20 Jun '04 08:19>
From memory this is (e to the power -i * pi) +1 = 0 (i may not have got that exactly right). This has been a source of amazement to me. Does anyone have any light to shed on the subject?
Originally posted by rspoddar82But is it really so simple? It seems to me that all you (and TheMaster37) have done is put forward one or more mathematical expressions as an explanation of another mathematical expression. but what does this EXPLAIN, really? equating two mathematical expressions is just to say that they are tautologous, as Wittgenstein would have said, i think. now i would ask: isn't it amazing that exp(i*x) = cos x + i sin x?
This is simply because,
exp(i*x) = cos x + i sin x.
Originally posted by dfm65Well, it can be shown easily by looking at the power series for each of those functions, and the amazing result it gives is a reflection of the unifying power of power series. In fact, no less a figure than Karl Weierstrass said, effectively, that power series are the answer to life, the universe and everything.
But is it really so simple? It seems to me that all you (and TheMaster37) have done is put forward one or more mathematical expressions as an explanation of another mathematical expression. but what does this EXPLAIN, really? equating two mathematical expressions is just to say that they are tautologous, as Wittgenstein would have said, i think. now i would as ...[text shortened]... ses seem unrelated - are actually related in such a simple way as expressed in Euler's theorem?
Originally posted by royalchickenby 'these functions' i take it you mean e to the power x, sin x and cos x
Well, it can be shown easily by looking at the power series for each of those functions, and the amazing result it gives is a reflection of the unifying power of power series. In fact, no less a figure than Karl Weierstrass said, effectively, that power series are the answer to life, the universe and everything.
Originally posted by dfm65No problem, and you have a good point. I don't think it's quite the same thing though; at some point it's more useful to consider the power series as defining the exponential and circular functions and ''forget'' about the conventional definition.
by 'these functions' i take it you mean e to the power x, sin x and cos x
but, again, equating the functions to their respective power series is just to produce another tautologous way of expressing them. in fact, it prompts me to say: isn't it amazing that sin x (when you consider how to get sin x on a diagram of a circle) can be expressed as a power se ...[text shortened]... sed to the power x as a power series?
btw, forgive my ignorance, but who is Karl Weierstrass?
Originally posted by TheMaster37I don't have a problem with definitions. However, some definitions are more suitable than others in different contexts. See the '0!' thread for more on this.
The second of my expressions, is as i said a definition. If you have a problem with definitions, you might as well stop calculating alltogether, you'd have no base for a mathematical structure to stand on...
Originally posted by TheMaster37M37, i'm sorry if my last post sounded dismissive of your position - it wasn't meant that way. i have no problem of definitions at all - in fact this has all to do with definitions. You said:
The second of my expressions, is as i said a definition. If you have a problem with definitions, you might as well stop calculating alltogether, you'd have no base for a mathematical structure to stand on...
Originally posted by dfm65in formal mathematics all results follow from certain basic premises, operations, and logic. As long as your logical analysis, basic tools of mathematical induction, deduction are sound , all the results U derive from the premises, are in a way already contained within the formalsystem. Remember Godel's theorem- no system which is consistent, can be complete, and no system which is complete, can be consistent.
M37, i'm sorry if my last post sounded dismissive of your position - it wasn't meant that way. i have no problem of definitions at all - in fact this has all to do with definitions. You said:
exp(2i*Pi/n) is a n-th root of unity. Visualise a unit circle in the complex plane, and divide the perimeter in a number (say n) of equal parts. The points you get t ...[text shortened]... e things one way, they seem unrelated; define them another way, they turn out to be connected...
Originally posted by dfm65On the complex plane, every differentiable function has a power series (at least locally, in the case of meromorphic functions). Now that's impressive. Sadly it's not true for real differentiable functions though, so we don't get the identity theorem amongst other things.
by 'these functions' i take it you mean e to the power x, sin x and cos x
but, again, equating the functions to their respective power series is just to produce another tautologous way of expressing them. in fact, it prompts me to say: isn't it amazing that sin x (when you consider how to get sin x on a diagram of a circle) can be expressed as a power se ...[text shortened]... sed to the power x as a power series?
btw, forgive my ignorance, but who is Karl Weierstrass?