Order of Operations and Number Properties

Order of operations (PEMDAS)

1. Parentheses and grouping

() [] {} | | \frac{◻}{◻}

innermost first

2. Exponents and roots

a^{n} \sqrt{a}

3. Multiply and divide

\times \div

left to right

4. Add and subtract

+ -

left to right

Commutative, addition

a + b = b + a

Commutative, multiplication

a \times b = b \times a

Associative, addition

(a + b) + c = a + (b + c)

Associative, multiplication

(a \times b) \times c = a \times (b \times c)

Distributive

a (b + c) = a b + a c

Distributive over subtraction

a (b - c) = a b - a c

Additive identity

a + 0 = a

Multiplicative identity

a \times 1 = a

Additive inverse

a + (- a) = 0

Multiplicative inverse

a \times \frac{1}{a} = 1

a ≠ 0

Zero property

a \times 0 = 0

Zero product property

a b = 0 \Rightarrow a = 0 or b = 0

Division algorithm

D = d \times q + r

0 ≤ r < d

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