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To get straight to the point: is $\pi$ defined as the ratio of the circumference and diameter of a circle, or as the first non-zero root of the power series of $\sin{x}$?

If the former, then $\pi$ would change with different geometries. If the latter, it would stay constant, and one would just need a different ratio for some non-Euclidean geometry?

Also on this topic: are $\sin$ and $\cos$ defined in terms of power series, or something else?, and is $e$ defined as the constant that satisfies $\frac{d}{dx}c^x=c^x$, or in terms of its power series?

Or are these simply all the same thing? It's just that we have recently been learning about power series, and now I am not too sure which definitions of functions are actual definitions, and which are simply equivalent statements of the idea that we wish to get across.

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The answer to the question in the title is, yes. –  Gerry Myerson May 9 '13 at 12:10
    
Since Euclidean geometry may be described easily with respect to the non-Euclidean ones (it suffices to say that there exist two triangles with the same angles but different sizes), I see no problem in saying that $\pi$ is the ratio of the circumference and diameter of a circle in Euclidean geometry. –  mau May 9 '13 at 12:15
    
For comparison: In spherical geometry (with unit curvature), circumference and area of a circle with radius $r$ are given by $C = 2\pi \sin r$ and $A = 4\pi \sin^2(r/2)$; in hyperbolic geometry, $C = 2\pi \sinh r$ and $A = 4 \pi \sinh^2(r/2)$. Simple ratios like $C/(2r)$ won't suffice to define $\pi$ in these circumstances. –  Blue May 9 '13 at 12:31
    
that's clear. I am saying that it is simple to define Euclidean geometry, and then find $\pi$ in this special geometry. –  mau May 9 '13 at 12:50
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If the definitions are equivalent, you can use whichever you like the most, since, after all, they are equivalent. –  Javier Badia May 9 '13 at 12:56
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1 Answer

up vote 4 down vote accepted

(1) $\pi$ defined as the ratio of the circumference and diameter of a circle in Euclidean plane geometry.

(2) $\pi$ is defined as the least positive zero of the power series $\sum_{n=0}^\infty (-1)^nx^{2n+1}/(2n+1)!$.

If (1) is the definition, the value in (2) is a theorem. If (2) is the definition, the value in (1) is a theorem.

Neither of these deals with non-Euclidean geometry.

Are sin and cos defined in terms of power series, or something else?

There are many possible definitions, but of course when you choose a defintion you must be able to show that the functions you get are the same as those obtained by the already-existing definitions. The reason for using a particular definition is to start developing the subject. After a while, all of the usual definitions should be obtained as theorems based on that definition. From that point on, we don't care which of these was used as the definition.

Sometimes, a student posts here a question, and asks that the proof be done from the definition. That student perhaps does not realize that the definition may be different in some other textbook. So (in order for us to answer here) the definition should be included.

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OK, I suppose my question would have been better worded as 'if we wanted to start from scratch with maths, and we had an idea of what we wanted a function called $\sin$ to satisfy, which definition 'comes first', for want of a better phrase'. Your answer about $\pi$ was very useful though, thank you! And I now see how my question might really be a bit of a poor question in semantic terms. –  Tim May 9 '13 at 14:47
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It is not a bad question at all. In some textooks, sin and cos are defined in terms of power series; in some as solutions of differential equations; in some in terms of integrals; perhaps also in still other ways. When YOU write a textbook at that level, you will have to choose what definition to use, and then deduce the properties you want from it. –  GEdgar May 9 '13 at 15:03
    
Well, historically the circle related definitions were the original ones for $\pi$, and I guess, $\sin$ and $\cos$ were first defined for acute angles, using right angle triangles (on the Euclidean plane)... –  Berci May 9 '13 at 16:52
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