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# Complex Number Conjugate

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Tool for calculating the value of the conjugate of a complex number. The conjugate of a complex number $$z$$ is written $$\overline{z}$$ or $$z^*$$ and is formed of the same real part with an opposite imaginary part.
Summary

## Complex Conjugate Calculator

### How to calculate the conjugate of a complex number?

The conjugate of a complex number $$z = a+ib$$ is noted with a bar $$\overline{z}$$ (or sometimes with a star $$z^*$$) and is equal to $$\overline{z} = a-ib$$ with $$a = \Re{z}$$ the real part and $$b = \Im{z}$$ the imaginary part.

Example: Consider $$z = 1 + i$$ then the conjugate is $$\overline{z} = 1-i$$

On a complex plane, the points $$z$$ and $$\overline{z}$$ are symmetrical (symmetry with respect to the abscissa x-axis), the 2 points are conjugate pairs.

### What are the properties of conjugates?

Using the complex numbers $$z, z_1, z_2$$, the conjugate has the following properties:

$$\overline{z_1+z_2} = \overline{z_1} + \overline{z_2}$$

$$\overline{z_1 \cdot z_2} = \overline{z_1} \times \overline{z_2}$$

$$\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}} \iff z_2 \neq 0$$

A number without an imaginary part is equal to its conjugate:

$$\Im(z) = 0 \iff \overline{z} = z$$

The modulus of a complex number and its conjugate are equal:

$$|\overline{z}|=|z|$$

### How to calculate the conjugate of a real number?

The conjugate $$\overline{a}$$ of a real number $$a$$ is the number $$a$$ itself: $$a=a+0i=a-0i=\overline{a}$$

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