Numbers & Operations
A complete, test-focused breakdown of every Numbers & Operations concept tested on the ACT — from integer rules to complex number arithmetic — with worked examples and strategy notes so you know exactly what to do on test day.
In This Lesson
Integers & Absolute Value
An integer is any whole number — positive, negative, or zero. No fractions, no decimals. The ACT tests your fluency with integer arithmetic, especially with negatives.
Arithmetic with Negative Numbers
| Operation | Rule | Example |
|---|---|---|
| Adding same sign | Add magnitudes, keep sign | (−4) + (−3) = −7 |
| Adding opposite signs | Subtract magnitudes, take sign of larger | (−7) + 3 = −4 |
| Subtracting | Add the opposite | 5 − (−3) = 5 + 3 = 8 |
| Multiplying / Dividing | Same signs → positive; Different signs → negative | (−2)(−5) = 10; (−6)÷2 = −3 |
Absolute Value
Absolute value |x| gives the distance from zero — always non-negative.
|x| = −x if x < 0
// |−8| = 8, |5| = 5, |0| = 0
ACT trap: When a negative sign sits outside absolute value brackets, apply it after resolving the absolute value. −|−9| = −9, not 9.
Fractions & Mixed Numbers
The Four Operations
a/b ± c/d = (ad ± bc) / bd
(a/b) × (c/d) = ac / bd
(a/b) ÷ (c/d) = (a/b) × (d/c)
2¾ = (2×4+3)/4 = 11/4
Simplifying Fractions
Always divide numerator and denominator by their GCF (Greatest Common Factor). On multiple-choice ACT, answers always appear in simplest form.
Speed trick: Cross-cancel before multiplying. In (3/4) × (16/9), cancel the 3 and 9 (÷3) and the 4 and 16 (÷4) to get 1/1 × 4/3 instantly.
Decimals & Place Value
Place Value Chart
| Position | Name | Value |
|---|---|---|
| … | 1 0 0 0 . | Thousands to Ones | ×1000, ×100, ×10, ×1 |
| . 1 | Tenths | ×0.1 |
| . 0 1 | Hundredths | ×0.01 |
| . 0 0 1 | Thousandths | ×0.001 |
Decimal ↔ Fraction Conversions
0.3333... = 1/3
0.125 = 1/8
// Memorize: 1/8=0.125, 1/6≈0.167, 1/3≈0.333, 2/3≈0.667
Rounding Rules
- Identify the digit to round to. Look at the digit immediately to its right.
- If the next digit is 0–4: round down (keep current digit).
- If the next digit is 5–9: round up (add 1).
Multiplication trap: When multiplying decimals, count the total decimal places in both factors — that's how many appear in the answer. 1.2 × 0.04 = 0.048 (3 decimal places total).
Ratios & Proportions
A ratio compares two quantities. A proportion states that two ratios are equal — and is your go-to tool for scaling problems.
Setting Up a Proportion
Part-to-Part vs. Part-to-Whole
If the ratio of boys to girls is 3:5, there are 3+5 = 8 parts total. Boys = 3/8 of the whole; girls = 5/8. The ACT frequently asks for part-to-whole when you're given part-to-part.
Rate Problems
Rate = Distance / Time (or more generally, quantity / unit). Always identify what the rate's units are — this alone often points to the correct setup.
D = R × T → R = D/T → T = D/R
Percents
Percent means "per hundred." The ACT tests three core percent calculation types, plus percent change and multi-step percent problems.
The Three Core Questions
| Question Type | Formula | Example |
|---|---|---|
| What is P% of W? | Answer = (P/100) × W | What is 30% of 80? → 24 |
| X is what % of W? | P = (X/W) × 100 | 18 is what % of 72? → 25% |
| X is P% of what? | W = X ÷ (P/100) | 15 is 20% of what? → 75 |
Percent Change
// Positive = increase; Negative = decrease
Successive Percents (Multiplier Method)
A 20% increase followed by a 20% decrease does NOT return to the original. Multiply the multipliers:
Original × 1.20 × 0.80 = Original × 0.96 → a 4% net decrease.
Shortcut: To increase by P%, multiply by (1 + P/100). To decrease by P%, multiply by (1 − P/100). Chain multiplications for multi-step problems.
