Arithmetic formulas and rules reference

Addition and Subtraction

$\begin{aligned} a + (b + c) & = (a + b) + c & a + b = b + a \\ a + 0 & = a & a + (- a) = 0 \end{aligned}$

$a - b = a + (- b) = - b + a$

$- (- a) = a$

Multiplication and Division

$\begin{aligned} 0 \times a = 0 & 1 \times a = a \end{aligned}$

$\begin{array}{r} a \times (b \times c) = (a \times b) \times c \end{array}$

$\begin{array}{r} (-) \times (-) = (+) \\ (+) \times (+) = (+) \\ (-) \times (+) = (-) \\ (+) \times (-) = (-) \end{array}$

$a \div b = \frac{a}{b} = a \times \frac{1}{b}$

Fractions

$\frac{a}{0} = Undefined/Does Not Exist$

$\frac{0}{0} = Undefined/Does Not Exist$

$\begin{aligned} \frac{0}{a} = 0, a \neq 0 & \frac{a}{a} = 1 \end{aligned}$

$\begin{aligned} \frac{a}{b} \times \frac{c}{b} = \frac{a \times c}{b \times b} & \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \end{aligned}$

$\begin{aligned} \frac{a}{b} \pm \frac{c}{b} = \frac{a \pm c}{b} & \frac{a}{b} \pm \frac{c}{d} = \frac{(a \times d) \pm (c \times b)}{b \times d} \end{aligned}$

$\begin{aligned} \frac{1}{\frac{a}{b}} = \frac{b}{a} & \frac{a}{\frac{b}{c}} = \frac{a \times c}{b} & \frac{\frac{a}{b}}{c} = \frac{a}{b \times c} \end{aligned}$

$\begin{array}{r} \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \end{array}$

$\begin{aligned} \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} & \frac{- a}{- b} = \frac{a}{b} \end{aligned}$

Exponents

$\begin{aligned} 1^{a} = 1 & a^{1} = a \end{aligned}$

$\begin{aligned} 0^{a} = 0, a \neq 0 & a^{0} = 1, a \neq 0 & 0^{0} = Undefined \end{aligned}$

$\begin{aligned} (a b)^{n} = a^{n} b^{n} & {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}} \end{aligned}$

$\begin{aligned} a^{n} = \frac{1}{a^{- n}} & \frac{1}{a^{n}} = a^{- n} \end{aligned}$

$\begin{aligned} a^{n} a^{m} = a^{n + m} & \frac{a^{n}}{a^{m}} = a^{n - m} \end{aligned}$

$\begin{array}{r} (a^{n})^{m} = a^{n m} \end{array}$

Radicals

$\begin{aligned} \sqrt[n]{0} = 0 & \sqrt[n]{1} = 1 \end{aligned}$

$\begin{aligned} \sqrt{a} = \sqrt[2]{a} = a^{\frac{1}{2}} & \sqrt[n]{a} = a^{\frac{1}{n}} \end{aligned}$

$\begin{array}{r} \sqrt[n]{a^{m}} = {\sqrt[n]{a}}^{m} = a^{\frac{m}{n}} \end{array}$

$\begin{aligned} \sqrt{a} \sqrt{a} = a & \sqrt[n]{a} \sqrt[n]{a} = a, a \geq 0 \end{aligned}$

$\begin{aligned} \sqrt[n]{a b} = \sqrt[n]{a} \sqrt[n]{b} a, b \geq 0 & \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{aligned}$

Logarithms

$\begin{aligned} \log_{a} (1) = 0 & \log_{a} (a) = 1 \end{aligned}$

$\begin{aligned} \log_{1} (a) = Undefined & \log_{a} (b), a or b \leq 0 = Undefined \end{aligned}$

$\begin{aligned} \log_{a} (c^{b}) = b \cdot \log_{a} (c) & \log_{a^{b}} (c) = \frac{1}{b} \log_{a} (c) \end{aligned}$

$\begin{aligned} \log_{a} (\frac{1}{b}) = - \log_{a} (b) & \log_{\frac{1}{a}} (b) = - \log_{a} (b) \end{aligned}$

$\begin{array}{r} \log_{a} (b) = \frac{\log_{c} (b)}{\log_{c} (a)} \end{array}$

$\begin{aligned} \log_{a} (a^{n}) = n & \log_{a} [{(\frac{1}{a})}^{n}] = - n & a^{\log_{a} (b)} = b \end{aligned}$

Factorials

$\begin{aligned} n! = n \times (n - 1) \times (n - 2) . . . 3 \times 2 \times 1 & 0! = 1 \\ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \end{aligned}$

Complex Numbers

$\begin{aligned} \sqrt{- 1} = i & \sqrt{- a} = \sqrt{a} \sqrt{- 1} = a i \end{aligned}$

$\begin{aligned} i & = \sqrt{- 1} & i^{2} = - 1 & i^{3} = - \sqrt{- 1} & i^{4} = 1 \\ i^{5} & = \sqrt{- 1} & i^{6} = - 1 & . . . & . . . \end{aligned}$

i^{n}

expanded

\begin{aligned} i & = \sqrt{- 1} & = i \\ i^{2} & = \sqrt{- 1} \sqrt{- 1} & = - 1 \\ i^{3} & = \sqrt{- 1} \sqrt{- 1} \sqrt{- 1} & = - \sqrt{- 1} & = - i \\ i^{4} & = \sqrt{- 1} \sqrt{- 1} \sqrt{- 1} \sqrt{- 1} & = (- 1) (- 1) & = 1 \\ i^{5} & = \sqrt{- 1} \sqrt{- 1} \sqrt{- 1} \sqrt{- 1} \sqrt{- 1} & = (1) \sqrt{- 1} & = i \end{aligned}

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