The conjugate reflects a complex number in the real axis. Multiplying a number by its conjugate gives a real number, which is how division by complex numbers is done: multiply the top and bottom by the conjugate of the denominator.
How to use the Complex Conjugate Calculator
- Real part (a)
- Imaginary part (b)
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How it works
The real part stays the same and the imaginary part changes sign. The modulus is the distance from the origin; the argument of the conjugate is the negative of the original argument.
Formula
z̄ = a − bi
|z| = √(a² + b²)
z · z̄ = a² + b²
Worked example
Inputs — Real part (a): 3; Imaginary part (b): 4.
Results — Conjugate z̄ = 3 − 4i; Absolute value |z| = 5; z · z̄ = 25; Argument of z̄ = −53.13010235°.
Key terms
- Complex conjugate
- a − bi for z = a + bi.
- Modulus
- The distance |z| of z from 0.
Frequently asked questions
What is the conjugate of 3 + 4i?
3 − 4i.
What is the conjugate of a real number?
The number itself, because its imaginary part is 0.
Why multiply by the conjugate?
z · z̄ = a² + b² is real, so it removes i from a denominator.
Do roots of real polynomials come in conjugate pairs?
Yes – non-real roots of a polynomial with real coefficients always come in conjugate pairs.