I was asked once why mathematicians invented things like imaginary and complex numbers even though they “don't exist”. When we see the equation x^2=-1 we are inclined to believe that no solution exists. This was my reply.
The Island of Even Numbers.
Far far away on an distant island exists a civilization that has never heard of odd numbers. Everything on this island comes in even numbers – always. The locals count 2,4,6,8... and so on. They know nothing about any of the odd numbers which we take for granted. The following took place in a math class in this exotic land.
“Some algebra problems have solutions.”
6x – 8 = 40
6x = 48
x = 8
“However, many do not.”
4x -6 = 14
4x = 20
but 4*4 =16
and 4*6 =24
but there is no number between 4 and 6
so there is no solution.
“The most fundamental theorem of this course is an obvious one: Given any number A there is no number you can multiply it by and get A back. Multiplication must always result in a different value unlike addition where we can add by zero.”
“There is no number x such that ax=a.”
“We can use this theorem to prove that certain problems do not have a solution”
“By our theorem we know that there is no number that can be equal to x-4 and therefore there can be no number that is equal to x.”
But one student asked the teacher:
“What if there was a number.... lets call it “1” such that A*1=A?”
The teacher dismissed the bizarre idea, but slowly the student found out the number was extremely useful and built an entire branch of math called the study of "odd" numbers.