Complex Number Calculator

Result

The working

1. To divide, multiply the top and the bottom by the conjugate of the divisor. That makes the bottom a real number.

   (3 + 4i)(1 + 2i) / (1 − 2i)(1 + 2i)

2. The bottom becomes c² + d², which is real.

   1² + (−2)² = 5

3. Multiply out the top.

   (3 + 4i)(1 + 2i) = −5 + 10i

4. Divide the real and imaginary parts by the bottom.

   (−5 + 10i) / 5 = −1 + 2i

Complex plane

How to Use the Complex Number Calculator

1. 1. Choose what you need. Arithmetic works on two numbers, z₁ and z₂. Polar form works on z₁. Powers and roots take z₁ and a whole number n from 2 to 24.
2. 2. Type the numbers. Write 3 + 4i, −2i, 1/2 − 3/4 i or even (1 + i)(2 − i). Use i, or j if you are used to engineering notation. Fractions and decimals are read exactly.
3. 3. Read the result. Sums, products, quotients and integer powers are exact fractions. Moduli, arguments and roots are shown as exact where possible and as four-decimal approximations otherwise.
4. 4. Follow the working and the diagram. Each step is written out, and the numbers are drawn as arrows on the complex plane.

Arithmetic in Rectangular Form

Multiplication is ordinary expansion with one extra fact, i² = −1: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i. Division uses the conjugate, because (c + di)(c − di) = c² + d² is a real number. For (3 + 4i) ÷ (1 − 2i) the bottom becomes 1² + (−2)² = 5 and the top becomes (3 + 4i)(1 + 2i) = −5 + 10i, so the quotient is (−5 + 10i)/5 = −1 + 2i.

Modulus, Argument and Polar Form

The modulus |z| = √(a² + b²) is the distance from the origin. The argument θ is the angle from the positive real axis. This calculator gives the principal value, with −π < θ ≤ π, and adjusts for the quadrant: arctan(b/a) in the first and fourth, plus 180° in the second, minus 180° in the third. For z = 3 + 4i:

|z| = √(3² + 4²) = √25 = 5
θ = atan2(4, 3) = 0.9273 rad = 53.1301°
Polar form:       5(cos 53.1301° + i sin 53.1301°)
Exponential form: 5 × e^(i × 0.9273)

On an axis or a diagonal the argument is a simple multiple of π/4, and the calculator writes it exactly: −1 + i has modulus √2 and argument 3π/4. Zero has modulus 0 but no argument, so it has no polar form.

Powers and Roots: De Moivre's Theorem

De Moivre's theorem says that (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ): raise the modulus to the power and multiply the angle. For example (1 + i)⁸ has modulus (√2)⁸ = 16 and angle 8 × 45° = 360°, so it equals 16 exactly. The calculator multiplies out the power exactly and shows the De Moivre calculation next to it as a check.

Every non-zero number has exactly n distinct n-th roots. They share the modulus r^(1/n) and have angles (θ + 360°k)/n for k = 0, 1, …, n − 1, so they sit at the corners of a regular polygon centred on the origin. The cube roots of −8, which has modulus 8 and argument π, have modulus 2 and angles π/3, π and 5π/3: 1 + 1.7321i, −2 and 1 − 1.7321i. The square roots of 3 + 4i are exactly 2 + i and −2 − i.

Powers of i

To find iⁿ, divide n by 4 and use the remainder: i⁵⁷ = i¹ = i because 57 = 4 × 14 + 1.

Common Mistakes

• Forgetting i² = −1 when multiplying: the bd term changes sign, so (ac − bd), not (ac + bd).
• Dividing without the conjugate. Multiplying the top and bottom by the conjugate of the bottom is what removes i from the denominator.
• Taking arctan(b/a) in every quadrant. A calculator's arctan only returns −90° to 90°, so numbers in the second and third quadrants need ±180°.
• Reporting only one root. z^(1/n) has n values; the principal root is only k = 0.
• Mixing degrees and radians inside cos and sin. 53.13° and 0.9273 rad are the same angle.

Frequently Asked Questions

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator. The new denominator is c² + d², a real number, and you divide the real and imaginary parts of the new numerator by it. For (3 + 4i) ÷ (1 − 2i) the answer is −1 + 2i.

What are the modulus and argument of a complex number?

The modulus is the length √(a² + b²), the distance from the origin on the complex plane. The argument is the angle from the positive real axis to the number, between −180° and 180° for the principal value. Together they give polar form r(cos θ + i sin θ).

What is the exponential form of a complex number?

By Euler's formula, e^(iθ) = cos θ + i sin θ, so z = r(cos θ + i sin θ) can be written re^(iθ), with θ in radians. For 3 + 4i that is 5e^(0.9273i). It makes multiplication easy: multiply the moduli and add the angles.

What is De Moivre's theorem?

It states that (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, and so zⁿ = rⁿ(cos nθ + i sin nθ). It turns powers into a change of modulus and angle, and with a fractional exponent it gives the n-th roots.

How many n-th roots does a complex number have?

Exactly n, if the number is not zero. They have the same modulus and angles spaced 360°/n apart, so they form a regular n-sided polygon. A real number such as −8 has three cube roots: one real and two complex.

What is the conjugate of a complex number?

The conjugate of a + bi is a − bi. Multiplying a number by its conjugate gives the real number a² + b², the square of the modulus. Conjugates are used for division and for pairing the complex roots of real polynomials.

Why is the argument of 0 undefined?

Zero is the origin, which lies in every direction at once, so no single angle describes it. Its modulus is 0, and the calculator tells you when a polar form or root is not available.