ACT Math — Number & Quantity: Deep Understanding Notes

Real and Complex Number Systems

What “number systems” are and why they matter

A number system is a way of classifying numbers based on what kinds of values they can represent and what operations make sense on them. On the ACT, number-system ideas show up whenever you’re asked to compare values, solve equations, interpret radicals, reason about sign and magnitude, or decide whether a solution “makes sense.” Understanding the hierarchy of number sets helps you avoid illegal moves—like taking the square root of a negative number while staying in the real numbers.

A key big-picture idea is that as you move to larger systems, you keep the old numbers and add new ones so that more equations have solutions. For example, the real numbers solve equations like x^2 = 2, but to solve x^2 = -1 you need complex numbers.

The real number hierarchy

The real numbers are all numbers that can be placed on a number line. Within the reals, there are important subsets:

Why this matters: Many ACT questions are really “classification” questions in disguise. For instance, if you’re told x is an integer, then x^2 is automatically a nonnegative integer. If you’re told x is rational, then \sqrt{x} might be rational or irrational depending on x.

Rational vs. irrational in practice

A rational number can be written as \frac{a}{b} where a and b are integers and b \ne 0. Rational numbers have decimals that terminate (like 0.75) or repeat (like 0.333\ldots).

An irrational number cannot be written as a ratio of integers. Common sources:

• Square roots of non-perfect squares, like \sqrt{18} = 3\sqrt{2} (still irrational because \sqrt{2} is irrational).
• Constants like \pi.

Common trap: Simplifying a radical does not “make it rational.” For example, \sqrt{50} = 5\sqrt{2} is simpler, but still irrational.

Absolute value and distance

Absolute value measures distance from zero on the number line:

|x| = \text{the nonnegative distance of } x \text{ from } 0

That leads to two common equation patterns:

• |x| = a with a > 0 has two solutions: x = a and x = -a.
• |x| < a describes an interval: -a < x < a.

What goes wrong: Students often forget the negative solution for |x| = a, or they mishandle inequalities by not writing the compound inequality.

Complex numbers: extending the reals

A complex number has the form

a + bi

where a and b are real numbers and i is the imaginary unit defined by

i^2 = -1

You can think of complex numbers as what you get when you allow square roots of negative numbers:

\sqrt{-1} = i

So,

\sqrt{-9} = 3i

Why complex numbers matter on the ACT: They let you simplify expressions like \sqrt{-16} correctly and solve some quadratic equations whose discriminant is negative.

Basic complex arithmetic

You add and subtract complex numbers by combining real parts and imaginary parts:

If z1 = a + bi and z2 = c + di, then

z_1 + z_2 = (a+c) + (b+d)i

Multiplication uses distribution (FOIL) and the fact that i^2 = -1:

(a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac-bd) + (ad+bc)i

Example (multiply):
Compute (2+3i)(4-i).

1. Distribute:

(2)(4) + (2)(-i) + (3i)(4) + (3i)(-i)

2. Simplify terms:

8 - 2i + 12i - 3i^2

3. Use i^2=-1 so -3i^2 = -3(-1)=3:

8 + 10i + 3 = 11 + 10i

Complex conjugates (useful for division)

The complex conjugate of a+bi is a-bi. Multiplying a complex number by its conjugate eliminates the imaginary part:

(a+bi)(a-bi) = a^2 - (bi)^2 = a^2 + b^2

This is especially useful for writing a quotient in standard form.

Example (divide):
Write \frac{3+2i}{1-i} in the form a+bi.

1. Multiply numerator and denominator by the conjugate of the denominator, 1+i:

\frac{3+2i}{1-i} \cdot \frac{1+i}{1+i} = \frac{(3+2i)(1+i)}{(1-i)(1+i)}

2. Denominator becomes a real number:

(1-i)(1+i) = 1 - i^2 = 1 - (-1) = 2

3. Expand numerator:

(3+2i)(1+i) = 3 + 3i + 2i + 2i^2 = 3 + 5i - 2 = 1 + 5i

4. Divide by 2:

\frac{1+5i}{2} = \frac{1}{2} + \frac{5}{2}i

Exam Focus

• Typical question patterns:
  • Classify a number (rational vs. irrational; real vs. nonreal) after simplification.
  • Simplify expressions involving \sqrt{-a} or basic operations with a+bi.
  • Use absolute value to solve an equation or interpret a distance on a number line.
• Common mistakes:
  • Treating \sqrt{-a} as -\sqrt{a} instead of i\sqrt{a}.
  • Forgetting that rational decimals can repeat (not only terminate).
  • Solving |x|=a and giving only x=a.

