Ninety percent of ACT test-takers miss complex number questions—yet mastering them takes about fifteen minutes. Complex numbers appear on just 1–2 questions per test, but they follow such predictable rules that they're practically free points once you know the playbook. Every complex number question boils down to one of four moves: cycling through powers of i, adding or subtracting, multiplying with FOIL, or dividing with conjugates. That's it. No surprises, no curveballs. By the end of this chapter, you'll actually look forward to seeing i on test day.
Ninety percent of ACT test-takers miss complex number questions—yet mastering them takes about fifteen minutes. Complex numbers appear on just 1–2 questions per test, but they follow such predictable rules that they're practically free points once you know the playbook. Every complex number question boils down to one of four moves: cycling through powers of i, adding or subtracting, multiplying with FOIL, or dividing with conjugates. That's it. No surprises, no curveballs. By the end of this chapter, you'll actually look forward to seeing i on test day.
Mathematicians needed a number whose square is negative. No real number does this—try squaring anything on a number line and you always get something positive (or zero). So they invented the imaginary unit i, defined by one simple rule:
i = √(-1), which means i² = -1
That single fact—i² = -1—is the engine that drives all of complex arithmetic. Whenever you see i² in a calculation, replace it with -1. That's the only new rule you need.
Every complex number is written in standard form a + bi, where a is the real part and b is the imaginary part. Think of it as a two-part number: one piece lives on the ordinary number line, and the other piece lives in the imaginary dimension.
Every year, thousands of ACT test-takers lose easy points on exponent questions because they forget one simple rule. Exponents directly appear in 1–2 questions per test, but exponent rules underpin dozens more across algebra, functions, and geometry—making this one of the highest-leverage topics you can study. And here's the thing: they aren't hard. They're elegant shortcuts that turn impossible-looking problems into 30-second victories. From negative exponents that flip fractions to rational exponents that unlock hidden roots, mastering these rules transforms confusion into confidence. By the end of this chapter, you'll see x²/³ and instantly think "cube root, then square"—no hesitation, no panic.
All of exponent math rests on five rules. Learn these and every exponent problem on the ACT becomes a matter of recognizing which rule to apply. The rules only work when the bases are the same—that's the one condition to watch for.
Product Rule: x^a · x^b = xa+b When you multiply same bases, add the exponents. Think of 2³ · 2⁴ as "three 2s multiplied by four more 2s"—that's seven 2s total, so 2⁷.
Quotient Rule: (x^a)/(x^b) = xa⁻b When you divide same bases, subtract the exponents. The x's in the denominator cancel some from the numerator.
Power Rule: (x^a)^b = xab A power raised to another power? Multiply the exponents. (3²)³ = 3⁶ because you're tripling the number of factors.
Zero Exponent: x⁰ = 1 for any x ≠ 0 Anything (except zero) raised to the zero power equals 1. Always. Even (million)⁰ = 1.
Negative Exponent: x⁻ⁿ = (1)/(xⁿ) A negative exponent flips the base into a fraction. It does NOT make the answer negative.
Imagine walking into the ACT math section and seeing a grid of numbers inside brackets. Your heart might skip a beat—but here's the secret: matrices are just organized lists of numbers, and the ACT only asks you to do basic arithmetic with them. You'll encounter at most 1–2 matrix questions on the entire test, and they're essentially free points if you know three simple operations. The math is nothing more than what you learned in elementary school, just arranged in a grid. By the end of this chapter, you'll see matrices as a gift on test day—a chance to earn quick points while everyone else panics.
A matrix is a rectangular grid of numbers arranged in rows and columns, wrapped in brackets. A 2 × 2 matrix has 2 rows and 2 columns—four numbers total. A 3 × 2 matrix has 3 rows and 2 columns—six numbers. We describe the size as "rows by columns" and refer to individual entries by their position: row i, column j.
Adding or subtracting matrices is as simple as it gets: combine the elements in matching positions. Top-left plus top-left, top-right plus top-right, and so on. The only rule: both matrices must have the same dimensions. You can't add a 2 × 2 matrix to a 3 × 3 matrix—the positions don't match up.
Imagine transforming 8⁵/³ into 32 in just three seconds. Sounds impossible? That's exactly the kind of shortcut this chapter teaches. Radicals and rational roots appear in 2–3 questions on every ACT, and mastering these conversions can mean the difference between rushing through the section and finishing with time to spare. The key insight: every radical is just a fractional exponent in disguise, and once you see that connection, these problems become mechanical. You'll learn to simplify √(72) on sight, rationalize any denominator in seconds, and combine radical expressions like a pro.
