Consider a set of equivalent objects. The elements a, b, and c are members of the set. From this we can construct an algebra assigning a binary operation "+" such that c = a + b.
For this to be a proper algebra, we must say that c is always a member of the set for all values of a and b.. This operation we call addition. An example of this set is the set of positive integers, 1,2,3,...
Furthermore, we can introduce additional operators -, *, and / for the operations of subtract, multiply and divide, which we will do now, one at a time.
Consider adding a second operator, "-", for subtraction. That is, when c = a - b, the set of positive integers is no longer suitable for our basis set of our algebra. We must allow the set to become the Integers, I, adding 0 and the negative integers.
Consider adding a third operation * for multiply. We find that the set of integers, positive, negative and zero are still suitable for our algebra.
However, when we add the operation "/", the integers are no longer satisfy the requirement that c, which is a/b, would not comply since 2/3 is not an integer. To have produced a proper algebra, our set must be the set of real numbers, R. What we have now is a division algebra across the real numbers.
There are certainly other division algebras. Consider a new set comprised of a pair of sets (R,R) with basis functions 1 and i, the square root of -1. This set, C, is the set of complex numbers. The complex numbers form a proper division algebra.
We can go further and define another division algebra by using a pair of complex numbers (C,C) and call this the set of quaternions, Q. We can also go on and define another set of pairs (Q,Q) We now have the set of octonions, O.
All four of the above sets are proper division algebras.
The astonishing thing is that according to Hurwitz's Theorem - there are no more division algebras!
The importance of these matters is that it has been said equations to describe the physical universe must be expressed in terms of the four division algebras described above.