e to the power of pi times i equals negative 1

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Reply Wed 14 Apr, 2010 01:13 am
It doesn't get much prettier. And this equation takes us right to i to the power of i, which is strangely a real number. Euler's identity - Wikipedia, the free encyclopedia

Wanting to understand this equation is one of the things that drew me into math. And it makes a hell of lot more sense now, while remaining amazing.

Anyone else checked this out?
 
Holiday20310401
 
Reply Wed 14 Apr, 2010 09:42 am
@Reconstructo,
How do you multiply pi and i together?
 
Owen phil
 
Reply Wed 14 Apr, 2010 10:27 am
@Holiday20310401,
Holiday20310401;151824 wrote:
How do you multiply pi and i together?


It might be a problem if we interpret multiplication as repeated addition.

There is no sense in saying that ((pi) added (i) times) or ((i) added (pi) times), means (pi)*(i).
 
Holiday20310401
 
Reply Wed 14 Apr, 2010 01:03 pm
@Owen phil,
Yes, but I mean, pi is a real number whereas i is not. I know this is what complex numbers are all about, but where does the canceling out happen to give the equation a value of -1 (a real number).
 
Amperage
 
Reply Wed 14 Apr, 2010 01:06 pm
@Holiday20310401,
it may be easier to think about it in terms of direction.....

where a variable would contain a value of pi protruding in the imaginary direction...

actually apparently it would be easier to think about in terms of a circle and in sines and cosines
 
Reconstructo
 
Reply Wed 14 Apr, 2010 01:41 pm
@Amperage,
Amperage;151918 wrote:

actually apparently it would be easier to think about in terms of a circle and in sines and cosines

Bingo. I believe that Euler wrote it terms of trig functions. Radians. But of course it works algebraically as well.

---------- Post added 04-14-2010 at 02:52 PM ----------

Holiday20310401;151824 wrote:
How do you multiply pi and i together?

You can't simplify pi times i without doing something else to both sides of the equation. Pi times i squared equals negative pi. John Derbyshire explains its well in Prime Obsession. It's pretty strange. I was drawn to the equation before it even made sense to me. I was dying to know how you take one transcendental to the power of another. One way to take e to the power of pi * i is to insert pi * i into the infinite series that defines e.

e^ pi*i = 1 + pi*i + pi*i^2/2! + pi*i^3/3! + pi*i^4/4!......


As this equation is computed for more and more terms, the real part closes in on negative one, and the imaginary part closes in on zero. (This infinite series converges for every number, according to Derbyshire.) note: that e^z = 1 + z + z^2/2! + z^3/3! + z ^4/4! ..... and that this works for all numbers. the number e continues to blow my mind. there is way to much to go into and i want to focus on this famous equation AND I am still wading through all the strangeness. ...



There are solid rules for taking any complex number to any complex power. Amazing, but true. (All this is new to me, I admit. I've been reading this stuff obsessively. Nothing is more fascinating. One can easily insert phi into this mix, as -1 is always equal to e^pi*i, and phi (the golden ratio/number) squared minus phi minus one = zero. So phi squared minus phi plus e^pi*i = 0. Strange how they all fit together.

phi^2 - phi + e^pi*i = 0

It looks so much better in the usual math symbols. I mean the Greek letters and the usual exponent form. But such is life.
 
Reconstructo
 
Reply Wed 14 Apr, 2010 07:18 pm
@Reconstructo,
If anyone is unfamiliar with particular terms (aimed at those who have not jumped in the thread yet --and it's just good stuff...), here's some background. I can only urge the non-math type to check out this particular equation. It's a piece of art.
e (mathematical constant) - Wikipedia, the free encyclopedia
Pi - Wikipedia, the free encyclopedia
Golden ratio - Wikipedia, the free encyclopedia
Complex number - Wikipedia, the free encyclopedia
Imaginary number - Wikipedia, the free encyclopedia
 
Reconstructo
 
Reply Thu 15 Apr, 2010 11:17 pm
@Holiday20310401,
Holiday20310401;151917 wrote:
Yes, but I mean, pi is a real number whereas i is not. I know this is what complex numbers are all about, but where does the canceling out happen to give the equation a value of -1 (a real number).


