Complex Numbers: Step-by-Step Guide
A complex number looks like a + bi. The number a is the real part, b is the imaginary part, and i is a special number where i^2 = -1. You add, subtract, multiply, and divide complex numbers a lot like you do with binomials, and then use i^2 = -1 to simplify.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Add or subtract the parts separately. Combine the real parts with each other, and combine the imaginary parts with each other.\[(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)\,i\]
- Multiply with FOIL, then replace i^2 with -1. Multiply (a + bi)(c + di) like two binomials. Then replace i^2 with -1 and combine like terms.\[(a+bi)(c+di) = (ac - bd) + (ad + bc)\,i\]
- Divide by using the conjugate. The conjugate of c + di is c - di. Multiply the top and bottom by it. The bottom becomes the real number c^2 + d^2.\[\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^{2}+d^{2}}\]
- Write the answer as a + bi. Write one real part and one imaginary part. Reduce any fractions.\[z = a + bi\]
Worked example
Compute (3 + 2i)(1 - 4i) and write in the form a + bi.
Step 1: Use FOIL: (3)(1) + (3)(-4i) + (2i)(1) + (2i)(-4i).
Step 2: Multiply each pair: 3 - 12i + 2i - 8i^2.
Step 3: Replace i^2 with -1: 3 - 12i + 2i - 8(-1) = 3 - 12i + 2i + 8.
Step 4: Combine the real parts: 3 + 8 = 11. Combine the imaginary parts: -12i + 2i = -10i.
Step 5: The answer is 11 - 10i.
Check: The real part is 3 x 1 - 2 x (-4) = 3 + 8 = 11. The imaginary part is 3 x (-4) + 2 x 1 = -12 + 2 = -10. So 11 - 10i.
Tips and common mistakes
- Always replace i^2 with -1. Do not leave i^2 in your answer.
- The powers of i repeat in a cycle: i, -1, -i, 1. So i^3 = -i and i^4 = 1.
- When dividing, use the conjugate. (c + di)(c - di) = c^2 + d^2, which is a real number.
- Write the final answer as a + bi, with the real part first.
- The conjugate of a + bi is a - bi.
- (a + bi)(a - bi) = a^2 + b^2. This is always a real number that is zero or positive.
- Keep the real parts and the imaginary parts separate as you work, so you do not mix them up.