Complex Number Calculator
All four operations, plus polar form.
Work out Complex Number. All four operations, plus polar form. Names the misconception directly.
A × B
-5 + 10i
A = 3 + 4i in polar form is 5∠53.1301°
Multiplication follows from i² = −1: (a+bi)(c+di) expands to (ac−bd) + (ad+bc)i. Division works by multiplying top and bottom by the conjugate of the denominator, which clears the imaginary part from below the line. The polar form is where complex numbers earn their keep. Multiplying two of them multiplies the moduli and adds the arguments, so multiplication is a rotation combined with a scaling — which is why they run through signal processing, AC circuit analysis and quantum mechanics. The conjugate flips the sign of the imaginary part. A number times its own conjugate is always real and equals the modulus squared, which is the identity every one of these operations rests on.
How the Complex Number Calculator works
Add, subtract, multiply and divide two complex numbers, with modulus, argument, conjugate and polar form. Division works by multiplying through by the conjugate, which clears the imaginary part from below the line.
Also known as: divide complex numbers · multiply imaginary numbers · convert to polar form · complex conjugate calculator
Frequently asked questions
How do you multiply complex numbers?
Expand as normal and use i² = −1: (a+bi)(c+di) becomes (ac−bd) + (ad+bc)i. The real part picks up a minus sign from the i² term, which is the step people miss.
How do you divide complex numbers?
Multiply top and bottom by the conjugate of the denominator. That makes the denominator real — it becomes the modulus squared — and the division becomes ordinary arithmetic.
What is polar form for?
Multiplying in polar form multiplies the moduli and adds the arguments, so multiplication becomes a rotation combined with a scaling. That is why complex numbers run through signal processing, AC circuit analysis and quantum mechanics.
What is the complex conjugate?
The same number with the sign of the imaginary part flipped. A number times its own conjugate is always real and equals the modulus squared — the identity that makes division work.
What is the modulus?
The distance from the origin in the complex plane, √(a² + b²). It is the complex equivalent of absolute value, and it behaves the same way under multiplication.
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