That's exactly what I wanted to know.. I couldn't believe my eyes when I first saw the equation. e to the power of pi? I hadn't heard of that..let along of an imaginary exponent. But as I quoted above, e to any number, real or complex, is apparently defined by an infinite series discovered by Euler. And Euler also calculated weird things like (((x to the power of x) to the power of x) to the power of x) and so on, infinitely, determining when such a series converges. He also did e to power of (e to power of e). He strikes me as an artistic type. Can i to the power of i have much practical use? Who knows? Non-Euclidean geometry was apparently useful after Einstein. Like I said, I'm just an eager student who wishes he had dug into all of this long long ago. Smile
 
Reconstructo
 
Reply Fri 16 Apr, 2010 05:56 pm
@Reconstructo,
Bump. Ode to the beauty of the style of mathema.......
 
Reconstructo
 
Reply Sun 18 Apr, 2010 08:33 pm
@Reconstructo,
Ok, I'm a maniac on a bullhorn, but I can't stop being amused at this equation. In its longer form we have the five most important numbers e, pi, i, 1, and 0 lined up in a clean simple equation that uses the three most fundamental operations: addition, multiplication, and exponentiation. It's a sculpture made of thought, that is absurdly true, to whatever degree math is true. And if it weren't true, it wouldn't be a good sculpture anymore. "Truth is beauty, and beauty truth." At least some of the time, at least in mathematics.
 
Holiday20310401
 
Reply Sun 18 Apr, 2010 09:50 pm
@Reconstructo,
I tried to figure it out for myself but only got this far.


e^(pi*i) + 1 = 0

You could natural log both sides, but then you are left with infinity on the right side, so you move the 1 over, and get an imaginary.

pi*i = ln(-1)

then I said, well why don't I square both sides, because then you get rid of i.

(pi*i)(pi*i) = [ln(-1)]^2

(pi^2)(-1) = [ln(-1)]^2

When I think of pi I think of trig identities, so I isolated the 1 on the right side.

(pi^2)(-1) = [ln - (1)]^2

But I just don't know what to do here.
 
Reconstructo
 
Reply Sun 18 Apr, 2010 09:59 pm
@Reconstructo,
Reconstructo;151939 wrote:
John Derbyshire explains its well in Prime Obsession. One way to take e to the power of pi * i is to insert pi * i into the infinite series that defines e.

e^ pi*i = 1 + pi*i + pi*i^2/2! + pi*i^3/3! + pi*i^4/4!......


As this equation is computed for more and more terms, the real part closes in on negative one, and the imaginary part closes in on zero. (This infinite series converges for every number, according to Derbyshire.) note: that e^z = 1 + z + z^2/2! + z^3/3! + z ^4/4! ..... and that this works for all numbers.

Here you go. I think a series is the only way for a case like this.

---------- Post added 04-18-2010 at 11:01 PM ----------

Holiday20310401;153829 wrote:

When I think of pi I think of trig identities, so I isolated the 1 on the right side.

I think it was written in trig functions and the pi value is just a special case. But then the pi * i value does indeed work algebraically, if inserted into that series that defines the exponentiation of complex numbers. (I'm a fan, learning as fast as I can, but certainly no expert. Prime Obsession is a great book. So is" e :The Story of a Number")
 
longknowledge
 
Reply Sun 18 Apr, 2010 10:37 pm
@Reconstructo,
For an interesting surface, try plotting x to the y*z, where y is an imaginary axis. Then find x=e, y=i, z=pi!

:flowers:
 
Reconstructo
 
Reply Mon 19 Apr, 2010 12:03 am
@longknowledge,
longknowledge;153832 wrote:
For an interesting surface, try plotting x to the y*z, where y is an imaginary axis. Then find x=e, y=i, z=pi!

