# Fundamental Mathematics/Algebra

## Number Systems

 Algebra number system Egytian's number system use digits from 0-9 N10 = {0,1,2,3,4,5,6,7,8,9} Roman's number system use letters I , V , X , L , M , C

## Number Types

From divisibility

A number which can be divided by 2 without remainder is called an Even number.

${\displaystyle \mathbb {2N} =\{0,2,4,6,8...\}}$

A number which cannot be divided by 2 without remainder is called an Odd number.

${\displaystyle \mathbb {2N+1} =\{1,3,5,7,9...\}}$

A number, greater than 1, which can be divided by 1 and itself without remainder (aka a number with exactly 2 natural factors) is called a Prime number.

${\displaystyle \mathbb {P} =\{2,3,5,7,11...\}}$

## Algebra operations

Operation Arithmetic
Example
Algebra
Example
≡ – means "equivalent to"
≢ – means "not equivalent to"
Addition ${\displaystyle (5\times 5)+5+5+3}$

equivalent to:

${\displaystyle 5^{2}+(2\times 5)+3}$

${\displaystyle (b\times b)+b+b+a}$

equivalent to:

${\displaystyle b^{2}+2b+a}$

{\displaystyle {\begin{aligned}2\times b&\equiv 2b\\b+b+b&\equiv 3b\\b\times b&\equiv b^{2}\end{aligned}}}
Subtraction ${\displaystyle (7\times 7)-7-5}$

equivalent to:

${\displaystyle 7^{2}-7-5}$

${\displaystyle (b\times b)-b-a}$

equivalent to:

${\displaystyle b^{2}-b-a}$

{\displaystyle {\begin{aligned}b^{2}-b&\not \equiv b\\3b-b&\equiv 2b\\b^{2}-b&\equiv b(b-1)\end{aligned}}}
Multiplication ${\displaystyle 3\times 5}$ or

${\displaystyle 3\ .\ 5}$   or   ${\displaystyle 3\cdot 5}$

or   ${\displaystyle (3)(5)}$

${\displaystyle a\times b}$ or

${\displaystyle a.b}$   or   ${\displaystyle a\cdot b}$

or   ${\displaystyle ab}$

${\displaystyle a\times a\times a}$ is the same as ${\displaystyle a^{3}}$
Division   ${\displaystyle 12\div 4}$ or

${\displaystyle 12/4}$ or

${\displaystyle {\frac {12}{4}}}$

${\displaystyle b\div a}$ or

${\displaystyle b/a}$ or

${\displaystyle {\frac {b}{a}}}$

${\displaystyle {\frac {(a+b)}{3}}\equiv {\tfrac {1}{3}}\times (a+b)}$
Exponentiation   ${\displaystyle 3^{\frac {1}{2}}}$
${\displaystyle 2^{3}}$
${\displaystyle a^{\frac {1}{2}}}$
${\displaystyle a^{3}}$
${\displaystyle a^{\frac {1}{2}}}$ is the same as ${\displaystyle {\sqrt {a}}}$

${\displaystyle a^{3}}$ is the same as ${\displaystyle a\times a\times a}$