The Complex Conjugate Calculator provided by Hesapstan finds the conjugate of a complex number written as z = a + bi. Only the sign of the imaginary part changes, and the result also shows z × z̄ and |z|².
What this calculator does
Enter the real part a and imaginary coefficient b. The calculator writes the original number and its conjugate a − bi.
It also calculates the product of the number and its conjugate as a² + b², which is the modulus squared.
Limitations
This tool works with numeric a and b values only. It does not solve symbolic expressions, polar-form conversion, or broader complex-number operations.
Frequently Asked Questions
What is a complex conjugate?
The conjugate of a + bi is a − bi. The real part stays the same, and the imaginary part changes sign.
What if b is negative?
The sign still flips. For example, the conjugate of 3 − 2i is 3 + 2i.
Why is z × z̄ equal to a² + b²?
When a complex number is multiplied by its conjugate, the imaginary terms cancel and the real value a² + b² remains.