Arithmetic & Number Sense

NUMBER SETS

ℕ Natural

1, 2, 3, 4, …

ℤ Integers

… −2, −1, 0, 1, 2, …

ℚ Rational

any p/q where q ≠ 0

Irrational

√2, π (no exact fraction)

ℕ ⊂ ℤ ⊂ ℚ — each set contains the one before it

ABSOLUTE VALUE

|a|

distance from 0 — always ≥ 0

|−7| = |7|

= 7

BEDMAS ORDER

Brackets (innermost first)

Exponents

Division & Multiplication (L→R)

Addition & Subtraction (L→R)

⚠ DM and AS have equal priority — always left to right

FRACTIONS

Add / Sub

same denominator (find LCD)

Multiply

(a/b)×(c/d) = ac/bd

Divide

Keep · Change · Flip (KCF)

KCF example

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8

Simplify

divide top & bottom by GCF

Mixed → Improper: 2¾ = (2×4+3)/4 = 11/4
Improper → Mixed: 11÷4 = 2 r3 → 2¾

ADDING FRACTIONS (LCD)

1/4 + 2/3: LCD = 12
= 3/12 + 8/12 = 11/12

FRACTION MISTAKES

1/2 + 1/3 ≠ 2/5 (must use LCD!)
Don't forget to flip when dividing
Simplify after multiplying

CONVERTING

Frac → Dec

divide numerator ÷ denominator

Dec → %

× 100

% → Dec

÷ 100

Frac → %

(frac × 100)%

0.75 = 75% = 3/4
0.06 = 6% = 6/100 = 3/50

PERCENTAGE FORMULAS

Find part

part = (%) / 100 × whole

Find %

% = (part / whole) × 100

Find whole

whole = part ÷ (% / 100)

Ex: 30% of 80 = 0.30 × 80 = 24

COMMON CONVERSIONS

Fraction	Decimal	Percent
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%
1/3	0.333…	33.3%

SQUARE ROOTS

Definition

√a = b means b² = a

Product rule

√(a·b) = √a · √b

WRONG

√(a+b) ≠ √a + √b

PERFECT SQUARES

n	n²	n	n²
1	1	7	49
2	4	8	64
3	9	9	81
4	16	10	100
5	25	11	121
6	36	12	144

COMMON MISTAKES

BEDMAS: multiply before adding
−3² = −9 but (−3)² = +9
√25 + √16 = 5 + 4 = 9, not √41
35% = 0.35, not 35
Flip when dividing fractions!