Numbers & Their Operations

Types of Numbers

Z

Integers ($\mathbb{Z}$)

$..., -3, -2, -1, 0, 1, 2, 3, 4, ...$

P

Prime Numbers

Integers divisible by $1$ and itself only. Smallest prime is $2$.

Q

Rational Numbers ($\mathbb{Q}$)

$\frac{\text{integer}}{\text{integer}}$ — e.g., $\frac{4}{7}, -3\frac{1}{8}, 0.3, 2.\dot{6}\dot{5}, 92, \sqrt{16}$

I

Irrational Numbers

Cannot be expressed as fractions — e.g., $\pi, \sqrt{2}, e$

R

Real Numbers ($\mathbb{R}$)

All rational and irrational numbers combined.

$A \times 10^n$

where $n$ is an integer, and $1 \leq A < 10$

Multiplication

$a^m \times a^n = a^{m+n}$

Division

$a^m \div a^n = a^{m-n}$

Power of Power

$(a^m)^n = a^{mn}$

Product Power

$(ab)^m = a^m b^m$

Quotient Power

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Negative Index

$a^{-n} = \frac{1}{a^n}$

Zero Index

$a^0 = 1$

Fractional Index

$a^\frac{1}{n} = \sqrt[n]{a}$

General Fractional

$a^\frac{m}{n} = (\sqrt[n]{a})^m$