Find the conjugate of a complex number, and its product with the original. Enter the real and imaginary parts.
Usage Tip
To divide complex numbers, multiply top and bottom by the conjugate of the denominator — it makes the denominator real.
z × conjugate = a² + b² (a real number)
It is the key to dividing complex numbers and shows up throughout signal processing and quantum mechanics.
A number times its conjugate is always real, equal to the modulus squared.
The conjugate has the same modulus as the original.
The conjugate of a complex number a + bi is a – bi, the same real part with the sign of the imaginary part flipped. Multiplying a number by its conjugate always gives a real number, equal to a squared plus b squared, which is also the square of the number’s modulus.
That property is what makes conjugates useful for dividing complex numbers: multiply the top and bottom of the fraction by the conjugate of the denominator and the denominator becomes real. Conjugates also show up in polynomial roots, where non-real roots of a polynomial with real coefficients always come in conjugate pairs.