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Less imaginary imaginary numbers

Some imagination may be required
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The definition of the imaginary unit is

i^2 = - r --- (1)

where r is the real unit (=1). Although I'm pretty comfortable working with imaginary numbers, I never really understood their necessity other than as a mathematical "fix" for negative roots. It was also never clear to me what made real and imaginary numbers particular real or imaginary. Would we even notice if we suddenly started using imaginary numbers for everyday counting? I recently became irked by the fact that you couldn't take the "complement" of (1) to give an identity for the real unit:

r^2 = - i --- (2)

which is false. This lack of symmetry is troublesome to me. After a bit of thought I realised that this problem can be resolved by re-writing (1) and (2) as

i^2 = - r^2 --- (1')

r^2 = - i^2 --- (2')

(1') and (2') are both true statements. The classical definition of i is preserved whilst having the advantage that r is similarly defined with pleasing symmetry. Adding them together

r^2 + i^2 = - (r^2 + i^2) --- (3)

Ignoring trivial solutions (r = i = 0) then we're left with

r^2 + i^2 = 0 --- (4)

I think that (4) is a better identity than (1) for the following reasons:

- It shows that the set of real numbers and the set of imaginary numbers are completely tied up with each other and are symmetric and interchangeable. Neither is more special, real, or imaginary than the other. Whether we choose to use real or imaginary numbers as everyday counting units is a completely arbitrary decision.

- It gives a geometric definition of the real-imaginary number space, describing a unit circle with zero radius.

- It can be reduced to the more familiar form (1) with trivial ease.

- It's more elegant.

I'd be astounded if I've stumbled upon some new mathematics here, but I can't find it written down in any of the obvious places. If someone can point me to a text that includes this definition I'd be grateful. Nevertheless I maintain that (4) is a superior way of defining and teaching imaginary numbers.

EnochLives, Dec 08 2014

"Imaginary Numbers" http://www.bbc.co.uk/programmes/b00tt6b2
BBC Radio 4, 'In Our Time' podcast. [DrBob, Dec 08 2014]

[link]






       Issac Asimov once got into an argument with a philosopher over imaginary numbers. The problem you are having actually relates to the meaning of "imaginary", not the meaning of "number". That is, a so-called "imaginary" number is just as real as a fraction. Consider the number "one half". Can you point to an actual all-by-itself thing with a value of "one half"? Not without including OTHER defined terms! A piece of a cake is always a whole piece of a cake, regardless of whether or not it is also half of the whole cake, see?
Vernon, Dec 08 2014
  

       Hi [Vernon] that's an interesting argument and is transferrable onto other numbers as well - imagine a universe consisting of only one thing - there would be no way to distinguish one from all, so, in order to make sense of one, you kind of need at least, another one, and that makes two. If "one" is contextual (i.e. it relies on a context of other things to understand), then is there anything that isn't?
zen_tom, Dec 08 2014
  

       [zen tom], if you have "one thing" and can imagine "no thing", then right there you are talking about a total of two situations, if not actually things....
Vernon, Dec 08 2014
  

       Although "one cup of coffee" and "no cup of coffee" isn't the same as "two cups of coffee" (q.v).
hippo, Dec 08 2014
  

       [hippo], my point is, even with just one thing you can need the concept of "two".
Vernon, Dec 08 2014
  

       I can't imagine no cup of coffee.
RayfordSteele, Dec 08 2014
  

       They're not symmetrical though, since r^2 > 0 and i^2 < 0   

       Unless you are going to argue that < and > and + and - are also arbitrary.
pocmloc, Dec 08 2014
  

       //a universe consisting of only one thing//

We are heading perilously close to Douglas Adams territory now, aren't we! "Any finite number divided by infinity..." etc.

//"one cup of coffee" and "no cup of coffee" isn't the same as "two cups of coffee" //

If someone were to invent a drinking vessel with the bowl separated into two chambers, both containing coffee, then one cup of coffee would also be two cups of coffee. Furthermore, as it would not be a standard cup it would therefore require a new name. So, once such a vessel was invented, the period between it ceasing to be a cup and starting to be whatever it's new designation was would necessarily render it both one, two & no cups of coffee.
DrBob, Dec 08 2014
  

       What does --- mean?
RayfordSteele, Dec 08 2014
  

       In Google type "half a cup of coffee" and click images and you will see many half coffee cups. Four of these would equal 2 cups of coffee. There are also cups that have side car for the tea bag.   

       --- means three fish bones. +
popbottle, Dec 09 2014
  

       //I can't imagine no cup of coffee.// But can you imagine an imaginary cup of coffee?   

       There's also the problem that
  

       i^2 = -r , and
(-i)^2 = -r
  

       If we discover that we picked wrongly, will we have to re-write all the textbooks and papers, substituting -i for i, and vice versa?
spidermother, Dec 09 2014
  

       I would suggest it is impossible not to imagine an imaginary cup of coffee. But how do you decide which i is + and which is -?
pocmloc, Dec 09 2014
  

       I thought in the imaginary set, it's a special quantity, two equal sides, that square to give -1.
wjt, Dec 16 2014
  

       Actually, I think the imaginary set is a perspective.   

       My logic goes like this. Take a Rubic's cube face, 9 one unit square areas. Remove the centre area twice. You are left with a negative one unit area in the centre surrounded by 8 one unit areas. Looking from the imaginary perspective (out towards the void rather than towards the solid) this negative one area unit has a side root of 1.
wjt, Dec 18 2014
  

       Made me take a Tylenol [+]
Voice, Dec 21 2014
  

       That's approaching the primary unit of a Menger Sponge, [wjt]
8th of 7, Dec 21 2014
  

       //But how do you decide which i is + and which is -?//   

       Good question. I actually find negative numbers at least as conceptually difficult as imaginary ones. If you add a dimension then it becomes straight-forward; for example it's easy to interpret +£1 as a credit and -£1 as a debt, but without the context of a dimension I struggle to know what -1 really means. I think the "root" of the problem is that numbers are not physical things and my brain doesn't like that. Apparently Pascal thought they were bollocks too.
EnochLives, Dec 21 2014
  

       From a visceral perception point of view, whole numbers >4 are a problem.
pocmloc, Dec 21 2014
  

       [bigsleep] That doesn't seem right. y and z stick out . i is just a clever directional explanation which, for a circle and sphere, changes. i is about the meaning of - .
wjt, Dec 22 2014
  

       You could have a 6, whole number, axis coordinate system - left , right, up, down, in and out, all starting at 0 going to infinity. this would have no negative roots but would make relating the quadrants of a circle very difficult.   

       If -1 = 1 is an accepted definition the negative roots are -1,1 -1,1 and + roots are 1,1 -1,-1 . A negative orange is exactly the same as a positive orange. They are both oranges.
wjt, Dec 22 2014
  
      
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