Question

What are quaternions?

Answer 1

A kind of number for which multiplication is not generally commutative.

Explanation:

Real numbers (`RR`) can be represented by a line - a one dimensional space.

Complex numbers (`CC`) can be represented by a plane - a two dimensional space.

Quaternions (H) can be represented by a four dimensional space.

In ordinary arithmetic numbers satisfy the following rules:

Addition

Identity: `EE 0 : AA a : a + 0 = 0 + a = a`

Inverse: `AA a EE (-a) : a + (-a) = (-a) + a = 0`

Associativity: `AA a, b, c : (a + b) + c = a + (b + c)`

Commutativity: `AA a, b : a + b = b + a`

Multiplication

Identity: `EE 1 : AA a : a*1 = 1*a = a`

Inverse of non-zero: `AA a != 0 EE 1/a : a * 1/a = 1/a * a = 1`

Associativity: `AA a, b, c : (a*b)*c = a*(b*c)`

Commutativity: `color(red)(AA a, b : a*b = b*a)`

Together

Distributivity: `{ (a*(b+c) = (a*b) + (a*c)), ((a+b)*c = (a*c)+(b*c)) :}`

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These rules work for the set of rational numbers `QQ`, the set of Real numbers `RR` and the Complex numbers `CC` and define what is called a field - a set equipped with operations of addition and multiplication satisfying these rules.

Quaternions (H) are what is called a skew field or associative division algebra - a set equipped with operations of addition and multiplication satisfying all of these conditions except the commutativity of multiplication.

Being also a `4` dimensional vector space over the Reals, they are the largest associative division algebra over the Reals, the only other two being `RR` and `CC`.

Apart from the Real axis, the units on the other three axes are called `i`, `j` and `k`. They are all square roots of `-1`.

These three imaginary units satisfy the following conditions:

`ij = k`

`jk = i`

`ki = j`

`ji = -k`

`kj = -i`

`ik = -j`

Quaternions can be represented by `2xx2` matrices with Complex values or by `4xx4` matrices with Real values.

They have applications in mechanics and theoretical physics.

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Footnote

Notice that I said associative division algebra. Beyond the Quaternions are the even stranger Octonions that drop the requirement that multiplication be associative.