BYTETOOLS

Complex Number Calculator

Add, subtract, multiply, divide and raise complex numbers to powers. Converts between rectangular a+bi and polar form with modulus, argument and conjugate.

z₁ = a + bi

Real part · imaginary part

z₂ = c + di

Real part · imaginary part

-5 + 10i
Result (a + bi)
11.18034
Modulus |z|
116.565051°
Argument (deg)
2.034444
Argument (rad)

Forms

Rectangular: -5 + 10i

Polar: 11.18034 ∠ 116.565051° = 11.18034(cos 2.034444 + i·sin 2.034444)

Conjugate: -5 − 10i

What is the Complex Number Calculator?

The ByteTools Complex Number Calculator performs arithmetic on complex numbers in rectangular form: addition, subtraction, multiplication, division and integer powers of z₁ = a + bi.

  • Add, subtract, multiply and divide complex numbers
  • Integer powers, including negative exponents
  • Rectangular a+bi and polar r∠θ forms shown together
  • Modulus, argument (degrees and radians) and conjugate
  • Division by zero detected and explained
  • 100% private — runs entirely in your browser

How to use the Complex Number Calculator

  1. 1

    Enter the real and imaginary parts of z₁.

  2. 2

    Choose the operation, and enter z₂ (or the exponent for powers).

  3. 3

    Read the result in a + bi form.

  4. 4

    Check the polar form: modulus, argument in degrees and radians.

  5. 5

    Copy the result with one click.

About the Complex Number Calculator

The ByteTools Complex Number Calculator performs arithmetic on complex numbers in rectangular form: addition, subtraction, multiplication, division and integer powers of z₁ = a + bi. Division uses the standard conjugate method, and powers are computed by fast binary exponentiation, with negative exponents handled through the reciprocal.

Every result is also converted to polar form: the modulus |z| = √(a² + b²), the argument in both degrees and radians from the two-argument arctangent, the r∠θ notation and the conjugate. Seeing rectangular and polar side by side is exactly what students need for electrical engineering phasors and complex analysis homework.

All arithmetic runs instantly and 100% locally in your browser — nothing is uploaded, and results can be copied with one click.

Frequently asked questions

How do you multiply complex numbers?

Expand like binomials and use i² = −1. For example (1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i² = 3 + 10i − 8 = −5 + 10i. In polar form the same operation is simpler: multiply the moduli and add the arguments.

How do you divide complex numbers?

Multiply the top and bottom by the conjugate of the denominator, which makes the denominator real. So (1 + 2i) ÷ (3 + 4i) = (1 + 2i)(3 − 4i) ÷ 25 = (11 + 2i) ÷ 25 = 0.44 + 0.08i. The calculator applies this method exactly.

What is the polar form of a complex number?

Instead of x and y parts, polar form gives a length and a direction: z = r(cos θ + i sin θ), often written r∠θ. For 3 + 4i, r = √(3² + 4²) = 5 and θ ≈ 53.13°. Polar form makes multiplication, division and powers much easier.

What are the modulus and argument?

The modulus |z| = √(a² + b²) is the distance from the origin to the point (a, b) — for 3 + 4i it is 5. The argument is the angle that line makes with the positive real axis, measured with atan2 so the quadrant is always correct.

What is the conjugate of a complex number used for?

The conjugate of a + bi is a − bi, mirroring the point across the real axis. Multiplying a number by its conjugate gives the real value a² + b², which is what makes complex division possible, and conjugates pair up as roots of real polynomials.

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