[Math Lair] Quaternions

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Quaternions are a set of hypercomplex numbers that were discovered by William Rowan Hamilton. For many years, Hamilton had brooded over the fact that the multiplication of a complex number can be interpreted as the rotation of a plane, and wondered whether it would be possible to invent a new kind of number such that a rotation of three-dimensional space would have a simple interpretation in terms of multiplying such numbers. Hamilton called these numbers "triplets", which were to be represented by points in three-dimensional space. However, the problem was quite difficult.

One day in 1843, while walking with his wife along a canal in Dublin, Hamilton discovered how to multiply triplets. It dawned on Hamilton that a fourth dimension was required in order to calculate with triplets. He took out a penknife and carved on Brougham Bridge the key to the problem, which was:

i² = j² = k² = ijk = −1
The letters i, j, and k represent hypercomplex numbers that Hamilton called quaternions.

The important part of understanding quaternions is that the commutative law of multiplication does not hold for them. For example, ij = k, but ji = −k. Here is the multiplication table for quaternions:

Are there hypercomplex numbers after quaternions, with 8, 16, 32, ... dimensions? The answer is yes. For example, the next step up is octonions.

Sources used (see bibliography page for titles corresponding to numbers): 7.