Naked Science Forum
General Science => General Science => Topic started by: DoctorBeaver on 31/07/2007 15:01:52
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I found this on a website (http://www.futilitycloset.com/2007/07/15/eulers-identity/ (http://www.futilitycloset.com/2007/07/15/eulers-identity/))
but I don't understand what's so special about it. Anyone know?...
You know these numbers:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fi148.photobucket.com%2Falbums%2Fs26%2FDoctorBeaver%2Fnumbers_01.jpg&hash=ada6eac18d67529ebc49fe24b749f3b2)
constants
On the surface they appear unrelated. e concerns natural logarithms, i is imaginary, π concerns circles. But, amazingly:(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fi148.photobucket.com%2Falbums%2Fs26%2FDoctorBeaver%2Fnumbers_02.jpg&hash=1e832dbed63cbdb42753e9277770b1e2)
Euler's identity
Harvard mathematician Benjamin Peirce told a class, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."
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Not quite as simple as that.
i is imaginary, but the number space occupied by real and imaginary numbers can be thought of as a circle (or at least as a 2 dimensional plane in which a circle can be drawn through the points 1, 1i, -1, and -1i).
If one draws a line, where positive real numbers are to the right, and negative real numbers are to the left; then one can draw a perpendicular line, where negative imaginary numbers are down, and positive imaginary numbers are up.
You can then draw a line where X2 + iY2 = 1, and this line will be a circle through the points where X = 1 and Y = 0, X = 0 and Y = 1, X = -1 and Y = 0, and X = 0 and Y = -1.
The eiπ = -1 is based on the more general formula that eiθ = cos(θ) + i sin(θ), and since cos(π) = -1 and sin(π) = 0, so cos(π) + i sin(π) = -1 + 0 i = -1.
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But why is it "amazing" that eiπ = -1? Is that any more special than eiπ = 27 or eiπ = -593.456?
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But why is it "amazing" that eiπ = -1? Is that any more special than eiπ = 27 or eiπ = -593.456?
You'll have to ask the Harvard Professor about why he was amazed by it.
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OK. Thanks anyway.
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But why is it "amazing" that eiπ = -1? Is that any more special than eiπ = 27 or eiπ = -593.456?
It's usually put in the form:
eiπ + 1 = 0
which involves the 5 most important numbers of mathematics: 0, 1, e, π, i.
it's amazing because the numbers e, π, i are defined in a way (seemingly) independent of one another: e is the basis of natural logaritms, π the area of a unit circle, i the square root of -1...who coluld think there is a relationship between them? If you think, it's a sort of magic!
The discover of their connection was quite amazing for me too, but it probably was much more to Euler and the matematicians of those times.
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Frankly, I always wondered what was so "natural" about "natural logaritms".
The number "e" seems rather artificial to me, not even related to anything natural (as the number π is related to the circle).
It seems to be invented just for the purpose of making ∫dx/x = ln x
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hmmm... I think I'm better off living in ignorance [:I]
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Frankly, I always wondered what was so "natural" about "natural logaritms".
The number "e" seems rather artificial to me, not even related to anything natural (as the number π is related to the circle).
It seems to be invented just for the purpose of making ∫dx/x = ln x
The number e is also:
e = lim(n → ∞) (1 + 1/n)n
e = ∑(n=0;n=∞) 1/n!
The first expression was discovered (as a constant) from Jakob Bernoulli (XVII century).
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The one I always remember is:
d/dxex = ex
In other words the gradient of the line ex is always the same as the value of the line at that point.
This is ofcourse consistent with the relationship between imaginary numbers and trig functions, since the differential of trig functions is the same function shifted by π/2, which, if one maps those trig functions onto the plane of real and imaginary numbers, has the effect of moving the function from real to imaginary.