Curriculum fit: US Algebra 2. Aligned to the Common Core Complex Number System standards (HSN-CN.A and HSN-CN.C.7). Check your own state standards and course.
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What Is It?
Some quadratic equations, such as x² = −1, have no real solution. To fix this, mathematicians define the imaginary unit i, where i² = −1.
A complex number has the form a + bi, where a is the real part and b is the imaginary part. You add and multiply them like algebraic expressions, replacing i² with −1.
To divide, multiply the top and bottom by the conjugate of the denominator. The conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a² + b², which is a real number.
Key Formulas and Ideas
- i² = −1, so i³ = −i and i⁴ = 1.
- Add: (a + bi) + (c + di) = (a + c) + (b + d)i.
- Multiply by expanding and replacing i² with −1.
- Conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a² + b².
- Divide by multiplying top and bottom by the conjugate of the denominator.
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. A negative discriminant gives complex solutions.
Worked Examples
Multiply Complex Numbers
Multiply (3 + 2i)(1 − 4i).
- Expand: 3 − 12i + 2i − 8i².
- Replace i² with −1: −8i² = +8.
- Combine: (3 + 8) + (−12 + 2)i = 11 − 10i.
✓ Answer 11 − 10i
Divide Complex Numbers
Simplify (5 + i) ÷ (2 − i).
- Multiply top and bottom by the conjugate 2 + i.
- Top: (5 + i)(2 + i) = 10 + 5i + 2i + i² = 9 + 7i. Bottom: (2 − i)(2 + i) = 4 + 1 = 5.
- Divide each part by 5.
✓ Answer 9/5 + (7/5)i
Complex Solutions of a Quadratic
Solve x² + 4x + 13 = 0.
- Quadratic formula: x = (−4 ± √(16 − 52)) ÷ 2 = (−4 ± √−36) ÷ 2.
- √−36 = 6i.
- x = (−4 ± 6i) ÷ 2 = −2 ± 3i.
✓ Answer x = −2 + 3i or x = −2 − 3i
Common Mistakes to Avoid
- Forgetting that i² = −1 after multiplying, which leaves an unwanted i².
- Using the same sign instead of the opposite sign to make the conjugate.
- Combining real and imaginary parts as if they were like terms.
- Writing √−36 as −6. It is 6i.
Practice Questions
Frequently Asked Questions
Are imaginary numbers real?
They are not on the real number line, but they are a consistent part of mathematics. They are used in engineering, such as in electrical circuits and signal processing.
What is the conjugate of a complex number?
It is the same number with the sign of the imaginary part changed. The conjugate of 4 − 7i is 4 + 7i.
Why do complex roots come in pairs?
For a quadratic with real coefficients, the ± in the quadratic formula gives a pair of conjugates, such as −2 + 3i and −2 − 3i.
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Reviewed by StudyMorrow editorial review on 6 Nov 2026.