Complex number calculator with an Argand diagram

A complex number has a real part and an imaginary part, z = a + bi, where i² = −1. You add them part by part, multiply them like brackets using i² = −1, and divide by multiplying top and bottom by the conjugate. On an Argand diagram z is an arrow whose length is the modulus |z| = √(a² + b²) and whose angle is the argument. This complex number calculator does all of these exactly, with fractions and square roots, and draws the answer on an Argand diagram.

It opens on z = 1 + √3i. Open the full complex number calculator over this page to type your own.

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Complex Number Calculator

Calculate with complex numbers and plot them on an Argand diagram

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How to use the complex number calculator

1. Choose an operation: One number, z + w, z − w, z × w, z ÷ w, Power or Roots.
2. Type z, and w if the operation needs it. Fractions and square roots work, such as 1 + √3i or 1/2 − 3i.
3. For Power or Roots, type n.
4. Read the exact answer, with the working underneath. One number gives the modulus, argument, conjugate, and polar and exponential forms.
5. Look at the Argand diagram: each number is an arrow from the origin, and roots sit evenly round a circle.
6. The Examples section loads polar form, a multiplication, a division, De Moivre, the cube roots of 1 and fourth roots. To practise, press Random problem in the Practice section.

Adding, multiplying and dividing complex numbers

Example 1: adding and multiplying

Take z = 5 − 2i and w = 3 + 4i. Adding works part by part: z + w = 8 + 2i. To multiply, expand the brackets and replace i² with −1:

Example 2: dividing

To divide, multiply the top and bottom by the conjugate of the bottom, 3 − 4i. The bottom becomes a real number, 3² + 4² = 25:

So the answer is 7/25 − (26/25)i. The calculator keeps it as exact fractions.

Modulus, argument and polar form

The argument θ is the angle from the positive real axis, usually given between −π and π. Watch the quadrant: for −1 − i, tan⁻¹(1) is 45°, but the arrow points down and left, so the argument is −135° (−3π/4).

Example 3: polar form of 1 + √3i

|z| = √(1 + 3) = 2, and the argument is tan⁻¹(√3) = π/3 (60°). So z = 2(cos π/3 + i sin π/3) = 2e^(iπ/3), and its conjugate is 1 − √3i. These are the calculator's opening answers. It gives exact arguments for multiples of π/6 and π/4, and decimals otherwise.

Powers and roots: De Moivre's theorem

Example 4: a power

(1 + √3i)⁶ in polar form is 2⁶(cos 6π/3 + i sin 6π/3) = 64(cos 2π + i sin 2π) = 64. Expanding the brackets six times would take much longer. Choose Power with n = 6, or load the De Moivre example.

Example 5: cube roots of −27

−27 = 27(cos π + i sin π). Its cube roots have modulus ³√27 = 3 and arguments (π + 2kπ) ÷ 3, which are π/3, π and −π/3. That gives 3/2 + (3√3/2)i, −3 and 3/2 − (3√3/2)i, evenly spaced round a circle of radius 3 on the Argand diagram. Choose Roots with z = −27 and n = 3.

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Common mistakes

• Forgetting that i² = −1 when multiplying, which flips the sign of that term.
• Dividing part by part. Multiply top and bottom by the conjugate of the bottom instead.
• Wrong-quadrant argument. tan⁻¹(b ÷ a) only gives the right angle when a is positive. Sketch the point first.
• Degrees and radians mixed up. Arguments are usually in radians.
• Missing roots. A number has n different n-th roots, not one.

Read next

Maths · AlgebraArgand diagram: plot complex numbers, modulus and argumentHow to plot complex numbers on an Argand diagram, find the modulus and argument, add and multiply geometrically, and draw loci, with to-scale diagrams.