# Quaternion Modulus of Conjugate

## Theorem

Let $z = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

Let $\overline z$ be the conjugate of $z$.

Let $\cmod z$ be the quaternion modulus of $z$.

Then:

$\cmod {\overline z} = \cmod z$

## Proof

 $\displaystyle \cmod z$ $=$ $\displaystyle a^2 + b^2 + c^2 + d^2$ Definition of Quaternion Modulus $\displaystyle \cmod {\overline z}$ $=$ $\displaystyle \cmod {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}$ Definition of Quaternion Conjugate $\displaystyle$ $=$ $\displaystyle a^2 + \paren {-b}^2 + \paren {-c}^2 + \paren {-d}^2$ Definition of Quaternion Modulus $\displaystyle$ $=$ $\displaystyle a^2 + b^2 + c^2 + d^2$

$\blacksquare$