Quaternion

In mathematics, the quaternion number system (represented using the symbol $ℍ$ ) 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.

The eight-dimensional octonions come after the quaternions.

Definition

A quaternion is a number written like this:

$a + b 𝐢 + c 𝐣 + d 𝐤$

where $a, b, c, d$ 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.

$i j = - j i = k$ $j k = - k j = i$ $k i = - i k = j$

Putting all those together,

$i j k = - 1$

↑ "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 ( $ℕ$ ) Integers ( $ℤ$ ) Rational numbers ( $ℚ$ ) Real numbers ( $ℝ$ ) Complex numbers ( $ℂ$ )
Hypercomplex numbers	Quaternions ( $ℍ$ ) Octonions ( $𝕆$ ) Sedenions ( $𝕊$ ) Trigintaduonions ( $𝕋$ )

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