Quaternions
Quaternions are an extension of the complex numbers. They are the sum of the four basic quaternions 1, i, j, and k. Addition is simple: simply add each part of the quaternion separately. For multiplication, these must be memorised: ij = k, jk = i, ki = j. However, multiplication is not commutative, as ji = -k, kj = -i, and ik = -j. Also, i^{2} = j^{2} = k^{2} = -1. However, mltiplication is associative. Thus (ij)k = i(jk). To show this, it can be evaluated: (ij)k = (k)k = -1 while i(k) = i(i) = -1. There are many, many interesting properties of quaternions. For example,
like complex numbers, the modulus or norm of X where X = a + bi + cj +
dk is ||X|| = (a^{2} + b^{2} + c^{2} + d^{2})^{1/2}.
There is also the conjugate X* = a - bi - cj - dk To define division we
must first define the reciprocal X = X*/||X||^{2}. However, there
are two ways to define division. X/Y = Y^{-1}X using left-division
and XY^{-1} using right-division. The vector product in 3 dimensions
(using i as the x-axis, j as the y-axis, and k as the z-axis) can be defined
using quaternions with no real part to be XxY = (XY - YX)/2. Related entries• Octonions• Complex Matters | ||
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