Integer and Rational Exponents

What exponents mean (beyond memorizing rules)

An exponent tells you repeated multiplication. For a positive integer exponent,

a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ factors}}

This definition is the “anchor” that makes the exponent rules feel logical instead of arbitrary. The ACT often tests exponent reasoning inside algebraic simplification, scientific notation, growth/decay patterns, or rewriting radicals.

Zero and negative exponents

To keep exponent rules consistent (especially a^m \div a^n = a^{m-n}), we define:

• Zero exponent (for a \ne 0):

a^0 = 1

• Negative exponent (for a \ne 0):

a^{-n} = \frac{1}{a^n}

Why this matters: Negative exponents are not “negative numbers.” They indicate reciprocals. A common ACT move is to rewrite everything with positive exponents before simplifying.

Example (negative exponent):
Simplify 2^{-3}.

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

The main exponent rules (and when they apply)

Exponent rules rely on multiplication/division of like bases:

• Product of powers:

a^m \cdot a^n = a^{m+n}

• Quotient of powers:

\frac{a^m}{a^n} = a^{m-n}

• Power of a power:

(a^m)^n = a^{mn}

• Power of a product:

(ab)^n = a^n b^n

• Power of a quotient:

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

What goes wrong: Students often apply these rules to addition, which is not allowed. In general,

(a+b)^n \ne a^n + b^n

A quick counterexample: (1+1)^2 = 4 but 1^2+1^2=2.

Rational exponents and radicals

A rational exponent is an exponent that is a fraction. It connects directly to roots:

a^{\frac{1}{n}} = \sqrt[n]{a}

and more generally,

a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m

Why this matters: ACT problems often ask you to rewrite radicals as exponents (or vice versa) to simplify, compare sizes, or solve equations.

Domain caution (important): For real-number answers, even roots require nonnegative radicands. For example, \sqrt[2]{-4} is not real, but \sqrt[3]{-8} is real because odd roots of negatives are allowed.

Simplifying expressions with rational exponents

Example (rewrite and simplify):
Simplify 16^{\frac{3}{4}}.

1. Use the meaning: 16^{\frac{3}{4}} = \left(16^{\frac{1}{4}}\right)^3.
2. The fourth root of 16 is 2 because 2^4=16.
3. So:

16^{\frac{3}{4}} = 2^3 = 8

Example (variable expression):
Simplify x^{\frac{1}{2}} \cdot x^{\frac{3}{2}} assuming x>0.

1. Same base, add exponents:

x^{\frac{1}{2}} \cdot x^{\frac{3}{2}} = x^{\frac{1}{2}+\frac{3}{2}} = x^{\frac{4}{2}}

2. Reduce:

x^{\frac{4}{2}} = x^2

Solving equations with exponent structure

On the ACT, many exponent equations are solvable by rewriting both sides with the same base.

Example (same base):
Solve 3^{x+1} = 27.

1. Rewrite 27 as a power of 3:

27 = 3^3

2. Set exponents equal:

x+1 = 3

3. Solve:

x = 2

What goes wrong: If bases are not the same, you usually cannot just “set exponents equal.” For example, 2^x = 3^x is only true for special cases, not because bases differ.

Scientific notation connections

Scientific notation is essentially exponent rules with base 10:

a \times 10^n

where 1 \le a < 10 and n is an integer. You multiply numbers by multiplying coefficients and adding exponents.

Example (multiply in scientific notation):
Compute (3 \times 10^4)(2 \times 10^{-3}).

1. Multiply coefficients: 3 \cdot 2 = 6.
2. Add exponents on 10:

10^4 \cdot 10^{-3} = 10^{4+(-3)} = 10^1

3. So the product is:

6 \times 10^1

Exam Focus

• Typical question patterns:
  • Simplify expressions with negative and rational exponents (often inside fractions).
  • Rewrite radicals as powers (or powers as radicals) to compare or evaluate.
  • Solve equations by expressing both sides with the same base.
• Common mistakes:
  • Treating a^{-n} as -a^n instead of \frac{1}{a^n}.
  • Using exponent rules across addition: turning (a+b)^2 into a^2+b^2.
  • Forgetting root restrictions (even roots require nonnegative radicands for real answers).