Here's the big idea: every radical can be rewritten as a fractional exponent, and every fractional exponent can be rewritten as a radical. They're two notations for the same operation. The index of the root becomes the denominator, and any power under the radical becomes the numerator:
sqrt[n]{a^m} = am/ⁿ
So √(x) is really x¹/². The cube root sqrt[3]{x} is x¹/³. And sqrt[3]{x²} is x²/³. Once you internalize this, you can switch freely between the two forms—and the ACT loves testing whether you can.
Simplifying a radical means pulling out the largest perfect square (or cube, etc.) hiding inside. The method:
1. Find the largest perfect square that divides evenly into the number under the radical. 2. Rewrite as a product: √(perfect) × text{leftover}. 3. Separate: √(perfect) × √(leftover). 4. Simplify the perfect square.
This works because √(ab) = √(a) · √(b)—you can always break a radical into pieces.
Imagine trying to split a pizza perfectly into equal slices—sometimes it works out cleanly, sometimes it doesn't. That's the essence of rational versus irrational numbers. Rational numbers are the clean cuts: fractions, integers, decimals that stop or repeat. Irrational numbers are the messy ones: they go on forever without a pattern and refuse to be written as fractions. On the ACT, you'll face 1–2 questions directly testing whether you can classify numbers and predict what happens when you combine different types, with these skills also surfacing inside algebra and function problems. The rules are surprisingly simple—and once you know them, these questions become automatic points.
A rational number is any number that can be written as a fraction (p)/(q) where p and q are integers and q ≠ 0. That's the definition, but here's the practical version: if the decimal terminates (stops) or repeats, it's rational.
Integers: -3, 0, 5 are all rational (write them as (-3)/(1), (0)/(1), (5)/(1)). Fractions: (2)/(3), -(7)/(4), (22)/(7) are obviously rational. Terminating decimals: 0.5 = (1)/(2), 0.125 = (1)/(8). Repeating decimals: 0.overline{3} = (1)/(3), 0.overline{142857} = (1)/(7). Perfect square roots: √(4) = 2, √(9) = 3, √(144) = 12.
Irrational numbers are the rebels of mathematics—they refuse to be tamed into fractions. Their decimal expansions go on forever without any repeating pattern.
The most common irrational numbers on the ACT: √(2) approx 1.41421356... √(3) approx 1.73205080... √(5) approx 2.23606797... π approx 3.14159265... e approx 2.71828182...
The quick test: is the number under a square root a perfect square? If not, the root is irrational. √(9) = 3 is rational, but √(8) is irrational. √(100) = 10 is rational, but √(101) is irrational.
Imagine arranging -3.5, π, and (1)/(2) on a number line without a calculator. Sounds tricky? On the ACT, you'll face 1–2 questions directly testing these skills—classifying numbers, applying algebraic properties, and ordering values of different types—though the underlying concepts power problems across the entire test. These aren't isolated skills—they're the foundation that every other math topic builds on. The student who can rearrange 25 × 37 × 4 into 100 × 37 using the commutative property finishes 15 seconds faster. Multiply that by a dozen problems and you've bought yourself an extra three minutes on test day.
Every number you've ever used lives somewhere in this hierarchy. Each layer adds something new:
Natural numbers: {1, 2, 3, 4, ...} — the counting numbers. No zero, no negatives, no fractions.
Whole numbers: {0, 1, 2, 3, ...} — add zero to the natural numbers.
Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} — add the negatives.
Rational numbers: anything you can write as (p)/(q) where p and q are integers and q ≠ 0. This includes all integers (put them over 1), all terminating decimals (0.75 = (3)/(4)), and all repeating decimals (0.overline{3} = (1)/(3)).
Irrational numbers: everything that's NOT rational. Their decimals go on forever without repeating: π, √(2), e.
Real numbers: rationals and irrationals together. Every point on the number line is a real number.
Five algebraic properties govern how real numbers behave under arithmetic. The ACT tests whether you can recognize them and, more importantly, use them strategically to simplify calculations.
Your smartphone GPS, every video game character, and the flight path of every airplane have something in common: they all rely on vectors. A vector is a quantity with both magnitude (how much) and direction (which way). The ACT has been testing vectors more frequently in recent years—you'll see 1–2 questions involving vector notation, component addition, or magnitude. The math is simple (just adding and multiplying coordinates), but the notation looks unfamiliar, which is why 37% of test-takers miss their first vector question. Master the notation, and these become the easiest points on the test.
A vector describes a movement or force with two pieces of information: how far and in which direction. We write vectors in component form as langle x, y rangle, where x is the horizontal component and y is the vertical component. You'll also see the i-j notation: xi + yj, where i means "one unit right" and j means "one unit up."
So langle 3, 4 rangle and 3i + 4j mean the same thing: go 3 units right and 4 units up.
Notice the dashed right triangle in the figure above. The horizontal leg is the x-component (3), the vertical leg is the y-component (4), and the vector itself is the hypotenuse. This triangle is the key to everything vectors do.