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Complex Number Calculator

Add, subtract, multiply and divide complex numbers, with modulus, argument, conjugate, polar form and integer powers.

How to use this calculator

  1. 1Enter the real and imaginary parts of both numbers. Use 0 for a purely real or purely imaginary value.
  2. 2Read the four operations, then the modulus, argument and conjugate.
  3. 3Set a power to raise the first number by, and switch angles between degrees and radians.

How the calculation works

(a + bi)(c + di) = (ac − bd) + (ad + bc)i (a + bi) ÷ (c + di) = ((ac + bd) + (bc − ad)i) ÷ (c² + d²) r = √(a² + b²) θ = atan2(b, a) zⁿ = rⁿ(cos nθ + i sin nθ)
i
The imaginary unit, defined by i² = −1
a + bi
A complex number in rectangular form: real part a, imaginary part b
r
Modulus — the distance from the origin on the complex plane
θ
Argument — the angle from the positive real axis, in (−180°, 180°]
conj(z)
Conjugate: a − bi, the reflection of z across the real axis

Multiplication is ordinary binomial expansion with one extra rule: wherever i² appears, replace it with −1. That single substitution produces the ac − bd in the real part.

Division works by multiplying numerator and denominator by the conjugate of the denominator. Because (c + di)(c − di) = c² + d², the denominator becomes real and the division reduces to two real divisions.

The argument uses atan2, not arctan(b ÷ a). Plain arctangent cannot distinguish −1 + i from 1 − i, since both give a ratio of −1; atan2 keeps the sign of each part and returns the correct quadrant.

De Moivre's theorem makes powers cheap: raise the modulus to the power and multiply the argument by it. Squaring a complex number squares its distance from the origin and doubles its angle.

Worked example

(3 + 4i) and (1 − 2i)

  1. 1.Multiplication: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i².
  2. 2.Since i² = −1, the −8i² becomes +8: (3 + 8) + (−6 + 4)i = 11 − 2i.
  3. 3.Division: multiply top and bottom by the conjugate 1 + 2i. The denominator becomes 1² + 2² = 5.
  4. 4.Numerator: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i. So a ÷ b = (−5 + 10i) ÷ 5 = −1 + 2i.
  5. 5.Modulus of a: √(3² + 4²) = √25 = 5 — the 3-4-5 triangle on the complex plane.
  6. 6.Argument of a: atan2(4, 3) ≈ 53.13°.
  7. 7.Square by De Moivre: modulus 5² = 25, angle 2 × 53.13° = 106.26°, giving −7 + 24i.

Result: a × b = 11 − 2i, a ÷ b = −1 + 2i

Why the argument needs atan2

  1. 1.a = −1 + i sits in the second quadrant; b = 1 − i sits in the fourth.
  2. 2.Both have imaginary ÷ real = −1, so plain arctan(−1) = −45° would be returned for each — wrong for a.
  3. 3.atan2 keeps both signs: arg(a) = 135°, arg(b) = −45°. They differ by 180°, as the picture demands.
  4. 4.Both have modulus √2 ≈ 1.414214.
  5. 5.a to the 4th: modulus (√2)⁴ = 4, angle 4 × 135° = 540°, which is 180° — giving −4 + 0i.

Result: arg(a) = 135°, arg(b) = −45°

One rule generates all of it

Complex numbers add exactly one rule to ordinary algebra: there is a quantity i whose square is −1. Everything else follows from treating a + bi as a binomial and simplifying with that substitution.

Multiply (a + bi)(c + di) by expanding as usual and you get ac + adi + bci + bdi². Replace i² with −1 and the last term flips sign, leaving (ac − bd) + (ad + bc)i. No new rule was needed — the minus sign in the real part is the i² doing its work.

The name "imaginary" is a historical accident and an unfortunate one. Descartes coined it dismissively, and it stuck long after the numbers proved indispensable. They describe alternating current, quantum states and signal processing as concretely as real numbers describe lengths.

The complex plane: numbers as arrows

Plotting the real part along one axis and the imaginary part along the other turns every complex number into a point — or an arrow from the origin. The modulus is the arrow's length, and the argument is its angle from the positive real axis.

This picture explains multiplication in a way the algebra hides: multiplying two complex numbers multiplies their lengths and adds their angles. Multiplying by i, which has length 1 and angle 90°, is therefore a pure rotation a quarter turn anticlockwise. Doing it twice rotates 180°, which is multiplication by −1 — a geometric restatement of i² = −1.

De Moivre's theorem is that observation applied repeatedly. Raising to the power n scales the length by the nth power and multiplies the angle by n, which turns a laborious repeated multiplication into two simple operations.

The conjugate, and why division works

The conjugate of a + bi is a − bi: the same number reflected across the real axis. Its usefulness comes from one identity — a number times its conjugate is always real, and equals the square of the modulus: (a + bi)(a − bi) = a² + b².

That is the whole trick behind complex division. Dividing by c + di looks impossible until you multiply top and bottom by c − di; the denominator collapses to the real number c² + d², and what remains is two ordinary real divisions. It is the same manoeuvre as rationalising a surd denominator, and for the same reason.

Conjugates also explain why polynomials with real coefficients have complex roots in matched pairs. If a + bi is a root, so is a − bi — which is why the quadratic formula produces conjugate roots whenever the discriminant is negative.

What this assumes, and where it stops

Assumptions

  • Numbers are entered in rectangular form as a real part and an imaginary part.
  • The argument is the principal value, in (−180°, 180°], as atan2 returns.
  • Powers are integers, evaluated by De Moivre's theorem.

Limitations

  • Two numbers at a time, with integer powers only. Fractional powers and roots are multi-valued — an nth root has n distinct answers — and are not reported here.
  • No complex logarithms, exponentials or trigonometric functions.
  • Results are floating point, so values that are exactly zero in theory may display as a very small number in edge cases.
  • Polar input is not accepted; enter rectangular components.

Common questions

How do you multiply complex numbers?

Expand like binomials, then replace i² with −1. For (a + bi)(c + di) that gives (ac − bd) + (ad + bc)i. Geometrically, multiplication multiplies the two moduli and adds the two arguments — so it scales and rotates at the same time.

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator. Since (c + di)(c − di) = c² + d² is real, the division becomes two ordinary real divisions. It is the same technique as rationalising a denominator containing a square root.

What is the modulus and argument of a complex number?

On the complex plane, the modulus is the distance from the origin, √(a² + b²), and the argument is the angle from the positive real axis. Together they are the polar form. The argument must be computed with atan2 rather than arctan, so that numbers in the second and third quadrants are not confused with those in the fourth and first.

What is De Moivre's theorem?

That (r(cos θ + i sin θ))ⁿ equals rⁿ(cos nθ + i sin nθ). Raising a complex number to a power raises its modulus to that power and multiplies its argument by it, which is far quicker than repeated multiplication and is how this calculator evaluates powers.

Sources

Formula and content last reviewed on .

Results are estimates for information only, not professional advice.

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