Question: how does i squared = negative 1

Cesar Lopez-Monsalvo answered on 11 Jun 2011:

Ok, this will be slightly elaborated and a bit long.

In mathematics, there are many kind of numbers. They all have a very good reason to be there and usually they have an interesting story behind. The type of numbers you are probably most familiar with are the REAL numbers. Those are the “every day” type of numbers like 1, 2, 3.14, -43.4555555…, 1/2, 0…basically all of them. For many centuries, mathematicians were quite happy with them. They satisfied pretty much all their needs, and the needs of a mathematician normally involve proving things or solving equations. So in particular, all this numbers are SOLUTIONS to many equations. Think of the following examples:

(a) If x – 1 = 0, then we know that x has to be exactly 1 to get the right answer ( 1-1 = 0 ).

A more complicated one would be

(b) If x^2 – 1 = 0 (Here x^2 means x squared) then we know that x can be either -1 or 1, since 1^2 -1 = 1 – 1 = 0, and (-1)^2 – 1 = 1 – 1 = 0 (again 1^2 means 1 squared and (-1)^2 means negative 1 squared which is just 1).

If you are still here, then we can go one step forward and try this one

(c) What is the solution (the value of x) of x^2 + 1 = 0?

To find out, think of how did you “know” the answers to the previous two cases. If x – 1 = 0, then you ADDED 1 to each side of the equal sign to get

x – 1 + 1 = 0 + 1,

which immediately gives you x = 1. In the second case you do exactly the same, so you have

x^2 – 1 + 1 = 0 + 1,

which gives x^2 = 1. Now here comes the real answer to your question. The square of ANY real number is ALWAYS a positive number, that is, greater than zero. Try it! What happens then in our third case? Following the same method as before we find out that

x^2 = -1,

that is, the square of a number which is negative!!! Surely that cannot happen! Surely we did something wrong! or, didn’t we? That is what puzzled mathematicians for quite a while. The answer is that there is nothing wrong in the method but that the REAL numbers are, in a sense, INCOMPLETE. They do not provide ALL the solutions to problems like the one we have. There is no REAL number whose square will give you negative 1. So why not “extending” the numbers you use so you have answers to every equation like the one we have? That is exactly what happened and “i” (i for Imaginary…perhaps) was DEFINED as the solution of the equation x^2 + 1 = 0, that is,

i^2 = -1 (i squared equals negative 1).

This literally ADDED an extra dimension to the real numbers and an extraordinary, powerful and useful type of numbers were born: the COMPLEX numbers. There is a lot to say about them but for the time I think this is all.