- Commutative property
- Associative property
- Distributive property
- Identity element
- Inverse element
- Other properties
Commutative property
Addition
\[a+b=b+a\]
Multiplication
\[a \cdot b=b \cdot a\]
Associative property
Addition
\[a + \left ( b+c \right ) = \left ( a+b \right ) +c\]
Multiplication
\[a \cdot \left ( b \cdot c \right ) = \left ( a \cdot b \right ) \cdot c\]
Distributive property
\[a \cdot \left ( b+c \right ) = a \cdot b + a \cdot c\]Identity element
Addition (0 is the identity element)
\[a+0=a\]
Multiplication (1 is the identity element)
\[a \cdot 1 = a\]
Inverse element
Addition (-a is the inverse element)
\[a + \left ( -a \right ) = 0\]
Multiplication (1/a is the inverse element)
\[a \cdot \frac{1}{a} = 1\]
Other properties
\[ \begin{split}\frac{\left( \frac{a}{b} \right)}{c} &= \frac{a}{bc}\\
\\
\frac{a}{b} + \frac{c}{d} &= \frac{ad+bc}{bd}\\
\\
\frac{a-b}{c-d} &= \frac{b-a}{d-c}\\
\\
\frac{ab+ac}{a} &= b+c \text{, } a \neq 0\\
\\
a \left(\frac{b}{c} \right) &= \frac{ab}{c}\\
\\
\frac{a}{\left ( \frac{b}{c} \right )} &= \frac{ac}{b}\\
\\
\frac{a}{b} – \frac{c}{d} &= \frac{ad – bc}{bd}\\
\\
\frac{a+b}{c} &= \frac{a}{c} + \frac{b}{c}\\
\\
\frac{\left( \frac{a}{b} \right)}{\left( \frac{c}{d} \right)} &= \frac{ad}{bc}
\end{split} \]
Reference:
https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx