Quaternion

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

The eight-dimensional octonions come after the quaternions.

Definition

A quaternion is a number written like this:

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

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

The square of every imaginary unit is equal to -1

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

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

$ij=-ji=k$ $jk=-kj=i$ $ki=-ik=j$

Putting all those together,

$ijk=-1$

↑ "On Quaternions; or on a new System of Imaginaries in Algebra". Letter to John T. Graves. 17 October 1843.
↑ Rozenfelʹd, Boris Abramovich (1988). The history of non-euclidean geometry: Evolution of the concept of a geometric space. Springer. p. 385. ISBN 9780387964584.

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v t e Number systems
Commonly used systems	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ )
Hypercomplex numbers	Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ ) Sedenions ( $\mathbb {S}$ ) Trigintaduonions ( $\mathbb {T}$ )

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