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February 28th, 2012, 03:10 AM  #1 
Newbie Joined: Jan 2012 Posts: 4 Thanks: 0  is there a proof that i squared = 1
As I remember, I learned that if i was assumed to have this property then it became a very useful mathematical number. But is there a proof for this? If there is a point in the plane with x,and y coordinates that revolves anticlockwise about the centre at 90 degree steps starting at (cx , iy) c being any nonzero constant can it be proved that 1 must be the square of i? Do we need to use the angle subtended by the point (theta) and its sine or tangent values to make the proof? Can anyone tell me whether a proof was used to find i in the first place (i.e. historically) or did it just fit into place serendipitously as being really convenient (as I was led to believe)? 
February 28th, 2012, 06:23 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
Proof of what exactly? If someone, knowing that 1 doesn't have any real square root, invents its square roots, calling them i and i, and they consider them useful, what is there to prove?

February 28th, 2012, 09:16 AM  #3 
Newbie Joined: Jan 2012 Posts: 4 Thanks: 0  Re: is there a proof that i squared = 1
I mean is there a circumstance (like the one I outlined at the start) where we can have an equation with one unknown and which we can resolve the unknown to being i ? I mean if we were not to use i and we were to move the x/y point (1,0) to (0,1) I assume there should be a fairly simple function (1,0) that would be equal to (0,1) a function that would move the point in the plane to another point corresponding to a 90 degree anticlockwise rotation. But if this function (or perhaps a different function)was written with both (1,0) and (0,1) as knowns would there be an unknown in the equation which would have to be i? I can see I am getting bogged down/befuddled here but isn't it reasonable to assume that something which has real applications in the real world should be able to be pinned down somehow? Would it be fair to say that all the mathematical operations that i is used for can be achieved in other ways so that the use of i is really only a short cut in the calculation? I can live with that .But if the use of i is actually essential to these basic operations then I would like to know why they work and not just that they do work thanks for the edit btw and I hope my reply made sense. 
February 28th, 2012, 10:51 AM  #4 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: is there a proof that i squared = 1
What is "i"? (This isn't an ignorant question; rather, I feel it is to the heart of the matter. If you tell me  or already know  what "i" is, then I can give you an equation and demonstrate that "i" is a solution. If you give me an equation like x^2 + 1 = 0 and ask what the solutions are, I can answer and show that the solutions are sufficiently "wellbehaved" numbers.) 
February 28th, 2012, 11:34 AM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,580 Thanks: 1038  Re: is there a proof that i squared = 1
SQRT(4) = 2 SQRT(4) = 2i Simply an indicator! And hang on to your socks; from 10 to 9: SQRT(n) = x SQRT(n) = xi 
February 28th, 2012, 11:44 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
If (x, 0) represents the real number x, and (x, 0) represents x, one can regard multiplication by 1 as a rotation by 180° (anticlockwise or clockwise). This suggests inventing "imaginary" numbers, multiplying by which is represented by rotation through 90° (anticlockwise and clockwise). For obvious reasons, these numbers have 1 as their square. The symbol "i" was adopted as an abbreviation of "imaginary". It turns out that rotating through a general angle ? corresponds to multiplication by a "complex" number that is part real and part imaginary, cos(?) + i sin(?) or (cos(?), sin(?)) if one (arbitrarily) associates i with anticlockwise and i with clockwise.


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