Complex Conjugate Calculator

Work out complex conjugate instantly with clear inputs, formula shown and shareable results.

About Complex Conjugate Calculator

The conjugate flips the sign of the imaginary part, reflecting the number across the real axis. Multiplying a number by its conjugate always gives the real value |z|², which is why conjugates clear denominators.

Formula

Conjugate

conj(a + bi) = a − bi, z·z̄ = a² + b² = |z|²

Worked example

With the default values below, Complex Conjugate Calculator returns Conjugate z̄ of 7 + 24i. Change any input above and every figure here is recalculated in your browser.

Inputs used

Real part: 7
Imaginary part: -24

Results

Conjugate z̄: 7 + 24i
z · z̄ = |z|²: 625.00000000
Modulus |z|: 25.00000000
z + z̄ = 2·Re(z): 14.000000

Step by step

1Values used
2Conjugate
3Conjugate z̄= 7 + 24i
4z · z̄ = |z|²= 625.00000000
5Modulus |z|= 25.00000000
6z + z̄ = 2·Re(z)= 14.000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Conjugate of 7 − 24i?

7 + 24i, and their product is 49 + 576 = 625 = 25².

Why do polynomial roots come in conjugate pairs?

For real coefficients, conjugating the whole equation shows that if z is a root then so is z̄.