h a l f b a k e r yThe word "How?" springs to mind at this point.
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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.
"Imaginary Numbers"
http://www.bbc.co.uk/programmes/b00tt6b2
BBC Radio 4, 'In Our Time' podcast. [DrBob, Dec 08 2014]
[link]
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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? |
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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? |
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[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.... |
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Although "one cup of coffee" and "no cup of coffee" isn't the same as "two cups of coffee" (q.v). |
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[hippo], my point is, even with just one thing you can need
the concept of "two". |
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I can't imagine no cup of coffee. |
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They're not symmetrical though, since r^2 > 0 and i^2 < 0 |
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Unless you are going to argue that < and > and + and - are also arbitrary. |
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//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. |
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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. |
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--- means three fish bones. + |
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//I can't imagine no cup of coffee.// But can you
imagine an imaginary cup of coffee? |
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There's also the problem that
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i^2 = -r , and
(-i)^2 = -r |
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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? |
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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 -? |
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I thought in the imaginary set, it's a special quantity, two equal sides, that square to give -1. |
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Actually, I think the imaginary set is a perspective. |
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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. |
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Made me take a Tylenol [+] |
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That's approaching the primary unit of a Menger Sponge, [wjt] |
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//But how do you decide which i is + and which is -?// |
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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. |
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From a visceral perception point of view, whole numbers >4 are a problem. |
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[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
- . |
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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. |
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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. |
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