Quaternions

In mathematics, a number system, or set of numbers, can be extended to form a more comprehensive number system. The following list describes one nested sequence of sets, where each number system extends the previous one. Wikipedia links are included with more information about the topics.

• The Integers are a proper subset of the Rational Numbers.
• The Rational Numbers are a proper subset of the Real Numbers.
• The Real Numbers are a proper subset of the Complex Numbers.
• The Complex Numbers are a proper subset of the Quaternions.

Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843. There are many uses for quaternions in pure mathematics and applied mathematics, including three-dimensional computer graphics.

Quaternions are generally represented in the form

a + bi + cj + dk

where a, b, c, and d are real numbers; and i, j, and k are the basic quaternions.

Multiplication of quaternions is not commutative in some cases. These are the results when basic quaternions are multiplied.

ii = jj = kk = ijk = -1

ij = -ji = k

jk = -kj = i

ki = -ik = j

The diagram depicts a Cayley Q8 graph showing the six cycles of multiplication by i, j and k.