I read something about an equation that links the four most important numbers in math: pi, i, 1 and 0. Does anyone know it?
Well, Grimey, I you are referring to the famous Euler Formula: e^(pi*i) = -1, which is explained by the linked page.
“I live in an apartment over a bowling alley, which is under another bowling alley”
IIRC, it’s something like e^(-i*pi)=1. I’ll see if I can dig up the proof.
Dammit, whitho beat me while I was trying to see if there was an ASCII code for “pi.”
Indeed, it is truly a magical equation. Probably the single most important factor contributing to what has become a lifelong love. If only I had the talent…sigh.
Thanks Strainger and whitetho.
I’ve never understood this one, and my advanced algebra skills have gotten rusted shut. Tell me some simpler ones:
What is e^i? I should probably be able to do this one, but I’ve totally forgotten how to raise to imaginary powers.
What is 1^(i x pi)?
Boris: To take exponential powers, you have to use the power series for e^x:
e^x = x^0/0! + x^1/1! + x^2/2! + x^3/3! + …
Or, you can use the shortcut a’la’ Euler (pronounced “Oiler”, btw):
e^ix = cos(x) + i*sin(x)
For complex exponents (not real or pure imaginary), you can break it up into its real and imaginary parts:
e^(a + bi) = (e^a)*(e^bi).
For bases other than e, use the formula
a^b = e^(ln(a)*b)
Does this help?
“There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe”
–A. Einstein
e^i:
0.540302305868 + 0.841470984808i
1^(Pi*i):
1 to any power is always 1…but we know you knew that.
“I don’t want to achieve immortality through my work… I want to achieve it through not dying.”
– Woody Allen
Another interesting one: i^i = .2078796 . That’s right, it’s pure real. Fun, eh?
“There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe”
–A. Einstein
Well … I did kinda know that, but I didn’t know imaginaries were included with “any power”.
And Chronos, I never woulda guessed that i^i was a real number. Thanks for blasting my world-view into nasty little splinters.
You guys are starting to make my brains wiggle. I’m going to head over to a nice pleasant gun control thread.
Of course, the form of the equation that answers the original question is
e^(i * pi) + 1 = 0
Arnold
Finally! And, of course, that’s the five most important numbers, not four…
It’s not really that magical. Remember that the pi stands for pi radians, which is just a human convention. You can also write e^(i*180 degrees) = -1.
Nothing but convention and hype.
Yeah, Konrad, but remember, it’s just convention that 1 stands for the unit, and that e stands for lim(x->inf) (1+1/x)^x, and that = stands for “is equal to”. Yes, it’s a human convention to refer to angles in radians, but it’s a very logical convention: were it not for that convention, all the trig differentials/integrals would be more complicated, the simple analogy wouldn’t exist between exponentials and trig functions, etc. For that matter, even if we DIDN’T use radian notation, e^(ipi) would STILL equal -1, since
sum(n=0 to inf) (ipi)^n/n! = -1 .
It’s only because of this that the pi in that equation is interpreted to mean pi radians in the first place.
“There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe”
–A. Einstein
Chronos: That’s true. I guess that does make the equation pretty interesting.
Also remember that radians are defined as arclength divided by radius (is the angle in radians that subtends the given arc).
That means that radians have no human-defined calibration (like degrees, etc.), but are a pure ratio.
I think you’re saying that radians are unitless, but degrees are unitless also. There’s just 360 of them in a circle instead of two pi of them.
The reason radians are used is because they’re more “natural” - equations become much simpler with them. Same thing with using “e” for logarithms.
By the way, no one has pointed out that the easy way to find e^(i*x) is this way:
e^(i*x) = cos(x) + i * sin(x).
cos(pi) is -1, and sin(pi) is zero, so e^(i*pi) is -1.