Perform addition, subtraction, multiplication, and division on complex numbers. Perfect for engineering, physics, and advanced mathematics.
Last updated: 2026-05-24T22:58:31.823Z | By ForgeCalc Engineering
z₁ =
3.0000 + 2.0000i
z₂ =
1.0000 + 4.0000i
Select Operation
Operation
3.0000 + 2.0000i + 1.0000 + 4.0000i
Result
4.0000 + 6.0000i
Enter numeric real and imaginary parts for both complex numbers. Division requires a non-zero denominator.
A complex number z = a + bi consists of a real part (a) and an imaginary part (bi). The imaginary unit i is defined by i² = -1. Complex numbers extend our number system to solve equations that have no real solutions.
Complex number operations are essential in electrical engineering (AC circuits), quantum mechanics, signal processing, and vibration analysis. They provide a unified framework for solving problems involving waves, rotations, and oscillations.
Formula
(a + bi) + (c + di) = (a+c) + (b+d)i
Example
(3 + 2i) + (1 + 4i) = 4 + 6i
Combine real and imaginary parts separately
Formula
(a + bi) - (c + di) = (a−c) + (b−d)i
Example
(3 + 2i) - (1 + 4i) = 2 − 2i
Subtract real and imaginary parts separately
Formula
(a + bi)(c + di) = (ac−bd) + (ad+bc)i
Example
(3 + 2i)(1 + 4i) = −5 + 14i
Use FOIL and remember i² = -1
Formula
(a+bi)/(c+di) = [(ac+bd)+(bc−ad)i] / (c²+d²)
Example
(3 + 2i) / (1 + 4i) ≈ 0.47 − 0.59i
Multiply by conjugate of denominator
z₁ + z₂ = z₂ + z₁
Order doesn't matter for addition
z₁ · z₂ = z₂ · z₁
Order doesn't matter for multiplication
(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
Grouping doesn't affect the sum
z₁(z₂ + z₃) = z₁z₂ + z₁z₃
Multiplication distributes over addition
z + 0 = z
Adding zero changes nothing
z · 1 = z
Multiplying by one changes nothing
The imaginary unit i is defined as the number whose square is -1 (i² = -1). This innovative definition allows us to extract square roots from negative numbers and solve previously impossible equations.
Absolutely! Electrical engineers use them for AC circuit analysis. Physicists use them in quantum mechanics. Signal processors use them for Fourier transforms. They're essential to modern technology.
Multiplying by the conjugate of the denominator (z̄) converts the denominator into a pure real number. This works because z · z̄ = |z|², which eliminates the imaginary part from the denominator.
Yes! Use the complex plane (Argand diagram) with the real part on the x-axis and imaginary part on the y-axis. This turns complex algebra into geometric operations like rotations and scaling.
Division by zero is undefined, just as it is with real numbers. In this calculator, dividing by z₂ = 0 produces an error message to alert you to the invalid operation.
Multiplying a complex number by another rotates it around the origin! The magnitude gets multiplied, and the angle (argument) gets added. This is crucial for understanding waves and oscillations.
Complex Number Format
z = a + bi
where a is the real part, b is the imaginary coefficient, and i² = -1
Magnitude (Modulus) of Complex Number
|z| = √(a² + b²)
The distance from the origin to the point (a, b)
Complex Conjugate
z̄ = a − bi
Same real part, opposite imaginary part
Useful Product Property
z · z̄ = a² + b² = |z|²
Multiplying a number by its conjugate gives a real number
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