# Definition:Quaternion/Mistake

## Source Work

Chapter $1$: Introductory Concepts
$1$. Basic Building Blocks

## Mistake

Every quaternion can be represented in the form
$q = q_0 1 + q_1 \lambda_1 + q_2 \lambda_2 + q_3 \lambda_3$
where the $q_i \, \paren {i = 0, 1, 2, 3}$ are real numbers and the $\lambda_1$ have multiplicative properties defined by
 $\displaystyle \lambda_0 \lambda_i$ $=$ $\displaystyle \lambda_i \lambda_0 = \lambda_i$ $\displaystyle i = 0, 1, 2, 3$ $\displaystyle \lambda_i \lambda_i$ $=$ $\displaystyle -\lambda_0$ $\displaystyle \lambda_1 \lambda_2$ $=$ $\displaystyle -\lambda_2 \lambda_1 = \lambda_3$ $\displaystyle \lambda_2 \lambda_3$ $=$ $\displaystyle -\lambda_3 \lambda_2 = \lambda_1$ $\displaystyle \lambda_3 \lambda_1$ $=$ $\displaystyle -\lambda_1 \lambda_3 = \lambda_2$

## Correction

The representation of $q$ should read:

$q = q_0 \lambda_0 + q_1 \lambda_1 + q_2 \lambda_2 + q_3 \lambda_3$

and the line:

$\lambda_i \lambda_i = -\lambda_0$

holds only for $i = 1, 2, 3$ because:

$\lambda_0 \lambda_0 = \lambda_0$

One suspects that the author conflated this presentation with one where $\lambda_0$ has been identified with the number $1$, and only partially has this been translated into its current form.