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The most amazing mathmatical statement ever.
croxis
I am the walrus
in Zocalo v2.0
e^(pi*i)=-1
......
CAN SOMEONE EXPLAIN THIS TO ME?!?!?!?!?!
......
CAN SOMEONE EXPLAIN THIS TO ME?!?!?!?!?!
Comments
Are "e" and "i" Voltage and Current or ?
The negative 1 value in a digital environment ?
Hmmmmm... almost looks like it's dealing with an area value in a plane.
eh... What do I know anyway...
:rolleyes:
(this can be shown using the series expansion of e, cos, and sin)
So,
eiπ = cos (π) + i sin (π)
= -1 + i * 0
= -1
--------------
Taylor series expansions:
ex = ∑(n=0..∞) xn/n! = 1 + x/1 + x2/2 + x3/6 + x4/24 + x5/120 + ...
Which means
eix = ∑(n=0..∞) (xi)n/n! = 1 + i*x/1 - x2/2 - i*x3/6 + x4/24 + i*x5/120 - ...
Which is the same as [b]cos (x) + i sin (x)[/b], where cos and sin are as follows:
cos (x) = ∑(n=0..∞) (-1)n z2n/[(2n)!] = 1 - x2/2 + x4/24 - ...
sin (x) = ∑(n=0..∞) (-1)n z2n+1/[(2n+1)!] = x - x3/6 + x5/120 - ...
Note also:
cos (x) = 1/2 * [eix + e-ix]
sin (x) = 1/(2i) * [eix - e-ix]
Note: Hope everyone can read the symbols in the above - I don't know if all browsers handle the full W3C spec on HTML codes. I have used ∞, π, and ∑ to simplify it, but some browsers might balk :)
Coincidentally, we derived E=mc^2 in Physics today. T'was fun.
But calculus doesn't *really* get fun until you're calculating the curl of the gradient of a function over the space curve defined by the intersection of an elipsoid and a hyperbolic paraboloid... :eek:
-Φ
pretty much sums it up for me.
Reading Kant is so much easier :P
[B]One of my favourite equations :) Thanks RC for doing it justice ;) Though I must say...Taylor series...*shudder* eeeeevil....
Coincidentally, we derived E=mc^2 in Physics today. T'was fun.
[/b][/quote]
Actually the proper reletivistic equation is E=γmc2
At rest, γ=1 thus the reason why E=mc2 gained prominence since it indicated the rest energy of any mass.
[quote][b]
But calculus doesn't *really* get fun until you're calculating the curl of the gradient of a function over the space curve defined by the intersection of an elipsoid and a hyperbolic paraboloid... :eek:
-Φ [/B][/QUOTE]
Fun. Gradient isn't hard. What I hated was calculating line, surface, and volume integrals over strangely defined (but common) objects.
[B]I thought it was "Pi * i " in parenthesis? [/B][/QUOTE]
Can you see the difference between:
eiπ
eiπ
If not, your browser is obsolete and doesn't handle the superscript HTML command.
Indeed...that would be the exam in two weeks. Whee :rolleyes:
-Φ
I had to do that stuff in just about every other physics course I took. Fun. :rolleyes:
In physics we're just doing Newtonian mechanics (linear and angular), plus a smattering of special relativity....I think our teacher is afraid of scaring people (most of whom aren't in math) so he tends to avoid anything more complicated than integraton.
-Φ
[B]But calculus doesn't *really* get fun until you're calculating the curl of the gradient of a function over the space curve defined by the intersection of an elipsoid and a hyperbolic paraboloid... :eek:[/B][/QUOTE]
AHH! AHHHHH! FLASHBACKS! MAKE THE HURTING STOP!
[B]Reading Kant is so much easier :P [/B][/QUOTE]:D
Stupid engineers/mathematicians and their "facts." I like fields of study where I can defend and answer with the proper support.
[B]Can you see the difference between:
eiπ
eiπ
If not, your browser is obsolete and doesn't handle the superscript HTML command. [/B][/QUOTE]
Yes, I can see the difference. It was just my ignorance that assmued it was the Value of Pi * i
:rolleyes:
[B]But calculus doesn't *really* get fun until you're calculating the curl of the gradient of a function over the space curve defined by the intersection of an elipsoid and a hyperbolic paraboloid... :eek: [/B][/QUOTE]
For that you would need Spock's Brain...
:p