Vectors

What a vector is (and why it’s different from a number)

A vector is a quantity with both magnitude (size) and direction. A plain number like 5 can tell you “how much,” but it cannot tell you “which way.” Vectors model displacement, velocity, forces, and any situation where direction matters.

On ACT-style problems, vectors most commonly appear in coordinate geometry or physics-flavored setups: moving right/left and up/down, combining movements, or finding distances and directions.

Representing vectors

In a coordinate plane, a 2D vector is often written as an ordered pair:

\langle a,b \rangle

You can interpret \langle a,b \rangle as “move a units in the x direction and b units in the y direction.”

A very common vector is the displacement from point P(x_1,y_1) to point Q(x_2,y_2):

\overrightarrow{PQ} = \langle x_2 - x_1, y_2 - y_1 \rangle

What goes wrong: Students sometimes subtract in the wrong order. The vector from P to Q is “terminal minus initial.” Swapping gives the opposite direction.

Vector addition and subtraction

Vectors add component-wise:

\langle a,b \rangle + \langle c,d \rangle = \langle a+c, b+d \rangle

Subtraction is similar:

\langle a,b \rangle - \langle c,d \rangle = \langle a-c, b-d \rangle

Why this works: The components represent independent horizontal and vertical moves. Combining moves means adding the horizontal parts and vertical parts separately.

Example (add displacements):
A person walks \langle 3, -2 \rangle then \langle -5, 4 \rangle. Find the net displacement.

\langle 3, -2 \rangle + \langle -5, 4 \rangle = \langle -2, 2 \rangle

Interpretation: overall they moved 2 units left and 2 units up.

Scalar multiplication

Multiplying a vector by a scalar (a real number) scales its magnitude and may reverse its direction:

k\langle a,b \rangle = \langle ka, kb \rangle

• If k>1, the vector gets longer.
• If 0<k<1, it gets shorter.
• If k<0, it reverses direction and scales by |k|.

Example (reverse direction):
-2\langle 1, -3 \rangle = \langle -2, 6 \rangle

Magnitude (length) of a vector

The magnitude of \langle a,b \rangle is the distance from the origin to the point (a,b), found using the Pythagorean theorem:

|\langle a,b \rangle| = \sqrt{a^2+b^2}

This connects vectors directly to the distance formula.

Example (magnitude):
Find the magnitude of \langle 6,8 \rangle.

|\langle 6,8 \rangle| = \sqrt{6^2+8^2} = \sqrt{36+64} = \sqrt{100} = 10

Unit vectors and direction

A unit vector has magnitude 1 and points in a direction. If \vec{v} is a nonzero vector, a unit vector in the same direction is:

\frac{\vec{v}}{|\vec{v}|}

This idea matters when you need “direction only,” separated from “how far.”

Example (unit vector):
Let \vec{v}=\langle 3,4 \rangle. Then |\vec{v}|=5, so a unit vector in its direction is:

\frac{\langle 3,4 \rangle}{5} = \langle \frac{3}{5}, \frac{4}{5} \rangle

Exam Focus

• Typical question patterns:
  • Find \overrightarrow{PQ} from two points, then compute its magnitude (distance).
  • Combine multiple displacement vectors to get a net vector.
  • Interpret scalar multiplication as changing speed/distance or reversing direction.
• Common mistakes:
  • Reversing subtraction when forming \overrightarrow{PQ}.
  • Adding magnitudes instead of adding components (only valid if vectors are collinear and same direction).
  • Forgetting to take the square root when computing magnitude.

Matrices and Matrix Operations

What a matrix is and why it shows up

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are useful because they organize information and make certain calculations systematic—especially when you have multiple equations, transformations, or repeated computations.

On ACT-style problems, matrices most often appear in a straightforward computational way: evaluating an expression, performing matrix addition/subtraction, multiplying a matrix by a scalar, or doing basic matrix multiplication. Occasionally, a matrix represents a transformation or a compact way to store data (like prices and quantities).

Matrix notation and dimensions

A matrix is usually named with a capital letter like A. Its dimensions are written as rows by columns.

If A has m rows and n columns, it is an m \times n matrix.

An entry is written as a_{ij} meaning “row i, column j.”

