Complex Number Calculator

A complex number has a real part and an imaginary part: z = a + bi, where i is the number whose square is −1. You can picture it as the point (a, b) on the Argand diagram, with real numbers along the horizontal axis and imaginary numbers up the vertical one. The same point can be described by its distance from 0 (the modulus r) and its angle from the positive real axis (the argument θ). That is polar form, r∠θ, and exponential form, re^(iθ).

The formula

add: (a + bi) + (c + di) = (a + c) + (b + d)i
multiply: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
divide: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
modulus r = √(a² + b²), argument θ = atan2(b, a)
polar: a + bi = r(cos θ + i sin θ) = re^(iθ)
De Moivre: (r∠θ)ⁿ = rⁿ∠nθ; nth roots = r^(1/n)∠(θ + 360°k) ÷ n

Worked example

Multiply z₁ = 3 + 4i by z₂ = 1 − 2i. Here a = 3, b = 4, c = 1 and d = −2. The real part is ac − bd = 3 × 1 − 4 × (−2) = 3 + 8 = 11. The imaginary part is ad + bc = 3 × (−2) + 4 × 1 = −6 + 4 = −2. So (3 + 4i)(1 − 2i) = 11 − 2i. Its modulus is √(11² + 2²) = √125 = 5√5 ≈ 11.1803, and its argument is atan2(−2, 11) ≈ −10.3048° (−0.1799 rad), so in polar form it is 5√5∠−10.3048°.

Check in polar form: |z₁| = √(9 + 16) = 5 and |z₂| = √(1 + 4) = √5, and 5 × √5 = 5√5. The angles are 53.1301° and −63.4349°, which add to −10.3048°. Multiplying complex numbers multiplies their sizes and adds their angles.

Dividing, step by step

To work out (3 + 4i) ÷ (1 − 2i), multiply the top and bottom by the conjugate of the bottom, 1 + 2i. The bottom becomes 1² + 2² = 5. The top becomes (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i. Divide each part by 5: −1 + 2i. Check: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i² = −1 + 4i + 4 = 3 + 4i.

Polar form and the quadrant rule

θ = tan⁻¹(b ÷ a) only works on the right half of the diagram. When a is negative, add 180° if b is positive (quadrant 2) or take away 180° if b is negative (quadrant 3). atan2(b, a) does this for you. The main (principal) argument is kept between −180° and 180° (−π to π).

Roots and powers (De Moivre)

To raise r∠θ to the power n, raise r to the power n and multiply the angle by n: (1 + i)⁸ = (√2)⁸∠(8 × 45°) = 16∠360° = 16. Every non-zero number has exactly n nth roots, spaced 360° ÷ n apart on a circle. The cube roots of 8 are 2∠0°, 2∠120° and 2∠240°, which are 2, −1 + √3i and −1 − √3i. For a power that is not a whole number, such as z^0.5, the calculator gives the principal value, which uses the main argument.

e^z and ln z

Euler’s formula says e^(iθ) = cos θ + i sin θ, so e^(a + bi) = eᵃ(cos b + i sin b), and e^(iπ) = −1. The natural log goes the other way: ln z = ln r + iθ. Adding any multiple of 2πi gives another log, so the calculator shows the principal one, with θ between −π and π. Complex powers use z₁^z₂ = e^(z₂ ln z₁): for example i^i = e^(−π/2) ≈ 0.2079, a real number.

Powers of i

Typing numbers

• Rectangular: 3 + 4i, 3 − 4i, −2i, i, 5, 1/2 + 3/4i, 4j − 1.
• Polar: 2∠30° or 2<30 (degrees), 2∠π/6 or 2∠0.5rad (radians), 2 cis 45°. Angles with no unit use the setting under the boxes.
• Exponential: 2e^(iπ/6) or e^(0.5i).
• Sums and functions: (1 + i)^3, √(−4), conj(2 + i), abs(3 + 4i), ln(i), sin, cos and tan.
• 1/2i means (1/2) × i. Write 1/(2i) if you mean one over 2i.

The answers are worked out with ordinary computer decimals (about 15 digits). When an answer is very close to a simple fraction, a square root like √3/2 or a fraction of π, it is shown in that exact form. Decimals are rounded to the places you choose.