In the set of real numbers the square root of a minus number is a problem because it does not exist.
However, √-1can be used to generate algebras which use higher dimensional numbers, or to put it the other way round, algebras with higher dimensional numbers may have one or more solutions to the square root of a minus number.
By 'higher dimensional numbers' we mean that each element in the algebra, each number, may contain multiple real values similar to vectors. However there are a lot of possible algebras using multidimensional elements, when we are talking about real normed algebras then we define multiplication so that:
- Inverse multiplication (division) always exists, i.e. we can always find the solution to a/b
- norms are preserved by multiplication |a * b| = |a| |b|.
Where |a| = the distance from origin.
It turns out that these requirements can only be met by algebras which have 1,2,4 or 8 dimensional elements. These 4 algebras, which satisfy this condition, have the following properties:
| dimension | √-1 exists | Multiplication commutative | Multiplication associative | |
| Real Numbers R | 1 | no | yes | yes |
| Complex Numbers C | 2 | yes | yes | yes |
| Quaternions Q | 4 | yes | no | yes |
| Octonions O | 8 | yes | no | no |
Complex Numbers contain a copy of the real numbers plus a copy of the real numbers multiplied by the square root of -1.
Quaternions contain a copy of the Complex Numbers plus another copy of the real numbers multiplied by another square root of -1 (going via a different dimension at 90 degrees to the first).
Octonions contain a copy of the Quaternions and double up again.
