What is the definition of Pure Mathematics?
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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 and addition repeated
Addition is combining these integers.
Multiplication is repeating addition.
Exponentiation is repeated Multiplication. $a^b \neq b^a.
==== Inverse ==== Subtraction is the inverse of addition. Here, negitive numbers and zero come in.
Division is the inverse of multiplication. Here, non-integer fraction comes in.
n roots are the inverse of exponentiation. Here, irrational numbers come in.
==== Negitive numbers ====$ (Error compiling LaTeX. Unknown error_msg)ab$are positive.
1.$ (Error compiling LaTeX. Unknown error_msg)(-a)(-b)=ab(a)(-b)=-ab$Proof for 1: This is, in fact, the reason why the negative numbers were introduced: so that each positive number would have an additive inverse. ... The fact that the product of two negatives is positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.
Proof for 2: Since ab is repeated addition then$ (Error compiling LaTeX. Unknown error_msg)(a)(-b)$is repeated subtraction. Therefore it is negative.
==== Fractions ==== A fraction a number that can be expressed as two numbers divided. For example, five divided by four is$ (Error compiling LaTeX. Unknown error_msg)\frac{5}{4}\frac{a}{b}+\frac{c}{b}=\frac{a+b}{c}\frac{a}{b}+\frac{c}{d}=\frac{bd}{ad+bc}\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}\frac{10}{5}=\frac{1}{2}$$ (Error compiling LaTeX. Unknown error_msg)\frac{15}{3}=\frac{5}{1}$$ (Error compiling LaTeX. Unknown error_msg)\frac{28}{2}=\frac{14}{1}$=== Exponent rules === Listed below are some important properties of exponents:
- $ (Error compiling LaTeX. Unknown error_msg) b^x\cdot b^y = b^{x+y}$#$ (Error compiling LaTeX. Unknown error_msg) b^{-x}=\frac 1{b^x} $#$ (Error compiling LaTeX. Unknown error_msg) \frac{b^x}{b^y}=b^{x-y} $#$ (Error compiling LaTeX. Unknown error_msg) (b^x)^y = b^{xy} $#$ (Error compiling LaTeX. Unknown error_msg) (ab)^x = a^x b^x $#$ (Error compiling LaTeX. Unknown error_msg) b^0 = 1 b \neq 00^0$is undefined.)
Here are explanations of the properties listed above:
- On both sides, we are multiplying '''b''' together '''x+y''' times. Thus, they are equivalent.
- This is described in the previous section.
- This results from using the previous two properties.
- We are multiplying$ (Error compiling LaTeX. Unknown error_msg)b^x$by itself '''y''' times, which is the same as multiplying '''b''' by itself '''xy''' times.
- After multiplying '''ab''' by itself '''x''' times, we can collect '''a''' and '''b''' terms, thus establishing the property.
- Hoping that property #1 will be true when$ (Error compiling LaTeX. Unknown error_msg)y=0b^x\cdot b^0b^xb^01$in order to make this be true.
=== Negitive numbers ===$ (Error compiling LaTeX. Unknown error_msg)ab$are positive.
1.$ (Error compiling LaTeX. Unknown error_msg)(-a)(-b)=ab(a)(-b)=-ab$Proof for 1: This is, in fact, the reason why the negative numbers were introduced: so that each positive number would have an additive inverse. ... The fact that the product of two negatives is positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.
Proof for 2: Since ab is repeated addition then$ (Error compiling LaTeX. Unknown error_msg)(a)(-b)$ is repeated subtraction. Therefore it is negative.
Algebra
One-variable linear equations
Definition
'A One-variable linear equation is an equation that comes in the form . , , and are constants and is the varible'.
The answer is always...
When there are fractions in the equation, you multiply both sides by the LMC of the fractions and then you solve. More at the number theory part of this article.
Quadratics
Defination
A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form is with , , and being constants, or numerical coefficients, and is an unknown variable.
The answer is always...
Since the left side of the equation right above is a perfect square, you can factor the left side by using the coefficient of the first term (x) and the base of the last term(b/2a). Add these two and raise everything to the second.
I am assuming you simplified it on your own.
i
.
.
Numbers like this are called imaginary numbers. Impossible, you say. But no. Solve . You get . is . So zero is both real and imaginary. (real means not imaginary)
Powers of i
The pattern repeats.
Complex numbers
A complex number is , where a and b are real. All numbers are complex becuase a and/or/never b can be zero.
Complex
Complex
Complex
Complex
Systems of equations
A system of equations is a set of equations which share the same variables. An example of a system of equations is
Solving Linear Systems
A system of linear equations is where all of the variables are to the power 1. There are three elementary ways to solve a system of linear equations.
Gaussian Elimination
Gaussian elimination involves eliminating variables from the system by adding constant multiples of two or more of the equations together. Let's look at an example:
Problem
Find the ordered pair for which
Solution
We can eliminate by adding twice the second equation to the first:
Thus . We can then plug in for in either of the equations:
Thus, the solution to the system is .
Substitution
The second method, substitution, requires solving for a variable and then plugging that variable into another equation therefore reducing the number of variables. We'll show how to solve the same problem from the elimination section using substitution.
Problem
Find the ordered pair for which
Solution
The first equation can be solved for :
Plugging this into the second equation yields
Thus . Plugging this into either of the equations and solving for yields .
Algebra
There are many more types of algebra: inequalities, polynomials, graphing equations, arithmetic, and geometric sequence.
Algebra is a broad and diverse area of math in which this is just a short introduction.