From the e to the i to the e to the i pi.
April 26, 2005 8:41 AM
Can someone give me a reasonably simple explanation of why e raised to the power of i times pi is equal to -1? I've always had trouble understanding this and thought I'd try the green.
I'm going to wax mystical here for a second.
We, with our hu-man brains can see why the area of a rectangle is it's length times it's width. If we define a unit (say, the centimetre), we can see that a 3cm X 5cm rectangle is made of 15 1cm squares. The same applies to triangles and other regular shapes. If you do the math, you can see why it is. It is very difficult to imagine a universe where this is not the case.
But why is the ratio of the area of a circle to the square of its radius 3.141.....? IT JUST IS. In another universe, it could (concievably) be something else, but here it's that strange irrational number. For me, this is like looking in to the very heart of creation.
It's the same for Euler's formula. Why? Because the universe is amazing and strange and yet math seems to describe it accurately more often than it doesn't. Wow.
posted by Capn at 9:06 AM on April 26, 2005
We, with our hu-man brains can see why the area of a rectangle is it's length times it's width. If we define a unit (say, the centimetre), we can see that a 3cm X 5cm rectangle is made of 15 1cm squares. The same applies to triangles and other regular shapes. If you do the math, you can see why it is. It is very difficult to imagine a universe where this is not the case.
But why is the ratio of the area of a circle to the square of its radius 3.141.....? IT JUST IS. In another universe, it could (concievably) be something else, but here it's that strange irrational number. For me, this is like looking in to the very heart of creation.
It's the same for Euler's formula. Why? Because the universe is amazing and strange and yet math seems to describe it accurately more often than it doesn't. Wow.
posted by Capn at 9:06 AM on April 26, 2005
Just wanted to say that this is my favorite equation of all time in the way it correlates the cerebral world of mathematics with the natural world. Clearly we're on the right track.
posted by Civil_Disobedient at 9:10 AM on April 26, 2005
posted by Civil_Disobedient at 9:10 AM on April 26, 2005
Tangential (arf) question. Can anyone explain how pi is calculated? I never did much geometry but it seems somehow circular (erk) --
empirical ways (measure circumference and diameter of a circular to 9 zillion significant digits?) obviously don't work, yet, a formula for calculating pi to that much precision must be unverifiable?
I vaguely remember it has something to do with triangles which seems even less intuitive.
posted by Rumple at 10:09 AM on April 26, 2005
empirical ways (measure circumference and diameter of a circular to 9 zillion significant digits?) obviously don't work, yet, a formula for calculating pi to that much precision must be unverifiable?
I vaguely remember it has something to do with triangles which seems even less intuitive.
posted by Rumple at 10:09 AM on April 26, 2005
One very important aspect of the question is how trig functions relate to rotating circles. Here is a picture showing how a rotating circle traces out a sine wave over time.
If you can understand the other links posted here then you could probably modify the question a bit. One might ask what is the difference between an imaginary number and a two dimensional vector. (I might try to answer this later...)
Anyway, no offence but... The idea that the ratio could be something else "in another universe" is pure crap. Really! There is nothing more or less intuitive about areas and circles than areas and squares.
As proof I offer... How do we know that a 1cm square has an area of 1 cm^2? The link between one dimensional and two dimensional measurements requires calculus.
posted by Chuckles at 10:26 AM on April 26, 2005
If you can understand the other links posted here then you could probably modify the question a bit. One might ask what is the difference between an imaginary number and a two dimensional vector. (I might try to answer this later...)
Anyway, no offence but... The idea that the ratio could be something else "in another universe" is pure crap. Really! There is nothing more or less intuitive about areas and circles than areas and squares.
As proof I offer... How do we know that a 1cm square has an area of 1 cm^2? The link between one dimensional and two dimensional measurements requires calculus.
posted by Chuckles at 10:26 AM on April 26, 2005
This is difficult to "dumb down" but keep in mind that these are not just any old numbers that are tied together.
The equation itself is a special case of e^pi*i=cos(x)+sin(x)i
But what makes e and cos and sin *already* similar is with respect to differentiation. The derivative (or rate of change) of the cosine is the (-)sine. The derivative of the sine is the (-)cosine. They form a unique pair this way.
Now, e is also special with respect to differentiation. The derivative of e^x is e^x. That is the rate of change of an exponential curve is an exponential curve.
Now, this equation ties all these special curves together: thats all it is doing. Can we build a function, using only sin and cos that differentiates to itself? It seems like we should be able to. But the negative sign gets in the way: (cos-sin) becomes (sin-cos) not the same thing.
