What is the definition of Pure Mathematics?

Oh, easy you say it is just the study of numbers.

That may be true for some areas of math. However, what about geometry, trigonometry, and calculus? And what is the definition of numbers? Now you go to the dictionary and say The relationship between measurements and quantities using numbers and symbols. This is, however, not fully true because this definition also uses applied mathematics. We want pure mathematics.

Also, most of these definitions miss one area of math. Chaos Theory. What is Chaos Theory? Chaos Theory is a recently discovered area of math where nothing can be predicted but nothing is random. We are only at the beginning of learning it. For example can a butterfly that flaps his wings is brazil trigger a tornado in Texas?

Some definitions hit almost all the areas of math, but some are too broad and logic often fits into the definition.

We can, however, define some areas of math but not the whole thing. For example, the definition of geometry is Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. Or the definition of probability is the extent to which an event is likely to occur.

Arithmetic

Definition

The branch of mathematics dealing with the properties and manipulation of Constants.

Operations

Arithmetic starts with one thing which without it no arithmetic can survive: Counting Positive Integers. 1,2,3,4,5...

Addition is combining these integers. Remember that $a+b=b+a$ .

Subtracting is taking integer difference and getting another integer. Here is where negative numbers and zero come in. Remember that $a-b \neq b-a$ .

Multiplication is repeating addition. Remember that $ab=ba$ .

Division is the inverse of multiplication. Remember that $\frac{a}{b} \neq \frac{b}{a}$ .

Exponentiation is repeated Multiplication.

Exponent rules

Listed below are some important properties of exponents:

$b^x\cdot b^y = b^{x+y}$
$b^{-x}=\frac 1{b^x}$
$\frac{b^x}{b^y}=b^{x-y}$
$(b^x)^y = b^{xy}$
$(ab)^x = a^x b^x$
$b^0 = 1$ (if $b \neq 0$ . $0^0$ is undefined.)

Here are explanations of the properties listed above:

On both sides, we are multiplying b together x+y times. Thus, they are equivalent.
This is described in the previous section.
This results from using the previous two properties.
We are multiplying $b^x$ by itself y times, which is the same as multiplying b by itself xy times.
After multiplying ab by itself x times, we can collect a and b terms, thus establishing the property.
Hoping that property #1 will be true when $y=0$ , we see that $b^x\cdot b^0$ should (hopefully) be equal to $b^x$ . Thus, we define $b^0$ to be equal to $1$ in order to make this be true.

Negitive numbers

$a$ and $b$ are positive.

1. $(-a)(-b)=ab$

2. $(a)(-b)=-ab$

Proof for 1: This is, in fact, the reason why the negative numbers were introduced: so that each positive number would have an additive inverse. ... The fact that the product of two negatives is positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.

Proof for 2: Since ab is repeated addition then $(a)(-b)$ is repeated subtraction. Therefore it is negative.