1. Definition

• A complex number has the form: $$a+ib$$

  where:

  • $$a$$ is the real part,

  • $$b$$ is the imaginary part,

  • $$i$$ is the imaginary unit, defined as $$i^2 = -1$$.

2. Basic Operations

Addition & Subtraction

• Combine like terms (real with real, imaginary with imaginary).

• Example:
  $$(3+4i)+(2-5i)=(3+2)+(4i-5i)=5-i$$

Multiplication

• Use distributive property and simplify using $$i^2 = -1$$.

• Example:
  $$(2+3i)(1-4i)=2-8i+3i-12i^2=14-5i$$

Division

• Multiply numerator and denominator by the conjugate of the denominator.

• Example:
  $$\frac{3+2i}{1-i}=\frac{3+2i}{1-i}\cdot\frac{1+i}{1+i}=\frac{(3+2i)(1+i)}{(1-i)(1+i)}=\frac{3+3i+2i+2i^2}{1-i^2}=\frac{1+5i}{2}$$​

3. Conjugates

• The conjugate of $$a + bi$$ is $$a – bi$$.

• Useful for division and simplifying.

• Property:
  $$(a+bi)(a−bi)= a^2 + b^2$$

4. Powers of i

• Powers of $i$ repeat every 4 steps:

  • $$i^1 = i$$

  • $$i^2 = -1$$

  • $$i^3 = -i$$

  • $$i^4 = 1$$
    Then repeat.

• Example:
  $-i\$\$$

5. ACT Problem Types

1. Simplify expressions
   Example: $$(2 + 3i)(2 – 3i) = 4 + 9 = 13$$

2. Division with conjugates
   Example: $$\frac{4 + i}{2 – i}$$

3. Powers of i
   Example: $i^{72} = 1$

4. Roots and quadratic equations

   • Quadratics with negative discriminants have complex roots.

   • Example: Solve $$x^2 + 4 = 0$$ → $$x = \pm 2i$$.

ACT Tip: The ACT doesn’t test advanced complex numbers (like polar form or De Moivre’s Theorem). You only need to master basic operations, conjugates, and powers of $i$. All the types of problems are available to you to practice in the school of mathematics ACT Qbank.