Numbers go from rational numbers to real numbers and then to complex numbers, and the expansion of numbers ends. Complex number is a point on the plane, and it is not enough to continue to expand to a point in space.

X+yi+zj, and then J 2 =-I, which is logical from I 2 =- 1, and then what is ij?

Hamilton's story, he thought for many years. I thought that j 2 =- 1, ij = 0, ij=k, ij=-ji, and it never works. There is a criterion: whether the module of the product of two ternary numbers is equal to the product of modules. It is said that one day, he went to a XX meeting, and he and his wife were walking together. His wife has been talking about trivial things, and he has been thinking about problems. I don't know whether the breeze by the river inspired him, or whether his wife's voice and a part of his brain shook. Hamilton took out a knife from his trouser pocket and carved the following formula on the bridge railing:

i^2=j^2=k^2=ijk=- 1

It may also be:

i^2=j^2=k^2=- 1

ij=k，jk=i，ki=j

ji=-k，kj=-I，ik=-j

Maybe neither. Anyway, he discovered "quaternion", that is, a number can be expressed as a+bi+cj+dk, which finally satisfies most laws (no multiplication exchange rate).

Starting with ternary numbers, I finally found quaternions.

Quaternion is a mathematical concept discovered by William Tam Hamilton (1805- 1865) in 1843. Quaternion multiplication does not conform to the commutative law, so it seems to break one of the most basic principles in scientific knowledge.

Specifically, quaternions are noncommutative extensions of complex numbers. If the set of quaternions is regarded as a multidimensional real number space, then quaternions represent a four-dimensional space, which is two-dimensional relative to complex numbers.

Quaternion is an example of division ring. Division rings and fields are similar except for commutative laws without multiplication. In particular, the associative law of multiplication still exists, and non-zero elements still have unique inverse elements.

Quaternions form a four-dimensional associative algebra (actually a division algebra) on real numbers, and contain complex numbers, but do not form associative algebras with complex numbers. Quaternions (as well as real numbers and complex numbers) are just finite-dimensional real number combination division algebras.

The noncommutativity of quaternions often leads to some unexpected results, for example, the n-order polynomial of quaternions can have more than n different roots.