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Difference between revisions of "What is the definition of Pure Mathematics?"

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== Systems of equations ==
 
== Systems of equations ==
 +
A '''system of equations''' is a set of [[equation]]s which share the same [[variable]]s.  An example of a system of equations is
 +
 +
{| class="wikitable" style="margin: 1em auto 1em auto"
 +
| <math>2a - 3b</math> || <math>= 4</math>
 +
|-
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| <math>3a - 2b</math> || <math>= 3</math>
 +
|}
 +
 +
=== 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 ====
 +
[[Gauss]]ian 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 <math>(x,y)</math> for which
 +
 +
{| class="wikitable" style="margin: 1em auto 1em auto"
 +
| <math>x - 12y</math> || <math>= 2</math>
 +
|-
 +
| <math>3x + 6y</math> || <math>= 6</math>
 +
|}
 +
 +
===== Solution =====
 +
We can eliminate <math>y</math> by adding twice the second equation to the first:
 +
 +
{| class="wikitable" style="margin: 1em auto 1em auto"
 +
| || <math>x - 12y= 2</math>
 +
|-
 +
| <math>+2(</math> || <math>3x + 6y = 6)</math>
 +
|-
 +
| || <math>\overline{7x + 0=14}</math>
 +
|}
 +
 +
Thus <math>x=2</math>.  We can then plug in for <math>x</math> in either of the equations:  <center><math> (2)-12y = 2 \Rightarrow y = 0</math>.</center>
 +
 +
Thus, the solution to the system is <math>(2,0)</math>.
 +
 +
==== 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 <math>(x,y)</math> for which
 +
 +
{| class="wikitable" style="margin: 1em auto 1em auto"
 +
| <math>x - 12y</math> || <math>= 2</math>
 +
|-
 +
| <math>3x + 6y</math> || <math>= 6</math>
 +
|}
 +
 +
===== Solution =====
 +
The first equation can be solved for <math>x</math>:  <center><math> x = 12y + 2. </math></center>
 +
 +
Plugging this into the second equation yields <center><math>3(12y + 2) + 6y = 6 \Leftrightarrow 42 y = 0. </math></center>
 +
 +
Thus <math>y=0</math>.  Plugging this into either of the equations and solving for <math>x</math> yields <math>x=2</math>.

Revision as of 15:40, 18 June 2019

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:

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

Here are explanations of the properties listed above:

  1. On both sides, we are multiplying b together x+y times. Thus, they are equivalent.
  2. This is described in the previous section.
  3. This results from using the previous two properties.
  4. We are multiplying $b^x$ by itself y times, which is the same as multiplying b by itself xy times.
  5. After multiplying ab by itself x times, we can collect a and b terms, thus establishing the property.
  6. 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.

One-variable linear equations

Definition

'A One-variable linear equation is an equation that comes in the form $ax+b=c$. $a$, $b$, and $c$ are constants and $x$ is the varible'.

The answer is always...

$ax+b=c$

$ax=c-b$

$x=\frac{c-b}{a}$

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 $ax^2 + bx + c = 0$ with $a$, $b$, and $c$ being constants, or numerical coefficients, and $x$ is an unknown variable.

The answer is always...

$ax^2+bx+c=0$

$x^2+\frac{b}{a}x+\frac{c}{a}$

$x^2+\frac{b}{a}=0-\frac{c}{a}$

$x^2+\frac{b}{a}+(\frac{b}{2a})^2=0-\frac{c}{a}+(\frac{b}{2a})^2$


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.

$(x+\frac{b}{2a})^2=0-\frac{c}{a}+\frac{b^2}{2a^2}=\frac{b^2-4ac}{4a^2}$

$x+\frac{b}{2a}=\frac{\sqrt{\pm b^2-4ac}}{\pm 2a}$

$x=\frac{\sqrt{\pm b^2-4ac}}{\pm 2a}+\frac{b}{2a}$

I am assuming you simplified it on your own.

i

$i=\sqrt{-1}$.

$xi=\sqrt{-x^2}$.

Numbers like this are called imaginary numbers. Impossible, you say. But no. Solve $x^2+1=0$. You get $i$. $oi$ is $o$. So zero is both real and imaginary. (real means not imaginary)

Powers of i

$i^0=1$ $i^1=i$ $i^2=-1$ $i^3=-i$ $i^4=1$ $i^5=i$ $i^6=-1$ $i^7=-i$

The pattern repeats.

Complex numbers

A complex number is $ai+b$, where a and b are real. All numbers are complex becuase a and/or/never b can be zero.

$ai+b+ci+d=$ Complex

$(ai+b)(ci+d)=$ Complex

$\frac{ai+b}{ci+d}=$ Complex

$(ai+b)-(ci+d)=$ 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

$2a - 3b$ $= 4$
$3a - 2b$ $= 3$

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 $(x,y)$ for which

$x - 12y$ $= 2$
$3x + 6y$ $= 6$
Solution

We can eliminate $y$ by adding twice the second equation to the first:

$x - 12y= 2$
$+2($ $3x + 6y = 6)$
$\overline{7x + 0=14}$

Thus $x=2$. We can then plug in for $x$ in either of the equations:

$(2)-12y = 2 \Rightarrow y = 0$.

Thus, the solution to the system is $(2,0)$.

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 $(x,y)$ for which

$x - 12y$ $= 2$
$3x + 6y$ $= 6$
Solution

The first equation can be solved for $x$:

$x = 12y + 2.$

Plugging this into the second equation yields

$3(12y + 2) + 6y = 6 \Leftrightarrow 42 y = 0.$

Thus $y=0$. Plugging this into either of the equations and solving for $x$ yields $x=2$.

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