# Arithmetic Operations Cheat Sheet

## Commutative property

$a+b=b+a$

Multiplication
$a \cdot b=b \cdot a$

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## Associative property

$a + \left ( b+c \right ) = \left ( a+b \right ) +c$

Multiplication
$a \cdot \left ( b \cdot c \right ) = \left ( a \cdot b \right ) \cdot c$

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## Distributive property

$a \cdot \left ( b+c \right ) = a \cdot b + a \cdot c$

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## Identity element

Addition (0 is the identity element)
$a+0=a$

Multiplication (1 is the identity element)
$a \cdot 1 = a$

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## Inverse element

Addition (-a is the inverse element)
$a + \left ( -a \right ) = 0$

Multiplication (1/a is the inverse element)
$a \cdot \frac{1}{a} = 1$

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## Other properties

$\begin{split} \frac{\left( \frac{a}{b} \right)}{c} &= \frac{a}{bc}\\ \\ \frac{a}{b} + \frac{c}{d} &= \frac{ad+bc}{bd}\\ \\ \frac{a-b}{c-d} &= \frac{b-a}{d-c}\\ \\ \frac{ab+ac}{a} &= b+c \text{, } a \neq 0\\ \\ a \left(\frac{b}{c} \right) &= \frac{ab}{c}\\ \\ \frac{a}{\left ( \frac{b}{c} \right )} &= \frac{ac}{b}\\ \\ \frac{a}{b} – \frac{c}{d} &= \frac{ad – bc}{bd}\\ \\ \frac{a+b}{c} &= \frac{a}{c} + \frac{b}{c}\\ \\ \frac{\left( \frac{a}{b} \right)}{\left( \frac{c}{d} \right)} &= \frac{ad}{bc} \end{split}$

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Reference:
https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx