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- Adds, subtracts, multiplies and divides complex numbers in a + bi form.
- Shows the modulus, argument and polar form of the result.
- Gives the complex conjugate.
- Shows each step, including i² = −1 and dividing by the conjugate.
How to use the complex number calculator
Enter the first number
Its real part a and imaginary part b, for 3 + 4i type 3 and 4.
Pick the operation
Add, subtract, multiply or divide.
Enter the second number
Its real part c and imaginary part d. Use a negative d for numbers like 1 − 2i.
Read the result
See the answer in rectangular and polar form with the working.
Complex number rules
Operations
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
(a + bi) ÷ (c + di) = ((ac + bd) + (bc − ad)i) ÷ (c² + d²)
Multiplying uses i² = −1. Dividing multiplies top and bottom by the conjugate c − di.
Modulus and argument
|z| = √(x² + y²)
arg z = atan2(y, x)
Polar form: z = |z|(cos θ + i sin θ)
Example 1: multiplying
(3 + 4i)(1 − 2i)
- ac − bd 3 − (−8) = 11
- ad + bc −6 + 4 = −2
- Result 11 − 2i
Example 2: dividing
(3 + 4i) ÷ (1 − 2i)
- Multiply by the conjugate 1 + 2i (−5 + 10i) ÷ 5
- Result −1 + 2i
Example 3: useful facts
Quick checks.
- i × i −1
- |3 + 4i| 5
- (3 + 4i) + (1 − 2i) 4 + 2i
Where complex numbers are used
Algebra and precalculus
Check imaginary number homework and see each step.
Electrical engineering
Work with impedance and AC circuits.
Signal processing
Represent waves with magnitude and phase.
Quadratic equations
Handle the complex roots of equations with a negative discriminant.
Find complex roots of a quadratic with the quadratic formula calculator, work out angles with the trig calculator, or square-root negative numbers with the square root calculator.
Complex number calculator FAQ
What is a complex number?
A number of the form a + bi, where a is the real part, b is the imaginary part and i is the square root of −1.
What is i squared?
i² = −1. This is the key rule when multiplying complex numbers.
How do you divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom number, which makes the denominator a real number.
What is the modulus of a complex number?
Its distance from 0 on the complex plane: √(a² + b²). For 3 + 4i it is 5.
What is the polar form?
A way of writing z as its modulus and angle: z = r(cos θ + i sin θ).
What is a complex conjugate?
The same number with the sign of the imaginary part flipped. The conjugate of 3 + 4i is 3 − 4i.