Euler's Equation

John_C

Euler's equation says exp(i*pi)=-1 and a lot of people are big fans of it since it relates a lot of the fundamental numbers in mathematics to each other. The (a?) proof of this roughly follows the pattern;

exp(ix)=cos(pi) + i.sin(pi) ... this can be proven by Taylor expansion

cos(pi) = -1 and sin(pi) = 0


I was talking about this with a friend recently and a question occurred to me about the use of pi in the equation. I understand it to be a measure of angle, i.e. pi radians equals half a circle. If that's the case, are we cheating by saying this equation links e with pi since the link depends on a man made unit of angle?

Does the equation not more correctly read; exp(i.[half a circle])=-1 and we could pick any unit we like to define the half circle?

Pherekydes

John_C wrote: »

Euler's equation says exp(i*pi)=-1 and a lot of people are big fans of it since it relates a lot of the fundamental numbers in mathematics to each other. The (a?) proof of this roughly follows the pattern;

exp(ix)=cos(pi) + i.sin(pi) ... this can be proven by Taylor expansion

cos(pi) = -1 and sin(pi) = 0


I was talking about this with a friend recently and a question occurred to me about the use of pi in the equation. I understand it to be a measure of angle, i.e. pi radians equals half a circle. If that's the case, are we cheating by saying this equation links e with pi since the link depends on a man made unit of angle?

Does the equation not more correctly read; exp(i.[half a circle])=-1 and we could pick any unit we like to define the half circle?

Well Pi suits nicely since the ratio of the circumference of a circle is 2Pi times its radius.

MathsManiac

A few points wort noting:

Pi is a real number that exists outside of any geometric context.

Even within the relevant geometric context, the use of radians is not an arbitrary measure (in the way that degrees or grads are). Radians express a ratio between arc-length and radius.

Your observation is correct initially, but when you say [half a circle], what you really need is [half the length of the circumference of a unit circle]. This means that your last assertion breaks down: "and we could pick any unit we like to define the half circle". There is actually no discretion here: since you are concerned with the unit circle, its length, (in the same length-units) is 2*Pi.

By the way, the equation is often quoted with the -1 on the other side, so that it reads: e^(i*Pi)+1=0. When written this way, it unites five fundamental numbers in mathematics, (0, 1, i, e, Pi) which makes it very attractive.

John_C

MathsManiac wrote: »

Your observation is correct initially, but when you say [half a circle], what you really need is [half the length of the circumference of a unit circle].

Are you sure? The cosine of an angle doesn't relate directly to the length of an arc, it's the ratio of 2 sides of a triangle. We only relate it to the arc length when we define radians. In the case of cos(pi) the 'triangle' has closed onto itself so the hypotenuse and adjacent side are the same line, giving them a ratio of 1. That would be true even if there were no such thing as a circle.

Also, I'm impressed with the prompt replies. Thanks.

Pherekydes

John_C wrote: »

Are you sure? The cosine of an angle doesn't relate directly to the length of an arc...

The cosine function is explicitly and directly related to the length of an arc. The cosine function is the horizontal coordinate of the arc endpoint of an angle measured counterclockwise from the x-axis. The definition in relation to 2 sides of a triangle only applies to a right-angled triangle.

MathsManiac

When one first encounters the cosine on an angle, it is indeed defined in terms of ratios of lengths in an acute triangle. However, when you later need to define cosines and sines of angles bigger than 90 degrees, cosine is redefined to be the x-coordinate of a point on the unit circle. In this case, you're still thinking about the cosine of an "angle" rather than a "number".

When you start moving into the territory of functions and calculus, cosine and sine need to be subtly redefined in a way that makes them functions of numbers rather than functions of angles. Most of the time, we gloss over this distinction, but some careful treatments of this material distinguish notationally between these two things.

Anyway, when cosine and sine are treated as functions (e.g. for the purposes of calculus) these are functions of numbers. The cosine as a function of numbers is defined so that it coincides with the cosine of the same number as an angle-measure in radians. If you did otherwise, then, for example, the derivative of cos(x) wouldn't be -sin(x), etc.

The definitions of sine and cosine can be extended to make them functions of complex numbers too, by redefining them in terms of the exponential function.

Anyway, the upshot of all this semi-invisible redefinition of these functions as you extend them from one set/domain to a bigger one, is that if you want to interpret them in a geometric way again, you have to stick to radians.

Also note that you can't multiply an "angle" by i, you can only multiply a "number" by i, so exp(i*theta) has no definition in the absence of a clear unambiguous definition as to how theta is to be considered to be a number rather than an angle. This is where you get locked into radians.

John_C

Thanks for the replies. I had a think about this myself last night and I have one very clear mistake in this line:

exp(ix)=cos(pi) + i.sin(pi) ... this can be proven by Taylor expansion

The Taylor expansions of sine and cosine are dependant on the measure of angle I use so the equality only holds for radians.

I think the more fundamental point you make is that I can't swap between degrees and radians as easily as I thought I could.

Pi^2

Man made or not... 2.71828183...^ i*3.14159265...=-1


That is pretty cool.

spacetweek

Pi^2 wrote: »

Man made or not... 2.71828183...^ i*3.14159265...=-1

That is pretty cool.

Yes it is and there are other equally weird places you can take this.

e^ipi = -1

e^ipi = i^2

ipi = 2 ln i

pi = 2 (ln i) / i

Giving a highly bizarre definition of pi.