Quaternion

In mathematics, the quaternion number system (represented using the symbol ${\displaystyle \mathbb {H} }$) extends the complex numbers into four dimensions. They were first described by Irish mathematician William Rowan Hamilton in 1843.^[1][2] They are often used in computer graphics to compute 3-dimensional rotations.

Quaternion multiplication table

	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

Cayley Q8 graph showing the 6 cycles of multiplication by i, j and k. (In the SVG file, hover over or click a cycle to highlight it.)

The eight-dimensional octonions come after the quaternions.

Definition

A quanternion is a number written like this:

${\displaystyle a+b\mathbf {i} +c\mathbf {j} +d\mathbf {k} }$

where ${\textstyle a,b,c,d}$ are real numbers, and ${\displaystyle i,j,k}$ are the symbols called imaginary units.

Multiplication

The square of every imaginary unit is equal to -1

${\displaystyle i^{2}=j^{2}=k^{2}=-1}$

When you multiply, changing the order of the numbers can give a different answer.

${\displaystyle ij=-ji=k}$ ${\displaystyle jk=-kj=i}$ ${\displaystyle ki=-ik=j}$

Putting all those together,

${\displaystyle ijk=-1}$