Operations with Complex Numbers

Core Idea

Complex numbers support all standard arithmetic. Addition/subtraction: combine real parts and imaginary parts separately. Multiplication: use FOIL and replace i² with -1. Division: multiply numerator and denominator by the conjugate of the denominator (a - bi). The conjugate of a + bi is a - bi, and (a + bi)(a - bi) = a² + b², a real number. The modulus |a + bi| = sqrt(a² + b²) gives the distance from the origin in the complex plane.

How It's Best Learned

Practice each operation separately, then mix them. For multiplication, emphasize the FOIL-then-simplify process. For division, show that multiplying by the conjugate eliminates i from the denominator. Connect modulus to the distance formula. Give problems that combine multiple operations.

Common Misconceptions

• Forgetting to replace i² with -1 after multiplying.
• Not using the conjugate when dividing (trying to "divide" complex numbers directly).
• Thinking (a + bi)² = a² + b² (it is a² + 2abi - b²).
• Confusing |a + bi| with a + b.

Explainer

Complex arithmetic follows exactly the same rules as polynomial arithmetic over the reals — the only difference is the single reduction rule i² = -1. Once you internalize that, complex arithmetic stops feeling special and starts feeling routine. Think of a + bi as a polynomial in i of degree 1, and every operation you learned for polynomials applies directly, with one extra simplification step at the end.

Addition is the simplest operation: add real parts together and imaginary parts together, just as you add like terms in polynomials. (3 + 2i) + (1 - 5i) = 4 - 3i. Multiplication is FOIL applied to the two binomials, then one substitution: wherever i² appears, replace it with -1. For example, (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i - 3(-1) = 5 + i. If you skip the substitution, you'll have a nonsensical i² term remaining — so the substitution is the entire trick, not an optional cleanup.

Division is the operation that surprises most students, because you cannot divide complex numbers directly. The strategy is to convert the division into multiplication: multiply numerator and denominator by the conjugate of the denominator. The conjugate of (a + bi) is (a - bi), and their product (a + bi)(a - bi) = a² + b² is always a real number — no imaginary part. This is because you're applying the difference of squares pattern: (a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² + b². So to compute (3 + i)/(1 + 2i): multiply by (1 - 2i)/(1 - 2i) to get (3 + i)(1 - 2i)/((1)² + (2)²) = (3 - 6i + i - 2i²)/5 = (3 - 5i + 2)/5 = (5 - 5i)/5 = 1 - i.

The modulus |a + bi| = √(a² + b²) measures the distance from the origin to the point (a, b) in the complex plane. This is just the Pythagorean theorem — the real and imaginary parts are the two legs, and the modulus is the hypotenuse. The modulus has a multiplicative property: |z₁ · z₂| = |z₁| · |z₂|. This geometric view of complex numbers — where moduli multiply under multiplication and distances behave like magnitudes — is the foundation for polar form and for understanding why complex numbers are so powerful in describing rotations and waves.