the two points will coincide and the secant line becomes a tangent line to the curve.

Applying the Definition of the Derivative

Let us apply the definition of a derivative to some functions:

Example 1.
A constant function: $\ y(x) = 3.1$

$\ \begin{align} \frac{d}{dx} \left(3.1\right) & = & \lim_{\Delta x \rightarrow 0}{\left(\frac{3.1 - 3.1}{\Delta x}\right)} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}{\frac{0}{\Delta x}} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}{0} \\ & & \\ & = & 0 \end{align}$

 Exercise:

 Prove that

Example 2.
A Linear function: $\ f(x) = x + 1$

$\begin{align} \frac{d}{dx} (x+1) & = & \lim_{\Delta x \rightarrow 0}\frac{x+\Delta x + 1 - (x+1)}{\Delta x} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}\frac{x+\Delta x + 1-x - 1}{\Delta x} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}\frac{\Delta x}{\Delta x} \\ \\ & = & \lim_{\Delta x \rightarrow 0}1 \\ & & \\ & = & 1 \end{align}$

Example 3.
A quadratic function: $\ h(x) = x^{2} - 7$

$\begin{align} \frac{d}{dx}(x^{2} - 7) & = & \lim_{\Delta x \rightarrow 0}\frac{(x + \Delta x)^{2} - 7 - (x^{2} - 7)}{\Delta x} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}\frac{x^{2} + 2x\Delta x +\Delta x^{2}-7-x^{2}+7}{\Delta x} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}\frac{2x\Delta x + \Delta x^{2}}{\Delta x} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}\frac{\Delta x(2x + \Delta x)}{\Delta x} \\ & & \\ & = & \lim_{\Delta x \rightarrow 0}2x + \Delta x \\ & & \\ & = & 2x + 0 \\ & & \\ & = & 2x \end{align}$

Derivatives of Monomials

Recall a monomial is an expression with one term, i.e. $f (x) = 2 x$ is a monomial, and $g (x) = 2 x + 1$ is not.

 Examples:

 

 

 

 

 

 

 

 In general (for a monomial) subtract one from the exponent 
 and that is your new exponent and the old coefficient times the old exponent is your new coefficient.

Linearity of the Differential Operator

The differential operator $\ \frac{d}{dx}$ is linear. That is to say we can correctly distribute it over a sum or difference.

$\begin{matrix} \frac{d}{dx}(x+1) & = & \frac{d}{dx}x + \frac{d}{dx}1 \\ & & \\ & = & 1 + 0 \\ & & \\ & = & 1 \end{matrix}$

Verify for this example. Apply the definition of the derivative:

$\begin{matrix} \lim_{\Delta x \rightarrow 0}{x+1} & = & \frac{(x+\Delta x + 1)-(x+1)}{\Delta x} \\ & & \\ & = & \frac{\Delta x}{\Delta x} \\ & & \\ & = & 1 \end{matrix}$

That is to say that by distributing the differential operator $\frac{d}{dx}$ , and by applying the definition we obtain the same result: $\frac{d}{dx}(x+1) = 1$ , both ways round.

Exercise

 It is a good exercise to prove this in general, i.e. Prove the statement:

Integrals

Examples:
$\ \int 1 dx = x + c$
$\ \int x dx = \frac{1}{2}x^2 + c$

Differential Equations

A differential equation is an equation that contains derivatives. For example $\ \frac{dy}{dx} = x + y$ is a differential equation.

First Order Separable

A simple differential equation is: $\ \frac{dy}{dx} = 1$ , we can solve this differential equation by seperating and integrating.

$\frac{dy}{dx} = 1$

$dy = 1\cdot dx$ ; multiply both sides by $\ dy$

$\int dy = \int 1\cdot dx$ ; integrate both sides of the equation.

$\ y = x + c$ ; the solution to the differential equation is the resulting function: $\ y = x + c$

Analysis

Complex Analysis

Abstract Algebra

Just as Algebra is the study of interactions between numbers, variables, functions and what those relationships mean or show; Abstract Algebra is the study of systems and what properties does a system need to contain to be classified as an algebra. The most common property is that of a set, an operation on elements of said set ( a group ), and sometimes a second operation that distributes across the first operation ( a ring ). The analogue to what we now and love as regular or high school algebra is the/a set of numbers like $\ R$ and the operations of addition $\ +$ and multiplication $\ \cdot$ .

Sets

A set is a collection of objects.

For example:
$\ \mathbb{C} = \lbrace a + bi\ |\ a,\ b \in R,\ i = \sqrt{-1} \rbrace$

The objects do not have to be numbers of course, like:
$\ F[x] = \lbrace$ the set of all functions of x $\ \rbrace$

And the objects do not even have to be "mathematical", like:
$\ A_i = \lbrace cow, david, time, a, 8, conjuctions, dots, word, star \rbrace$

Basically a set is just a collection of things. Now presumably if someone went through the trouble of constructing a set, they did it for some reason, such as there is a relationship between the objects. Like they all have a common owner or they all share some property.

