Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Operations)
(→‎Addition and addition repeated)
Line 22: Line 22:
 
Multiplication is repeating addition. <math>ab=ba</math>
 
Multiplication is repeating addition. <math>ab=ba</math>
  
−
Exponentiation is repeated Multiplication. <math>a^b \neq b^a.
+
Exponentiation is repeated Multiplication. <math>a^b \neq b^a</math>.
  
 
==== Inverse ====
 
==== Inverse ====
Line 33: Line 33:
 
==== Negitive numbers ====
 
==== Negitive numbers ====
  
−
</math>a<math> and </math>b<math> are positive.
+
<math>a</math> and <math>b</math> are positive.
  
−
1. </math>(-a)(-b)=ab<math>
+
1. <math>(-a)(-b)=ab</math>
  
−
2. </math>(a)(-b)=-ab<math>
+
2. <math>(a)(-b)=-ab</math>
  
 
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 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 </math>(a)(-b)<math> is repeated subtraction. Therefore it is negative.
+
Proof for 2: Since ab is repeated addition then <math>(a)(-b)</math> is repeated subtraction. Therefore it is negative.
  
 
==== Fractions ====
 
==== Fractions ====
−
A fraction a number that can be expressed as two numbers divided. For example, five divided by four is </math>\frac{5}{4}<math>.  
+
A fraction a number that can be expressed as two numbers divided. For example, five divided by four is <math>\frac{5}{4}</math>.  
 
===== Adding frations =====
 
===== Adding frations =====
−
Here is how you add fractions. Is they have the same bottom half then </math>\frac{a}{b}+\frac{c}{b}=\frac{a+b}{c}<math>. However, if the two or three or n frations do not have the same bottom half you make them. For example, </math>\frac{a}{b}+\frac{c}{d}=\frac{bd}{ad+bc}<math>. Subtracting frations is the same exept everything has a minus symbol.
+
Here is how you add fractions. Is they have the same bottom half then <math>\frac{a}{b}+\frac{c}{b}=\frac{a+b}{c}</math>. However, if the two or three or n frations do not have the same bottom half you make them. For example, <math>\frac{a}{b}+\frac{c}{d}=\frac{bd}{ad+bc}</math>. Subtracting frations is the same exept everything has a minus symbol.
 
===== Multilpying fractions =====
 
===== Multilpying fractions =====
−
</math>\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}<math>.  
+
<math>\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}</math>.  
  
−
</math>\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}<math>.
+
<math>\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}</math>.
 
===== Simplifing fractions =====
 
===== Simplifing fractions =====
 
Remember to reduse frations to simplpest form after finishing a fration involved problem.
 
Remember to reduse frations to simplpest form after finishing a fration involved problem.
  
−
</math>\frac{10}{5}=\frac{1}{2}<math>
+
<math>\frac{10}{5}=\frac{1}{2}</math>
  
−
</math>\frac{15}{3}=\frac{5}{1}<math>
+
<math>\frac{15}{3}=\frac{5}{1}</math>
  
−
</math>\frac{28}{2}=\frac{14}{1}<math>
+
<math>\frac{28}{2}=\frac{14}{1}</math>
  
  
 
=== Exponent rules ===
 
=== Exponent rules ===
 
Listed below are some important properties of exponents:
 
Listed below are some important properties of exponents:
−
# </math> b^x\cdot b^y = b^{x+y}<math>
+
# <math> b^x\cdot b^y = b^{x+y}</math>
−
# </math> b^{-x}=\frac 1{b^x} <math>
+
# <math> b^{-x}=\frac 1{b^x} </math>
−
# </math> \frac{b^x}{b^y}=b^{x-y} <math>
+
# <math> \frac{b^x}{b^y}=b^{x-y} </math>
−
# </math> (b^x)^y = b^{xy} <math>
+
# <math> (b^x)^y = b^{xy} </math>
−
# </math> (ab)^x = a^x b^x <math>
+
# <math> (ab)^x = a^x b^x </math>
−
# </math> b^0 = 1 <math> (if </math>b \neq 0<math>.  </math>0^0<math> is undefined.)
+
# <math> b^0 = 1 </math> (if <math>b \neq 0</math>.  <math>0^0</math> is undefined.)
  
 
Here are explanations of the properties listed above:
 
Here are explanations of the properties listed above:
Line 74: Line 74:
 
# This is described in the previous section.
 
# This is described in the previous section.
 
# This results from using the previous two properties.
 
# This results from using the previous two properties.
−
# We are multiplying </math>b^x<math> by itself '''y''' times, which is the same as multiplying '''b''' by itself '''xy''' times.
+
# We are multiplying <math>b^x</math> 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.
 
# 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 </math>y=0<math>, we see that </math>b^x\cdot b^0<math> should (hopefully) be equal to </math>b^x<math>.  Thus, we ''define'' </math>b^0<math> to be equal to </math>1<math> in order to make this be true.
+
#Hoping that property #1 will be true when <math>y=0</math>, we see that <math>b^x\cdot b^0</math> should (hopefully) be equal to <math>b^x</math>.  Thus, we ''define'' <math>b^0</math> to be equal to <math>1</math> in order to make this be true.
  
 
=== Negitive numbers ===
 
=== Negitive numbers ===
−
</math>a<math> and </math>b<math> are positive.
+
<math>a</math> and <math>b</math> are positive.
  
−
1. </math>(-a)(-b)=ab<math>
+
1. <math>(-a)(-b)=ab</math>
  
−
2. </math>(a)(-b)=-ab<math>
+
2. <math>(a)(-b)=-ab</math>
  
 
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 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 </math>(a)(-b)$ is repeated subtraction. Therefore it is negative.
+
Proof for 2: Since ab is repeated addition then <math>(a)(-b)</math> is repeated subtraction. Therefore it is negative.
  
 
= Algebra =
 
= Algebra =

Revision as of 10:42, 19 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 and addition repeated

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

Multiplication is repeating addition. $ab=ba$

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

$a$ 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.

Fractions

A fraction a number that can be expressed as two numbers divided. For example, five divided by four is $\frac{5}{4}$.

Adding frations

Here is how you add fractions. Is they have the same bottom half then $\frac{a}{b}+\frac{c}{b}=\frac{a+b}{c}$. However, if the two or three or n frations do not have the same bottom half you make them. For example, $\frac{a}{b}+\frac{c}{d}=\frac{bd}{ad+bc}$. Subtracting frations is the same exept everything has a minus symbol.

Multilpying fractions

$\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$.

$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}$.

Simplifing fractions

Remember to reduse frations to simplpest form after finishing a fration involved problem.

$\frac{10}{5}=\frac{1}{2}$

$\frac{15}{3}=\frac{5}{1}$

$\frac{28}{2}=\frac{14}{1}$


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.

Algebra

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 $x=2$.

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.

Number Theory

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=What_is_the_definition_of_Pure_Mathematics%3F&oldid=106648"