Arithmetic Operations

Basic Operations

Addition and Subtraction

Operation	Rule
$a + b = b + a$	Commutative Property
$(a + b) + c = a + (b + c)$	Associative Property
$a + 0 = a$	Identity Property
$a + (-a) = 0$	Inverse Property

Multiplication and Division

Operation	Rule
$a \times b = b \times a$	Commutative Property
$(a \times b) \times c = a \times (b \times c)$	Associative Property
$a \times 1 = a$	Identity Property
$a \times \frac{1}{a} = 1$, $a \neq 0$	Inverse Property
$a \times 0 = 0$	Zero Property

Distributive Property

$a(b + c) = ab + ac$

$a(b - c) = ab - ac$

$(a + b)(c + d) = ac + ad + bc + bd$

Order of Operations (PEMDAS)

1. Parentheses (innermost first)
2. Exponents (powers and roots)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Fractions

Addition and Subtraction

$\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b} \quad \text{(same denominator)}$

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \quad \text{(different denominators)}$

Multiplication and Division

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Simplifying Fractions

$\frac{ac}{bc} = \frac{a}{b} \quad \text{(cancel common factors)}$

Signed Numbers

Rules for Multiplication and Division

Operation	Result
$(+)(+)$	$+$
$(+)(-)$	$-$
$(-)(+)$	$-$
$(-)(-)$	$+$

Rules for Addition

• Same signs: Add absolute values, keep the sign
• Different signs: Subtract absolute values, take sign of larger absolute value

Properties of Zero and One

Property	Rule
$a \times 0 = 0$
$\frac{0}{a} = 0$ $(a \neq 0)$
$\frac{a}{0}$ is undefined
$a^1 = a$
$a^0 = 1$ $(a \neq 0)$
$0^0$ is indeterminate