Nothing

Empty Set

$\emptyset = \lbrace \rbrace$

The empty set is a collection of no objects; it is not nothing, it is a something with no objects in it.

Full Set

$\ \chi = \lbrace$ e v e r y t h i n g $\ \rbrace$

The "Full Set" is a collection of all objects.

Full Set Exercises

 1.  What sets are subsets of the full set?      

 2.  What sets are not subsets of the full set?

 3.  Is three a member of the full set? 

 4.  True or false:   e v e r y t h i n g

Groups

A group is a set and an operation on the elements of the set. For example $\ R$ and the operation $\ +$ is a group; we denote this group like this: $\ \langle R,\ + \rangle$

Operations

Identity Elements

Rings

Fields

Prime Fields

Number Theory

Glossary of Symbols

This is a list of symbols used in the texts and there associated english meanings.

Set and Logic Symbols

$\implies$ ; implies

$\iff$ ; iff ; if and only if.

$\exist$ ; there exists

$\ \forall$ ; For all.

$\ \neg$ ; Logical not indicates the negative of a statement.

$\ \and$ ; Logical and.

$\ \or$ ; Logical or.

$\ \subset$ ; Proper subset.

$\ \supset$ ; Proper superset.

$\ \subseteq$ ; Subset.

$\ \supseteq$ ; Superset

$\ \in$ ; Stylized lowercase Greek epsilon, means in or is a member of, i.e. $\ you \in \lbrace Human Race \rbrace$

$\ \notin$ ; not in: i.e., $\ you \notin \lbrace AlienRace \rbrace$

$\ \mathbb{Z}$ ; the integers.

$\ \mathbb{Q} = \lbrace \ \frac{a}{b}\ \vert \ a,b \in Z,\ b \ne 0\ \rbrace$ ; the rational numbers, i.e. fractions.

$\ \mathbb{I} = \lbrace\ a\ \vert\ a \notin Q\ \and a \in R \rbrace$ ; the irrational numbers, i.e. $\ \sqrt{2}$

$\ \mathbb{T} = \lbrace\$ Irrational numbers with the additional property of not being a root of a polynomial with rational coefficients $\ \rbrace$ ; the trancendental numbers, i.e. $\ \pi,\ e$

$\ \mathbb{R}$ ; the real numbers.

$\ \mathbb{C} = \lbrace\ a + bi\ \vert\ a,b \in R\ \rbrace$ ; the complex numbers, i.e. $\ 2 - 3.7i$

$\ \infty$ infinity. Latex code: \infty

$\ i = \sqrt{-1}$ ; symbol for an imaginary number.

$\ \theta$ ; Greek letter theta, usually used to denote angles.

$\ f(x)$ ; read: f of x ; Functional notation, used to put numbers into a function.

$\ \Delta$ ; Capital Greek letter delta, often used to denote change in a quantity.

$\ m$ ; slope: $\ \frac{\Delta y}{\Delta x}$

$\ \frac{d}{dx}$ ; Derivative: rate of change of a function at a given point $\ (x,\ y)$ .

$\ D.N.E.$ ; Does Not Exist

$\ \sum$ ; add some stuff up.

$\ \int f(x) dx$ ; Integral: read: integral of f of x with respect to x; the inverse of the derivative.

$\ f^{-1}(x)$ ; f inverse of x.

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Math

From Js

Contents

Negative One

Arithmetic

Addition

Subtraction

Multiplication

Division

Exponentiation

Order of Operations

Fractions and Arithmetic

Adding fractions

Probability

Algebra

Number Sets

Exercises for Number Sets

Dots

Lines

Slope Intercept Form

Solving Linear Equations

Exercises Solving Linear Equations

Functions

Domain

Range

Excercises: Functions, Domain and Range

Inequalities

One Dimensional

Two Dimensional

Systems

Substitution

Elimination

Quadratics

Standard Form

Vertex Form

Exercises: Vertex/Standard

Factoring

Greatest Common Factors

Factoring Trinomials

Completing the Square

Quadtratic Formula

Degree of a Polynomial

Exercises for Degree

Fractional Exponents (Roots)

Exercises

Inverse Functions

Identity Elements

Addition

Multiplication

Identity Function

Cubics

Quartics

Quintics

Insolvability of the Quintic

Polynomials

FTA - The Fundamental Theorem of Algebra

Rational Functions

Vertical Asymptotes

Singularities

Discontinuities

Horizontal Asymptotes

Exponential Functions

Exponential Growth

Exponential Decay

Logarithmic Functions

Trigonometric Functions

Hyperbolic Functions

Linear Algebra

Scalars

Vectors

Addition of Vectors

Scalar Multiplication

Vector Multiplication

Dot Product

Cross Product

Matrices

Addition of Matrices

Scalar Multiplication

Multiplication of Matrices

Matrix Equations