Complex numbers extend reals with i, where i² = −1. Form: a + bi.
Drag the complex number
2 + 3i
real part 2 · imaginary part 3
modulus |z| = √(2² + 3²) = 3.61
argument ≈ 56.31°
Arithmetic with a + bi
- Add/subtract: combine real parts and imaginary parts separately.
- Multiply: FOIL it out, then replace i² with −1.
- Divide: multiply top and bottom by the conjugate of the denominator to clear the i below.
- The modulus |a + bi| = √(a² + b²) is the point's distance from the origin — drag the point to watch it change.
Your turn
Compute (3 + 2i) + (1 − 5i).
Watch out
i² = −1, not 1 or −i. After expanding a product, hunt down every i² and swap it for −1 before collecting terms.
Operations
- Add: (a+bi) + (c+di) = (a+c) + (b+d)i
- Multiply: use FOIL, replace i² with −1.
- Conjugate: of a+bi is a−bi.
Try it
(2 + 3i)(1 − i)
2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i.