You're so special
Numbers like pi and e are usually regarded as being special.
Are there any numbers that are not special?
Prove your assertion.
Are there any numbers that are not special?
Prove your assertion.
Labels: mathschallenge, SharedPuzzle
posted by Chris at
9:38 AM
18 Comments:
42
Pi, strictly speaking, is not a number. It's an expresion, or ratio of a circles curcumfirance divided by it's diameter.
It has an approximate value that is not finite.
...which makes Pi a Number, which express this ratio ^^
So after your arguing all numbers that doesn't have finite value would be like pi. which are all real number except the rational ones, which are infinitely more than the rationals (which are by themselves infinite).
Anonymous - so you're saying 3/4 isn't number (because it's a ratio)? And none of the even integers are numbers because they can be written as x/2?
pi actually turns up all over the place, often from paths that have no obvious connection with circles.
pi is an exact value. I think you're vaguely aware that it is not possible to write it down as an exact digit sequence - you got that bit right.
tha b:H, I assume you're talking to Anonymous - he/she won't understand what you're saying.
Take my first bit of cruelty to the limit, leaves us with prime numbers only - I expect Anonymous might have a problem with those too.
I think a number could be "special" for at least two reasons. "Special" is probably how you think about them. Maybe
1/3 is special because you can buy three for one! "Special" is how we think them.
The first could be special because you like them, or maybe they are "lucky" for you.
The second is probably "special" to you because you use them more often than others. If you don't know what pi or e is, and never need, or use them. Then they are not special.
To me, 7 and 12 are the most special numbers... Don't ask me why though.
[not the same anonymous by the way]
i'd figure that there are some numbers that are 'special'... and then there are ones that are just irrational :P
I think e and pi are special because e^(i*pi)+1=0 you take two completely irrational numbers, take them out of there context (in a manner they were found for totally different problems) plunge them together in some strange way and can describe the whole set of complex numbers. Kinda freaky, kinda special! ^^
t:b:H, Aug 20, 1:15am - I'm so pleased that you wrote the identity
in that form. I think the e^(i*pi) = -1 version less attractive.
The problem was posted tongue-in-cheek. I'm slightly disappointed
that it wasn't treated as a "mathschallenge" though.
Assuming that there are some non-special numbers. Define a set
whose elements are the absolute values of the non-special numbers.
The set must contain a lowest member. That makes that number special after all, it must be removed from the set. Now we have a new smallest member etc.
So the set can be emptied and so all numbers are special.
Purists will say this is not a proof - they're very right.
t:b:H. Even more amazing is that e and pi are not merely irrational, they're transcendental.
More trivia, e^pi is transcendental. It is not known if pi^e is transcendental.
I'm pretty sure you already knew that. Greetz.
TO cap it off, the proof is a hoax. It is not possible to define a set in such a manner. Bertrand Russell saw to that when working out how to the avoid the paradoxes present in the older formulation of set theory.
Sets must be well defined. The non-special number set isn't well-defined.
All you people who say pi isn't a number are wrong. Pi is a number. Secondly the only numbers that arn't special are non real numbers. P.S. 14 is a very very very very very very very very very very very very very very special number. TRUST ME!
Anonymous: I'm intrigued by your assertion that the non-reals aren't special, please tell me why. Have you found a flaw in my "proof" that let's you say that? I'd love to see it. Seriously, I love that sort of thing.
All: Also, I had just realised (that's why I'm back here) that my "proof" wan't quite as slick as it could have been. The first lowest non-special number being special after all was a contradiction. So there cannot be such a set.
2, and 22.
2x2=4
2+2=4
2^2+4
and 22 is special for a different reason...
You can add 2↑2 = 4 to that list. ↑ is the Knuth up arrow.
I'm really goofing off. I forgot to say, the set isn't properly defined because "special number" isn't properly defined. If we had defined "non-special" by a list of well-defined properties (positively or negatively) then each non-special number would have been well-defined and then the set would have been too.
non real numbers arn't real so they are not special. For instance: what is the square root of -5... There is no way to multiply two similar numbers in order to get -5 unless its a non real number, therefore non real numbers are not special. Non real numbers have never been used to help human kind in a big way, at least to my knowledge.
I would like to hear from all of you why 14 is such a special number. We all know that it is, but why?
Hi Anonymous before last. Electronic engineers would say that the non-reals (i.e. the complex numbers) have been of enormous benefit. With them they can turn linear differential equations into algebraic equations and so make their calculations much easier.
When dealing with ac circuits with angular frequency w, the equivalence is d/dt = i*w. d/dt is the time rate of change differential operator, and i is sqrt(-1).
I'd happily explain how the above comes about, but at present feel that it would inspire incredibly stupid comments from a couple of people on this site. I'll explain (the iw stuff) further if you request it.
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