But, *if* we could do it, then, since the exp curve is the only one that diferentiates to itself, that curve would be equal to the exp curve.
In a sense, the e curve is a "sum" of the cos and sin curve, or rather it behaves in a way such that it can be "built" using these two mirror-like functions along with a "switch" that allows us to take the right elements from each curve and also account for the negative sign that happens when sin goes to cos, or cos goes to sin. The fact that i when squared equals -1 allows us to combine these differentiable curve-pairs into the single self-differentiable curve.
You can think of it as a trick if you like but you can also think of it as: The exponential curve of a complex number is simply the sum of the real cosine plus the imaginary sine. The latter two are actual geometric entities.
Finally, we plug pi into all this and get the final equation.
posted by vacapinta at 10:26 AM on April 26, 2005
The equation itself is a special case of e^pi*i=cos(x)+sin(x)i
But what makes e and cos and sin *already* similar is with respect to differentiation. The derivative (or rate of change) of the cosine is the (-)sine. The derivative of the sine is the (-)cosine. They form a unique pair this way.
Now, e is also special with respect to differentiation. The derivative of e^x is e^x. That is the rate of change of an exponential curve is an exponential curve.
Now, this equation ties all these special curves together: thats all it is doing. Can we build a function, using only sin and cos that differentiates to itself? It seems like we should be able to. But the negative sign gets in the way: (cos-sin) becomes (sin-cos) not the same thing.
But, *if* we could do it, then, since the exp curve is the only one that diferentiates to itself, that curve would be equal to the exp curve.
In a sense, the e curve is a "sum" of the cos and sin curve, or rather it behaves in a way such that it can be "built" using these two mirror-like functions along with a "switch" that allows us to take the right elements from each curve and also account for the negative sign that happens when sin goes to cos, or cos goes to sin. The fact that i when squared equals -1 allows us to combine these differentiable curve-pairs into the single self-differentiable curve.
You can think of it as a trick if you like but you can also think of it as: The exponential curve of a complex number is simply the sum of the real cosine plus the imaginary sine. The latter two are actual geometric entities.
Finally, we plug pi into all this and get the final equation.
posted by vacapinta at 10:26 AM on April 26, 2005
Sorry, I meant to say "what is the difference between a complex number and a two dimensional vector"... Argh!
posted by Chuckles at 10:27 AM on April 26, 2005
posted by Chuckles at 10:27 AM on April 26, 2005
In fact, vacapinta's excellent descritption has important physical significance: it's intimate to our understanding of how electromagnitism: light, phonons, exitons, etc..., propogate. Maxwell's equations would be very different in a universe with a different Euler's equation.
posted by bonehead at 10:41 AM on April 26, 2005
posted by bonehead at 10:41 AM on April 26, 2005
I find the infinite series explanation to be the most compelling. If you can write out e(x) as a formula, swap in pi*i, and arrange to get the cosine terms, then it's like solving any other equation (waving off the issues of infinite series).
Rumple: a not-very-good way to calculate pi is to recognize that the arctangent of 1 is pi/4, then use a formula for the arctangent: 1 - x^3/3 + x^5/5 - x^7/7 ... so
pi = 4 * ( 1 - 1/3 +1/5 - 1/7 + 1/9 .. )
I think this is one of the slowest convergences you can find, though. (see also)
posted by fleacircus at 10:43 AM on April 26, 2005
Rumple: a not-very-good way to calculate pi is to recognize that the arctangent of 1 is pi/4, then use a formula for the arctangent: 1 - x^3/3 + x^5/5 - x^7/7 ... so
pi = 4 * ( 1 - 1/3 +1/5 - 1/7 + 1/9 .. )
I think this is one of the slowest convergences you can find, though. (see also)
posted by fleacircus at 10:43 AM on April 26, 2005
/me notes too late π = π
posted by fleacircus at 10:51 AM on April 26, 2005
posted by fleacircus at 10:51 AM on April 26, 2005
This thread is closed to new comments.
This is about the simplest explanation I have found. If you can't understand what's behind the link, then a simpler explanation would be:
Complex exponents don't make any intuitive sense in terms of tangible realities that we see every day; rather, their meaning falls out of a set of equations that describe the unit circle. Integrating and differentiating these equations lead to the equation (Euler's equation) describing the value of complex exponents.
But that's handwaving. The answer is in the equations.
posted by ikkyu2 at 8:53 AM on April 26, 2005