Why is i^2 = -1?

[Kyteland]

So I wrote this "proof" up on my board at work showing that 1=-1. It is a classic math fallacy, but it looks good. The trick is knowing which step the fallacy is in.

I told someone where it was (when you say sqrt(x/y) = sqrt(x)/sqrt(y)) and explained why, but they said "no it is where you claim i^2 = -1." I laughed at him but then realized I didn't know how to explain it to him. It's just a symbol with that property.

My calculus book has this:

A complex number can be represented by an expression of the form a+b*i, where a and b are real numbers and i is a symbol with the property that i^2 = -1

http://mathworld.wolfram.com/i.html also lists it as a definition.

Anyone have a better way to explain it, or some kind of proof? Although I have no idea how you would prove it, since it is a basic definition. Kind of like rigorously prove (-1)*(-1) = 1.

 

[gopunk]

you don't claim it, that is the definition of i. i is defined as the square root of -1

 

[dighn]

it's by definition

 

[hdeck]

^ what they said ^

and your co-worker is a moron =)

 

[her209]

i=sqrt(-1)

 

[NogginBoink]

You can't prove it.

Prove that the word "chair" means that thing that your fanny is in right now. You can't prove it. The word "chair," by definition, means that thing your fanny is in.

That's what "by definition" is all about.

 

[Kyteland]

Cool, that's what I told him before. That's just how it's defined. He didn't believe me... 🙁

 

[dighn]

Originally posted by: Kyteland
Cool, that's what I told him before. That's just how it's defined. He didn't believe me... 🙁

he is suck, show him a text book

 

[Zugzwang152]

it's an imaginary number, i forgot the usefullness it actually serves.

 

[MegaloManiaK]

Linky">http://www.math.toronto.edu/mathnet/answers/imagexist.html[/S</a>

i is a convention to represent the sqrt of -1, thats why i^2=-1, it is just the definition of i. Now to see how i works is a differnt matter.

as with most things mathmatical it can't be explained verbaly. The math explanation is

(0,1)(0,1) = ( (0)(0) - (1)(1), (0)(1)+(1)(0) ) = (-1,0).

(from the link)

when you square 0,1 you get -1, so the square root of -1 = 0,1 which is by convention given the name i.


The site does a good job explaining the different number systems and how some things dont exist in other number systems, i.e. 1/2 does not exist in natural numbers. If you change from a single dimension coordinate system to a 2 dimensions you can see how this shows up. 0,1 in a one dimensional coordinate system is 0 and therefore does not exist. But using a diff system you can see that 0,1 is differnet than 0,0 and therefore contains information that is lost in 1 dimmension. The same goes for 0,0,1 is not the same as 0,0,0. Im sure theres got to be someone in this forum that knows it better than me, but thats how ive understood it in the past.


Edit*

The proof works (even though it is incorrect) because when you square -1 in 1 dimmension you get back 1. But when you square 1,0 you maintain the second dimmension and get back i, which is 0,-1. You get the same thing if you take a float number and store it as an integer, 1.4 becomes 1.