Arithmetic formulas and rules reference

Addition and Subtraction

\[\begin{align*}a + ( b + c ) &= (a + b) + c & a + b = b + a \\\\ a + 0 &= a & a + (-a) = 0 \\\end{align*}\]

\[a - b = a + (-b) = -b + a\]

\[-(-a) = a\]

Multiplication and Division

\[\begin{align*} 0 \times a = 0 && 1 \times a = a\end{align*}\]

\[\begin{align*}a \times (b \times c) = (a \times b) \times c\end{align*}\]

\[\begin{align*}\\(-) \times (-) = (+)\\ (+) \times (+) = (+) \\\\ (-) \times (+) = (-) \\ (+) \times(-) = (-)\\\end{align*}\]

\[a \div b = \frac{a}{b} = a \times \frac{1}{b}\]

Fractions

\[\frac{a}{0} = \text{Undefined/Does Not Exist}\]

\[\frac{0}{0} = \text{Undefined/Does Not Exist}\]

\[\begin{align*}\frac{0}{a} = 0, a \ne0 \quad&& \frac{a}{a} = 1\end{align*}\]

\[\begin{align*}&& \frac{a}{b} \times \frac{c}{b} = \frac{a\times c}{b \times b} \quad&& \frac{a}{b}\times \frac{c}{d} = \frac{a\times c}{b\times d}\end{align*}\]

\[\begin{align*} \frac{a}{b} \pm \frac{c}{b} = \frac{a \pm c}{b} \quad&& \frac{a}{b} \pm \frac{c}{d} = \frac{(a\times d) \pm (c\times b)}{b \times d}\end{align*}\]

\[\begin{align*}\frac{1}{\frac{a}{b}} = \frac{b}{a} \quad&&\frac{a}{\frac{b}{c}} = \frac{a\times c}{b} \quad&& \frac{\frac{a}{b}}{c} = \frac{a}{b\times c}\end{align*}\]

\[\begin{align*} \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b}\div \frac{c}{d} = \frac{a}{b}\times\frac{d}{c} = \frac{a\times d}{b\times c} \end{align*}\]

\[\begin{align*} \frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b} && && \frac{-a}{-b} = \frac{a}{b} \end{align*}\]

Exponents

\[\begin{align*} 1^a = 1 && && a^1 = a \end{align*}\]

\[\begin{align*} 0^a = 0,\, a\ne 0 && && a^0 = 1, a\ne 0 && && 0^0 = \text{Undefined}\end{align*}\]

\[\begin{align*} (ab)^n = a^nb^n && && \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \end{align*}\]

\[\begin{align*} a^n = \frac{1}{a^{-n}} && && \frac{1}{a^n} = a^{-n} \end{align*}\]

\[\begin{align*} a^na^m = a^{n+m} && && \frac{a^n}{a^m} = a^{n-m} \end{align*}\]

\[\begin{align*} (a^{n})^{m}=a^{nm} \end{align*}\]

Radicals

\[\begin{align*} \sqrt[n]{0} = 0 && && \sqrt[n]{1} = 1 \end{align*}\]

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

\[\begin{align*} \sqrt[n]{a^m} = \sqrt[n]{a}^m = a^{\frac{m}{n}} \end{align*}\]

\[\begin{align*} \sqrt{a}\sqrt{a} = a && \sqrt[n]{a} \sqrt[n]{a} = a, \,\,\,a\ge 0 \end{align*}\]

\[\begin{align*} \sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}\,\,\,\, a,b \ge 0 && \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{align*}\]

Logarithms

\[\begin{align*} \log_a(1) = 0 && \log_a(a) = 1 \end{align*}\]

\[\begin{align*}\log_1(a) = \text{Undefined} && \log_a(b), \,\,\,a \text{ or } b \le 0= \text{Undefined} \end{align*}\]

\[\begin{align*} \log_a(c^b) = b\cdot\log_a(c) && \log_{a^b}(c) = \frac{1}{b} \log_a(c)\end{align*}\]

\[\begin{align*} \log_a\left(\frac{1}{b}\right) = -\log_a(b) && \log_{\frac{1}{a}}(b) = -\log_a(b) \end{align*}\]

\[\begin{align*} \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \end{align*}\]

\[\begin{align*} \log_a(a^n) = n && \log_a\left[\left(\frac{1}{a}\right)^n\right] = -n && a^{\log_a(b)}=b \end{align*}\]

Factorials

\[\begin{align*} n! = n\times (n-1) \times (n-2) ... 3\times 2 \times 1 && && 0! = 1 \\\\ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\end{align*}\]

Complex Numbers

\[\begin{align*} \sqrt{-1} = i && \sqrt{-a} = \sqrt{a}\sqrt{-1}=ai\end{align*}\]

\[\begin{align*} i &= \sqrt{-1} && i^2 = -1 && i^3 = -\sqrt{-1} && i^4 = 1\\ i^5 &= \sqrt{-1} && i^6 = -1\quad \,\,\ && .\,\,\,.\,\,\,. \quad && .\,\,\,.\,\,\,\,. \\ \,\end{align*}\]