# Basic properties of quaternions

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### Imaginary part

The imaginary units of a quaternion are defined as follows:

Unlike the multiplication of real or complex numbers, the multiplication is not commutative:

### Conjugate

The conjugate of a quaternion is obtained by negating the imaginary or vector part, respectively:

Multiplying a quaternion with its conjugate like described in the subarticle about multiplication of quaternions consequently leads to

The conjugate can be used to easily determine the real part and the imaginary part of a quaternion:

The most important properties of the conjugate are:

### Norm

The norm of a quaternion corresponds to its euclidean length in four-dimensional space and is computed as follows:

It holds:

### Inverse

The inverse of a quaternion is defined as follows:

This can be proven by multiplying a quaternion with its inverse using the definition of the norm: