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Complex Numbers calculator →

Complex Numbers guide

The Complex Numbers Calculator performs arithmetic, converts between polar and rectangular forms, and computes powers and roots. It visualizes every operation on the complex plane (Argand diagram) so you can see the geometry behind the math.

Worked examples

Multiply two complex numbers

  1. Select the "Operations" tab.
  2. Enter z₁ as 3 (real) and 4 (imaginary).
  3. Enter z₂ as 1 (real) and -2 (imaginary).
  4. Select the × operator.
  5. Read the result on the right.
Result: z₁ × z₂ = 11 - 2i

Using FOIL: (3)(1) + (3)(-2i) + (4i)(1) + (4i)(-2i) = 3 - 6i + 4i - 8i². Since i² = -1, this becomes 3 - 2i + 8 = 11 - 2i.

Convert rectangular to polar form

  1. Select the "Polar ⇄ Rect" tab.
  2. Enter z₁ as 1 (real) and 1 (imaginary).
  3. Look at the Analysis section for magnitude (r) and angle (θ).
Result: r = 1.414, θ = 45°

Magnitude r = √(1² + 1²) = √2 ≈ 1.414. The angle θ = atan2(1, 1) = 45°.

Raise a complex number to a power

  1. Select the "Powers & Roots" tab.
  2. Set the Base z₁ to 1 + 1i.
  3. Select "Raise to power" and set the exponent to 4.
  4. Click Calculate.
Result: (1 + i)⁴ = -4

Using De Moivre’s theorem: r⁴(cos(4θ) + i·sin(4θ)). Since r = √2 and θ = 45°, r⁴ = 4 and 4θ = 180°, giving 4(cos 180° + i·sin 180°) = -4.

Tips

  • •Use the Argand diagram to visually verify addition (parallelogram rule) and multiplication (rotation and scaling).
  • •The conjugate of a + bi is a - bi. Multiplying a number by its conjugate always gives a real number.
  • •Division by zero (0 + 0i) is undefined and rejected by the calculator.

Frequently asked questions

What is the imaginary unit i?
The imaginary unit i is defined as the square root of -1. It extends the real number system into a 2D plane, allowing us to solve equations like x² + 1 = 0.
What is an Argand diagram?
It is a 2D graph where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number.
How do you add complex numbers?
Add the real parts together and the imaginary parts together separately. (a+bi) + (c+di) = (a+c) + (b+d)i.
How do you multiply complex numbers?
Use the FOIL method (First, Outer, Inner, Last) and remember that i² = -1. For example, (a+bi)(c+di) = (ac-bd) + (ad+bc)i.
How do you divide complex numbers?
Multiply the numerator and denominator by the complex conjugate of the denominator. This eliminates the imaginary part from the denominator.
What is a complex conjugate?
The conjugate of z = a + bi is z̄ = a - bi. It is a reflection of the complex number across the real axis.
What is the magnitude (or modulus) of a complex number?
It is the distance from the origin to the point on the complex plane, calculated as |z| = √(a² + b²).
What is the argument of a complex number?
The angle the complex number makes with the positive real axis, usually measured counterclockwise in radians or degrees.
What is De Moivre’s Theorem?
A formula used to raise complex numbers to a power or extract roots: z^n = r^n(cos(nθ) + i·sin(nθ)).
How many roots does a complex number have?
Every non-zero complex number has exactly n distinct n-th roots, which are evenly spaced around a circle in the complex plane.
What is the Euler form of a complex number?
It uses the constant e and the imaginary unit: z = re^(iθ). It is highly useful for advanced engineering and physics calculations.
What does it mean for a complex number to be purely imaginary?
It means the real part is exactly zero. For example, 5i is purely imaginary.
What does it mean for a complex number to be purely real?
It means the imaginary part is exactly zero (e.g., 7 is a complex number with an imaginary part of 0).
Can you compare complex numbers using inequality (e.g., z > w)?
No. Because complex numbers exist on a 2D plane, there is no natural ordering. You can compare their magnitudes, but not the numbers themselves.
How are complex numbers used in electrical engineering?
They are used to represent AC circuit impedance. Resistance is the real part, and reactance (capacitors/inductors) is the imaginary part.
What happens to i raised to a power?
The values cycle every four powers: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then it repeats.
Does the calculator support 3D complex numbers?
No. Complex numbers are strictly 2-dimensional (one real, one imaginary). For 3D math, use the Vector calculator.
How precise are the polar calculations?
Results are rounded to 4 decimal places for readability, but the internal math uses full floating-point precision.
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