Robby, who is interested in almost everything

[Robert Tulip]

I think I am being unfairly maligned here. The algebra is very simple. And I have misparaphrased my compatriot and famous mathematician Crocodile Dundee. He actually said 'that's not a knife, that's a knife'.

[Randall R. Young]

Robert,

Upon consideration, I thought your contribution was very much what Robby was asking for, locally.

But the gist of our previous interaction at Amazon was about how we go about coming up with these ideas. Yours was so concise and dense that it left no cracks into which we could wedge any explanations--a very much "take it or leave it" kind of proof, with little trace of the process which got you there.

[Robert Tulip]

it left no cracks

Yes true, but you did talk about deductive logic, which is inherently a 'take it or leave it' type of thing aiming for no cracks.

What I like about this proof is how we can test it with all types of number, natural, integer, irrational, real, imaginary, complex. I still don't know if an imaginary number can really be considered even or odd. Are odd and even just properties of integers?

[Randall R. Young]

Robert,

I still don't know if an imaginary number can really be considered even or odd. Are odd and even just properties of integers?

I guess you must mean complex numbers, since 2i is even, and 3i is odd, just as in the real integers. But the Gaussian integers (aka complex integers) have a couple of ways one might define parity. One is: Can you factor out a 2, and still have an integer? If so, we'd likely want to call a number of the form 2a+2bi "even" (where a, b ∈ Z). Oddly, that would make any dense finite set of Gaussians have three times as many "odds" as "evens"--although countably infinite sets of each would maintain identical cardinality.

But you could also do something like divide the Z-plane (aka, the discrete, integer part of C) into alternating quadrants, and get back a few useful symmetry properties of regular parity in that way, similar to the way we define odd & even functions. Neither of these ideas gets the simple sort of parity we know and love from Z.

Perhaps there are other ways, but I don't know about them, offhand.

[johnson1010]

Robby,

what i do is right click on an image that i want, click properties, copy the "address URL" paste it in your post, then click on the "full editor" button below the quick reply window.

Now you have to highlight the URL that you pasted in the window, find the "IMG" button on the top of the full editor reply window and press it.

That should put your image in a formating bracket which tells the program to grab the image from the url.

[lady of shallot]

Robby,

Greetings to the Lady of Shallot. Why would you want to be her? She died young. It's too late for either you or I to do that.

Hi Robby, obviously I don't want to be Lady of Shallot, I just love the imagery of the mirror cracking, the web flying or whatever and all for love of Lancelot; and her floating down the river.

I too don't care if you believe in God. I just don't want to be proselytized to. It is also my belief that there is a psychological difference between those who do believe and those who do not. No one else seems to talk about this or be interested in it but this is what is of the greatest interest to me in the divide.

Randell when I want to attach a photo, I click on "full edition" at the bottom of the reply box and there it says "choose file" which I click on and select my desired photo or image.

[johnson1010]

lady's way will probably be quicker, seeing as you can just post them from your own harddrive that way.

The way i outlined above was for grabbing images off the net, which is where i got the image of the esteemed Professor Farnsworth.

[Robert Tulip]

Robert,

I still don't know if an imaginary number can really be considered even or odd. Are odd and even just properties of integers?

I guess you must mean complex numbers, since 2i is even, and 3i is odd, just as in the real integers. But the Gaussian integers (aka complex integers) have a couple of ways one might define parity. One is: Can you factor out a 2, and still have an integer? If so, we'd likely want to call a number of the form 2a+2bi "even" (where a, b ∈ Z). Oddly, that would make any dense finite set of Gaussians have three times as many "odds" as "evens"--although countably infinite sets of each would maintain identical cardinality.

But you could also do something like divide the Z-plane (aka, the discrete, integer part of C) into alternating quadrants, and get back a few useful symmetry properties of regular parity in that way, similar to the way we define odd & even functions. Neither of these ideas gets the simple sort of parity we know and love from Z.

Perhaps there are other ways, but I don't know about them, offhand.

yeah, that is what I was wondering. I suppose you can have a definite number of imaginary things, except that a 'thing' is real, so I wonder about whether even and odd apply to seemingly indefinite numbers such as the imaginary. I find this interesting as a way to examine how our mundane use of words like real and imaginary maps on to the realm of number. Logically, it is clear that imaginary numbers can be even or odd, but then in some sense i is like any operator, changing the value of the natural numbers, such that squares and oddness have an intrinsic relation, for all n even where n is a complex formula rather than a simple number.

[Randall R. Young]

I suppose you can have a definite number of imaginary things, except that a 'thing' is real, so I wonder about whether even and odd apply to seemingly indefinite numbers such as the imaginary.

Personally, I don't think the set Im has any less 'reality' to it than Re does. In fact, only by including both of them together into C do get to have all these wonderful real world applications, and all these simplifications of the algebra that come with having closure under exponentiation. These extensions of the natural numbers are all about making a number system that is as closed as we can make it. We want every possible arithmetic operation to result in a number of the same type as we began with.

I also see nothing indefinite about them. I can only surmise that you don't have a use for C in your work. If you did, you would see them as exceptionally real, even hyper-real. I don't think it's possible to do filter design or anything with AC current without C, and with Im keeping track of the phase relations for us. I can't imagine Fourier would ever have thought up his transforms without them.