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Complex Number Conjugate

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.

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Complex Number Conjugate -

Tag(s) : Geometry

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# Complex Number Conjugate

## Complex Conjugate Calculator

### What is the conjugate of a complex number? (Definition)

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.

In other words, the conjugate of a complex is the number with the same real part but with opposite imaginary part.

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

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

From the algebraic form of a complex number $z = a+ib$, the conjugate is calculated $\overline{z} = a-ib$.

Example: Determine the conjugate of $z = 1+i$ is to calculate $\overline{z} = 1-i$

In other words, to find the conjugate of a complex number, take that same complex number but with the opposite (minus sign) of its imaginary part (containing $i$).

### What is a complex conjugate pair?

The set of 2 elements: a complex number $z$ and its conjugate $\overline{z}$, form a pair of conjugates.

### 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:

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

### What is the product of a complex number and its conjugate?

The multiplication of a complex number $z = a + ib$ and its conjugate $\overline{z} = a-ib$ gives: $$z \times \overline{z} = a^2+b^2$$

This number is a real number (no imaginary part $i$) and strictly positive (addition of 2 squares values necessarily positive)

### What is the conjugate of i?

The conjugate of the number $i$ is the number $-i$

### How to calculate the conjugate of a real number (without i)?

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

Example: $\overline{1 + 0 \times i} = 1$

### Does the conjugate change the real part of a complex number?

The real part of a complex number always remains unchanged when calculating the conjugate. The real part never changes. $\Re(z) = \Re(\overline{z})$

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