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.

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

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

\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.

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

Algebra operations

Operation	Arithmetic Example	Algebra Example	Comments ≡ – means "equivalent to" ≢ – means "not equivalent to"
Addition	$(5\times 5)+5+5+3$ equivalent to: $5^{2}+(2\times 5)+3$	$(b\times b)+b+b+a$ equivalent to: $b^{2}+2b+a$	${\begin{aligned}2\times b&\equiv 2b\\b+b+b&\equiv 3b\\b\times b&\equiv b^{2}\end{aligned}}$
Subtraction	$(7\times 7)-7-5$ equivalent to: $7^{2}-7-5$	$(b\times b)-b-a$ equivalent to: $b^{2}-b-a$	${\begin{aligned}b^{2}-b&\not \equiv b\\3b-b&\equiv 2b\\b^{2}-b&\equiv b(b-1)\end{aligned}}$
Multiplication	$3\times 5$ or $3\ .\ 5$ or $3\cdot 5$ or $(3)(5)$	$a\times b$ or $a.b$ or $a\cdot b$ or $ab$	$a\times a\times a$ is the same as $a^{3}$
Division	$12\div 4$ or $12/4$ or ${\frac {12}{4}}$	$b\div a$ or $b/a$ or ${\frac {b}{a}}$	${\frac {(a+b)}{3}}\equiv {\tfrac {1}{3}}\times (a+b)$
Exponentiation	$3^{\frac {1}{2}}$ $2^{3}$	$a^{\frac {1}{2}}$ $a^{3}$	$a^{\frac {1}{2}}$ is the same as ${\sqrt {a}}$ $a^{3}$ is the same as $a\times a\times a$

Number Systems

Number Types

Algebra operations

Further reading