20 Jun 04
As a mathematician i've been dealing with these thing for quite some time now, it's a beautifull concept.
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 that way are called the n-th roots of unity.
All those roots have this property (root)^n = 1
Try is for yourself in the case n=2 (the only roots then are on the horizontal axis; 1 and -1) or n=4 (1, -1, i, -i).
the coordinate of such a root is (cos(2*Pi/n) , sin(2*Pi/n)). Or written as a complex number: cos(2*Pi/n) + i*sin(2*Pi/n). This last expression is the definition of exp(2i*Pi/n).
With the exponential, taking the n-th power is easy:
(exp(2i*Pi/n))^n = exp(2i*Pi)
Again, using the definition; exp(2i*Pi) = cos(2*Pi) + i*sin(2*Pi) = 1
26 Jun 04
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.
the point will perhaps be clearer if i express it this way: isn't it amazing that the base of natural logarithms, the ratio of a circle's circumference to its diameter, and the square root of -1 - three entities whose senses seem unrelated - are actually related in such a simple way as expressed in Euler's theorem?
26 Jun 04
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?
26 Jun 04
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.
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 series? ditto for cos. isn't it amazing that you can express the base of natural logarithms raised to the power x as a power series?
btw, forgive my ignorance, but who is Karl Weierstrass?
26 Jun 04
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?
Karl Weierstrass was a late 19th-century German mathematician who is credited with putting mathematical analysis in a more rigourous form (the earlier analysts, like Euler and Lagrange, operated on a more informal basis).
26 Jun 04
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...
27 Jun 04
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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...
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 that way are called the n-th roots of unity.
but at this point the crucuial step has already been taken: e, pi and implicitly sqrt(-1) are already linked in a mathematical expression. i'd like to point out again that these quantities can be defined in a way in which they appear on the surface to have no connection: the base of natural logs, the ratio of a circle's circumference to its diameter, the square root of -1. now, we follow the procedure you outline, and voila! they are all deeply and intimately connected after all! is this not already amazing? no, it is sublime!
And then, as RC points out, e to the x, sin x and cos x can also be alternatively defined in terms of power series. so again, things that aren't obviously connected turn out to be. work had to be done by mathematicians before these connections became apparent. and of course, it took the brilliance of an Euler to come up with his theorem, linking them in such an elegant way.
my point might be clear if i refer to an example used by Frege to illustrate the difference between sense and reference. the ancient Babylonians were keen astronomers, and plotted the celestial movements of the Morning Star, and also the Evening Star (and many other heavenly bodies). Both of these, we realise today, are the planet Venus, but the Babylonians did not know this. so the sense of 'Morning Star' and 'Evening Star' were different: they meant two different objects in their conceptual framework. Yet both terms refer to the same object. perhaps the first to realise they were the same thing was amazed that two hitherto unconnected things, turned out to have the most intimate relation of all (identity)...
define things one way, they seem unrelated; define them another way, they turn out to be connected...
27 Jun 04
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...
Thus ther is nothing imaginary about sqrt(-1).And the number pi has the peculiar properties, on accout of its formal relation to trigonometric functions. The trigonometric functions owe their peculiar power series to the Taylor series expansion,..so is the case of exponential series. So all are bound to "converge". Had it been otherwise it would have put a question mark on the tools of mathematics. Yes mathematics, brings out hidden relations which look miraculous. But that's the beauty of maths.
27 Jun 04
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?
If you're going to say e is 'the base of natural logarithms', you might as well write exp(x) for 'a function which is its own derivative' and derive things like e and the power series from there (show that the power series works, and then show it's unique).