Exponents & Roots
Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product Rule | xᵃ · xᵇ = xᵃ⁺ᵇ | x³ · x⁴ = x⁷ |
| Quotient Rule | xᵃ / xᵇ = xᵃ⁻ᵇ | x⁶ / x² = x⁴ |
| Power Rule | (xᵃ)ᵇ = xᵃᵇ | (x³)² = x⁶ |
| Zero Exponent | x⁰ = 1 (x ≠ 0) | 7⁰ = 1 |
| Negative Exponent | x⁻ⁿ = 1/xⁿ | 2⁻³ = 1/8 |
| Fractional Exponent | x^(m/n) = ⁿ√(xᵐ) | 8^(2/3) = (∛8)² = 4 |
Square Roots & Simplifying Radicals
√(a/b) = √a / √b
√(a²) = |a|
// Simplify: √72 = √(36·2) = 6√2
Common mistake: (−3)² = 9, but −3² = −9. Parentheses change everything. The exponent only applies to what it directly touches.
Order of Operations
Parentheses → Exponents → Multiplication & Division (left to right) → Addition & Subtraction (left to right)
Factors, Multiples & Prime Numbers
Key Definitions
Factors of 12: 1, 2, 3, 4, 6, 12.
Multiples of 4: 4, 8, 12, 16, …
GCF(18, 24) = 6
LCM(4, 6) = 12
Prime Numbers
A prime has exactly two distinct factors: 1 and itself. 1 is NOT prime. 2 is the only even prime.
// Memorize these — the ACT tests them often
Prime Factorization
Break any integer into its prime building blocks. This is the foundation for finding GCF and LCM.
Fast GCF trick: GCF × LCM = Product of the two numbers. So GCF(18,24) = (18 × 24) / 72 = 432/72 = 6.
Sequences & Patterns
Arithmetic Sequences
Each term increases (or decreases) by a constant amount called the common difference d.
// aₙ = nth term, a₁ = first term, d = common difference
Sum of first n terms: Sₙ = n/2 × (a₁ + aₙ)
Geometric Sequences
Each term is multiplied by a constant called the common ratio r.
// Example: 2, 6, 18, 54… → r = 3 → a₅ = 2 × 3⁴ = 162
Pattern Recognition
- Look at differences between consecutive terms first (arithmetic).
- If differences aren't constant, look at ratios (geometric).
- If neither, look at second differences (quadratic) or other patterns.
- For repeating decimal/digit patterns, divide out the cycle length.
Complex Numbers
The ACT (especially at higher score levels) tests basic arithmetic with imaginary and complex numbers.
i = √(−1) i² = −1 i³ = −i i⁴ = 1 then the cycle repeats every 4.
Powers of i — Cycle of 4
i² = −1
i³ = −i
i⁴ = 1
i⁵ = i (cycle restarts)
// To find iⁿ: divide n by 4, use the remainder
Arithmetic with Complex Numbers
A complex number takes the form a + bi where a is the real part and b is the imaginary part.
| Operation | Method | Example |
|---|---|---|
| Addition | Add real parts, add imaginary parts | (3+2i)+(1−4i) = 4−2i |
| Subtraction | Subtract real parts, subtract imaginary parts | (5+3i)−(2+i) = 3+2i |
| Multiplication | FOIL, then replace i²=−1 | (2+i)(3−2i)=6−4i+3i−2i²=8−i |
Number Line & Inequalities
Number Sets
| Set | Symbol | Contains |
|---|---|---|
| Natural Numbers | ℕ | 1, 2, 3, 4, … |
| Whole Numbers | 𝕎 | 0, 1, 2, 3, … |
| Integers | ℤ | …, −2, −1, 0, 1, 2, … |
| Rational Numbers | ℚ | Any fraction a/b (b≠0), including repeating decimals |
| Irrational Numbers | √2, π, e — non-repeating, non-terminating decimals | |
| Real Numbers | ℝ | All rational + irrational |
Inequality Rules
When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
Example: −2x > 6 → x < −3 (sign flipped when dividing by −2).
Absolute Value Inequalities
|x| > k → x < −k OR x > k (OR — two intervals)
Memory trick: "Less than" → Less than k is a single connected interval (think: small, together). "Greater than" → Greater than k splits into two separate pieces (think: big, apart).
What to Do on Test Day
Common ACT Traps
- Forgetting to flip the inequality sign when multiplying/dividing by a negative.
- Applying percent change to the wrong base (always use the original).
- Treating 1 as a prime number (it isn't).
- Ignoring order of operations — especially with exponents outside parentheses.
- Confusing part-to-part ratios with part-to-whole fractions.
Must-Know Facts
- Powers of i cycle every 4: i, −1, −i, 1.
- Percent change always divides by the original.
- GCF × LCM = product of the two numbers.
- √2 ≈ 1.41, √3 ≈ 1.73, √5 ≈ 2.24 — useful for estimation.
- Any number to the 0 power = 1 (except 0⁰, which is undefined).
Put It to the Test
You've reviewed the concepts. Now lock them in with timed ACT-style questions. Start with Quiz 1 and work your way through — each quiz adds difficulty.