Example matrix A:

A = \begin{pmatrix}1 & 2 & 3 \ 4 & 5 & 6\end{pmatrix}

This is a 2 \times 3 matrix with a_{21}=4.

Matrix addition and subtraction

You can add (or subtract) matrices only if they have the same dimensions. Addition is entry-by-entry:

If A and B are both m \times n, then

A+B = (a_{ij}+b_{ij})

Why dimension matching matters: If one matrix has 2 rows and the other has 3 rows, their entries don’t line up, so “add them” is not defined.

Example (addition):

A = \begin{pmatrix}2 & -1 \ 0 & 3\end{pmatrix}

B = \begin{pmatrix}5 & 4 \ -2 & 1\end{pmatrix}

Then

A+B = \begin{pmatrix}2+5 & -1+4 \ 0+(-2) & 3+1\end{pmatrix} = \begin{pmatrix}7 & 3 \ -2 & 4\end{pmatrix}

Scalar multiplication

Multiplying a matrix by a scalar multiplies every entry:

kA = (ka_{ij})

Example (scalar multiplication):

-3\begin{pmatrix}1 & -2 \ 4 & 0\end{pmatrix} = \begin{pmatrix}-3 & 6 \ -12 & 0\end{pmatrix}

Matrix multiplication (the key idea)

Matrix multiplication is not entry-by-entry. It combines rows with columns.

If A is m \times n and B is n \times p, then AB is defined and has dimension m \times p.

The entry in row i, column j of AB is the dot product of row i of A with column j of B:

(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}

Why the inner dimensions must match: You need the row length of A (which is n) to equal the column length of B (also n) so the dot product makes sense.

What goes wrong: A very common mistake is assuming AB = BA. Matrix multiplication is generally not commutative, meaning the order matters.

Worked example of matrix multiplication

Let

A = \begin{pmatrix}1 & 2 \ 3 & 4\end{pmatrix}

B = \begin{pmatrix}5 & 6 \ 7 & 8\end{pmatrix}

Compute AB.

• The top-left entry uses row 1 of A and column 1 of B:

1\cdot 5 + 2\cdot 7 = 5 + 14 = 19

• The top-right entry uses row 1 of A and column 2 of B:

1\cdot 6 + 2\cdot 8 = 6 + 16 = 22

• The bottom-left entry uses row 2 of A and column 1 of B:

3\cdot 5 + 4\cdot 7 = 15 + 28 = 43

• The bottom-right entry uses row 2 of A and column 2 of B:

3\cdot 6 + 4\cdot 8 = 18 + 32 = 50

So

AB = \begin{pmatrix}19 & 22 \ 43 & 50\end{pmatrix}

If you compute BA you will get a different result, reinforcing that order matters.

Identity matrix (the “do nothing” matrix)

The identity matrix acts like the number 1 for multiplication. For a 2 \times 2 identity matrix:

I = \begin{pmatrix}1 & 0 \ 0 & 1\end{pmatrix}

For any compatible matrix A,

AI = A

and (when dimensions match)

IA = A

This matters when simplifying expressions or recognizing that a matrix leaves vectors unchanged under a transformation.

Matrices as transformations (a useful interpretation)

A common way to connect matrices to geometry is to multiply a matrix by a vector. If you treat a vector as a column matrix,

\begin{pmatrix}x \ y\end{pmatrix}

then multiplying by a 2 \times 2 matrix can represent a transformation of the plane.

For example,

\begin{pmatrix}2 & 0 \ 0 & 3\end{pmatrix}\begin{pmatrix}x \ y\end{pmatrix} = \begin{pmatrix}2x \ 3y\end{pmatrix}

This stretches x-coordinates by a factor of 2 and y-coordinates by a factor of 3.

What goes wrong: Students sometimes mix up whether vectors are rows or columns. On many tests, vectors for multiplication are written as columns, and dimension compatibility is your safety check.

Exam Focus

• Typical question patterns:
  • Perform matrix addition/subtraction or scalar multiplication and read off a requested entry.
  • Multiply two small matrices (often 2 \times 2) or multiply a matrix by a vector.
  • Determine whether a product like AB is defined based on dimensions.
• Common mistakes:
  • Adding matrices of different dimensions.
  • Doing entry-by-entry multiplication instead of row-by-column multiplication.
  • Assuming AB=BA or ignoring dimension checks when choosing the order of multiplication.