Complex Numbers Calculator

A complex number pairs a real part with an imaginary part in the form a + bi, where i is the imaginary unit with the property i² = −1. The free CalcNova complex numbers calculator adds, subtracts, multiplies, and divides numbers in a + bi form, and also shows the modulus and the conjugate of your first number.

Complex numbers calculator

Enter two complex numbers in a+bi form, pick an operation, and see the result instantly. Everything runs in your browser.

First number z1
+i
Second number z2
+i

How the complex numbers calculator works

Type the real and imaginary parts of two complex numbers, z1 and z2, into the four boxes. Choose Add, Subtract, Multiply, or Divide. The calculator applies the arithmetic rule for that operation and prints the full expression together with its answer, so you can see exactly which numbers were combined.

Below the answer you always see two extra facts about z1: its modulus |z1|, the distance from the origin on the complex plane, and its conjugate, the same number with the sign of the imaginary part flipped. Results update live as you type, and any box left empty or non-numeric is treated as zero.

Division needs a non-zero second number. If z2 is 0 + 0i, the calculator shows a message asking you to enter a non-zero number instead of attempting the division, because dividing by zero is undefined.

Formulas for complex arithmetic

For z1 = a + bi and z2 = c + di, the calculator uses these rules:

  • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i — add the real parts and add the imaginary parts.
  • Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i — subtract the real parts and subtract the imaginary parts.
  • Multiplication: (a + bi) × (c + di) = (ac − bd) + (ad + bc)i — expand the brackets and use i² = −1 to turn the i² term into a real number.
  • Division: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²) — multiply top and bottom by the conjugate c − di, which makes the denominator the real number c² + d².
  • Modulus: |a + bi| = √(a² + b²) — the straight-line distance from the origin.
  • Conjugate: the conjugate of a + bi is a − bi.

Worked examples

1. Addition: (1 + 2i) + (3 + 4i). Add the real parts: 1 + 3 = 4. Add the imaginary parts: 2 + 4 = 6. The answer is 4 + 6i.

2. Multiplication: (1 + 2i) × (3 + 4i). Real part: 1 × 3 − 2 × 4 = 3 − 8 = −5. Imaginary part: 1 × 4 + 2 × 3 = 4 + 6 = 10. The answer is −5 + 10i.

3. Subtraction: (1 + 2i) − (3 + 4i). Subtract the real parts: 1 − 3 = −2. Subtract the imaginary parts: 2 − 4 = −2. The answer is −2 − 2i.

4. Division: (1 + 2i) ÷ (3 + 4i). The denominator is 3² + 4² = 25. Real part: (1 × 3 + 2 × 4) ÷ 25 = 11 ÷ 25 = 0.44. Imaginary part: (2 × 3 − 1 × 4) ÷ 25 = 2 ÷ 25 = 0.08. The answer is 0.44 + 0.08i.

5. Modulus and conjugate: z1 = 1 + 2i. The modulus is √(1² + 2²) = √5 ≈ 2.2360679775. The conjugate is 1 − 2i.

Common mistakes

  • Forgetting that i² = −1 when multiplying. The product (a + bi)(c + di) contains a bd·i² term, which becomes −bd, not +bd.
  • Adding the real part of one number to the imaginary part of the other. Real parts only combine with real parts, and imaginary parts only with imaginary parts.
  • Treating any zero part as a zero divisor. A complex number is zero only when both parts are zero, so 0 + 5i is a perfectly valid divisor — only 0 + 0i is not.
  • Flipping the wrong sign in the conjugate. The conjugate of a + bi is a − bi; only the imaginary part changes sign.
  • Reading the modulus as the larger part. The modulus √(a² + b²) is always at least as large as either part alone, and it is not simply |a| + |b|.

Privacy and accuracy

The complex numbers calculator runs entirely in your browser. Your numbers are never sent to a server, stored online, or shared with anyone. Results are computed with standard double-precision arithmetic and shown rounded to ten decimal places with trailing zeros removed, so long decimals are shortened for display while the calculation itself keeps full precision.

Frequently asked questions

What is a complex number?

A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = −1. Ordinary real numbers are the special case where b = 0.

Why does the calculator show a modulus and a conjugate?

The modulus |z1| = √(a² + b²) measures the size of the number as a distance on the complex plane, and the conjugate a − bi is used in division and in many physics and engineering formulas. They are shown for z1 so you get them without a second tool.

Can I divide by a complex number whose real part is zero?

Yes. Only 0 + 0i is a problem. A number like 0 + 5i gives c² + d² = 25, so division works normally — the calculator only blocks the case where both parts of the divisor are zero.

What happens if I leave a box empty?

An empty or non-numeric box is treated as zero, so leaving the imaginary box of z2 empty turns it into the real number you typed. Type 0 explicitly if you want to be sure.

How many decimal places does the answer show?

Up to ten decimal places, with trailing zeros stripped. A result like 0.4400000000 appears as 0.44, and a repeating decimal such as 1 ÷ 3 = 0.3333333333 stops at ten digits.

Are my numbers sent to a server?

No. Every calculation happens in your own browser. Nothing you type leaves your device, and nothing is stored online.

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