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What is the definition of Pure Mathematics?

Revision as of 22:48, 19 June 2019 by Riben (talk | contribs) (→‎What is the definition of Pure Mathematics?)

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 its wings in Brazil trigger a tornado in Texas?

Some definitions hit almost all the areas of math, but some are too broad and logic, for example, 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. To make this a well-defined function everywhere, negative numbers and zero were defined.

Division is the inverse of multiplication. To make this a well-defined function everywhere, non-integral fractions were defined.

$n^{\text{th}}$ roots and logarithms are the inverses of exponentiation. To make these well-defined functions everywhere, irrational numbers were defined.

Negative 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 $\frac{5}{4}$.

Simplifing fractions

Find common factors in each half of the fration and then divide top and bottam of the fraction by that factor.

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

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

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

Adding fractions

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 fractions do not have the same bottom half you make them. $\frac{a}{b}+\frac{c}{d}=\frac{bd}{ad+bc}$. Subtracting fractions is the same exept everything has a minus symbol. Remember to fully simplify.

Multilpying fractions

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

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

Remember to fully simplify.

Irrational numbers

Most square roots are irrational. An irrational number is a number that can not be expresed a a fraction.

Proof that any nonperfect square positive integer is irrational

Let us assume that $\sqrt{n}$ is rational where $n$ is a nonperfect square positive integer. Then it can be written as $\frac{p}{q}=\sqrt{n} \Rightarrow \frac{p^2}{q^2}=n \Rightarrow (q^2)n=p^2$. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore $\sqrt{n}$ is irrational.

pi and e

Some irrational numbers are limits. That means they are the sum of smaller and smaller fraction going infinitely long. Pi or $\pi$ (go to the geometry part of this article) is the limit $\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}...=\frac{\pi}{4}$. e is the limit $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}...$ More at the counting part of the article.

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.

Algebra

Definition

The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.

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...

$ax^2+bx+c=0$

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

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

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


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.

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

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

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

When simplified the Quadratic Formula is ${x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}$

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

Definition

The branch of mathematics that deals with the properties and relationships of numbers, especially the positive integers.

Primes

A prime number (or simply prime) is a positive integer $p>1$ whose only positive divisors are 1 and itself. Note that $1$ is usually defined as being neither prime nor composite because it is its only factor among the natural number numbers.

There are an infinite number of prime numbers. A standard proof attributed to Euclid notes that if there are a finite set of prime numbers $p_1, p_2, \ldots, p_n$, then the number $N = p_1p_2\cdots p_n + 1$ is not divisible by any of them, but $N$ must have a prime factor, which leads to a direct contradiction.

Techniques to Check for Prime Numbers

Divisibility

A prime number is only divisible by one or itself, so a number $n$ is prime if and only if $n$ is not divisible by any integer greater than $1$ and less than $n$. One only needs to check integers up to $\sqrt{n}$ because dividing larger numbers would result in a quotient smaller than $\sqrt{n}$.

Modular Arithmetic

Modular arithmetic can help determine if a number is not prime.

  • If a number not equal to $2,3$ is congruent to $0,2,3,4,6 \pmod{6}$, then the number is not prime.
  • If a number not equal to $2,5$ ends with an even digit or $5$, then the number is not prime.


Importance of Primes

According to the Fundamental Theorem of Arithmetic, there is exactly one unique way to factor a positive integer into a product of primes. This unique prime factorization plays an important role in solving many kinds of number theory problems.


Mersenne Primes

A Mersenne prime is a prime of the form $2^n-1$. For such a number to be prime, n must itself be prime. Compared to other numbers of comparable sizes, Mersenne numbers are easy to check for primality because of the https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test Lucas-Lehmer test, severely reducing the amount of computation needed.

Twin Primes

Two primes that differ by exactly 2 are known as twin primes. The following are the first few pairs of twin primes:
3, 5
5, 7
11, 13
17, 19
29, 31
41, 43

It is not known whether or not there are infinitely many pairs of twin primes. This is known as the Twin Prime Conjecture, which is a specific instance of the Hardy-Littlewood conjecture.

Gaussian Primes

A Gaussian prime is a prime that extends the idea of the traditional prime to the Gaussian integer. One can define this term for any ring, especially number rings.

Advanced Definition

When the need arises to include negative divisors, a prime is defined as an integer p whose only divisors are 1, -1, p, and -p. More generally, if R is an integral domain, then a nonzero element p of R is a prime if whenever we write $p=ab$ with $a,b\in R$, then exactly one of a and b is a unit.

LCM and GCD

LCM

Definition

The LCM or Least Common Multiple of two or three or n numbers is the smallest integer in which both numbers are factors of it. For example, the LCM of 10 and 5 is 10.

Problems

Problem 1
Problem

What is the LCM of 8 and 5?

Solution

The way to solve LCM's with no common factors is to multiply them. The answer is just 40.

Problem 2
Problem

What is the LCM of 15 and 10.

Solution

The Way to solve this is see that the answer would be the first multiple of 5 that is divisble by 15 and 10. That is just 30.

Problem 3
Problem

Find the LCM of 2, 3 and 5.

Solution

It is the same with three numbers. Multiply them to get 30. Same as last time.

GCD

Bases

Note

This article is not finished. Colin Friesen or Colball is the only person allowed to edit. The plan for me is to make this page longer than Gmaas's. (Sorry, Gmaas).

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