Search for a tool
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.

Results

Complex Number Conjugate -

Tag(s) : Geometry

Share dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Complex Number Conjugate' tool, so feel free to write! Thank you!

# Complex Number Conjugate

## 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: 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 complex conjugate pair?

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

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

### 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 $i$ is $-i$

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

## Source code

dCode retains ownership of the online "Complex Number Conjugate" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Complex Number Conjugate" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Complex Number Conjugate" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, copy-paste, or API access for "Complex Number Conjugate" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!