Quaternions

Quaternions were discovered by William R. Hamilton in 1843 and also independently by Olinde Rodriguez. When Hamilton discovered them, he was so excited that he carved a formula onto the stone of Brougham Bridge: "i² = j² = k² = ijk = -1". He had previously been working on triples to represent 3-dimensional rotations. However, when his sons Archibald and Edward asked him "Papa, can you multiply triples?" he had to say "No, I can only add and subtract them." However, he found a way to multiply quaternions, which is why he used them.

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, i² = j² = k² = -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|| = (a² + b² + c² + d²)^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^-1X 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.

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