Oct 16, 2008

what are complex numbers

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.

Teacher:

“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”

4x-6=14

4x=20

4x-16=4

4(x-4)=4

“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.