:flowers:

I have downloaded the "Euler" program, but haven't learned to use it yet. Do you have a link? I would like to see. Smile

---------- Post added 04-19-2010 at 01:04 AM ----------

Quote:

Euler's identity is considered by many to be remarkable for its mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:


[*]This is from Wiki, just to add more information to the thread.
[/LIST]
 
longknowledge
 
Reply Mon 19 Apr, 2010 08:22 am
@Reconstructo,
Reconstructo;153847 wrote:
I have downloaded the "Euler" program, but haven't learned to use it yet. Do you have a link? I would like to see. Smile

I plotted it by hand years ago on 3-dimensional graph paper. I'll see if I can find it.

:flowers:
 
kennethamy
 
Reply Mon 19 Apr, 2010 08:36 am
@Reconstructo,
Reconstructo;153006 wrote:
Bump. Ode to the beauty of the style of mathema.......


Ah. Pythagoras still lives. But it is math. Not philosophy, except maybe aesthetics. "Euclid alone has looked on beauty bare".
 
Twirlip
 
Reply Mon 19 Apr, 2010 09:39 am
@Reconstructo,
Reconstructo;153830 wrote:
Here you go. I think a series is the only way for a case like this.

Here's one way of making Euler's identity plausible without using series, or at least without using them directly.

(It's based on an impromptu post I made to sci.math on 19 May 2008, but I think it holds up OK.)

The act of faith required is to believe that: (i) there exists a differential calculus of complex-valued functions of complex variables; (ii) there exists a complex-valued generalisation of the exponential function, exp; (iii) even when understood as a complex-valued function, exp is equal to its own derivative.

I'll write exp(z) as e^z (with the understanding that the power notation for complex numbers is not well-defined in general).

Consider u(t) = e^{it}, as a complex-valued function of the real variable t.

Given the above acts of faith (which can all be justified, of course), it is reasonable to guess, correctly, that u satisfies the differential equation:

u'(t) = iu(t)

with the initial condition:

u(0) = 1.

If you are happy with the idea that multiplication by i means turning through a right angle anticlockwise in the complex plane, then this means that if u(t) is considered as a vector in the complex plane, its derivative is the same vector turned through an anticlockwise right angle.

But this is precisely the condition satisfied by the point u(t) on a circle of unit radius, centre the origin, at arc length t anticlockwise from the point (1, 0), which is as usual identified with the number 1 on the real axis.

(You can picture it: an infinitesimally small change in u(t) is almost exactly equal in length to the infinitesimal increment in the arc length t, and is tangential to the circle, therefore the derivative u'(t) is the unit vector tangent to the circle in the anticlockwise direction at the point u(t). Ignore Bishop Berkeley turning in his grave - we're talking acts of faith here!)

So it is reasonable to suppose that this gives the solution of the differential equation.

Therefore, u(pi) should be the point on the unit circle at arc length pi anticlockwise from the point (1, 0), i.e. it is the point (-1, 0), which is identified with the number -1 on the real axis.

Hence (by this plausible hand-waving argument):

e^{i*pi} = -1.

Series might have to be used in the rigorous justification of this hand-waving argument, but I hope that that that doesn't take away all of its charm! The full original thread in sci.math is here:
sci.math | Google Groups
 
Reconstructo
 
Reply Mon 19 Apr, 2010 03:17 pm
@longknowledge,
longknowledge;153967 wrote:
I plotted it by hand years ago on 3-dimensional graph paper. I'll see if I can find it.

:flowers:


That's sounds wild. I've never used 3-d graph paper. I've scribbled my approximation of an ascending and widening equiangular spiral. But I couldn't figure out how to write this spiral down in polar cylindrical coordinates. How does one avoid being drowned in possible solutions? I want a curve/spiral that both rises and widens in proportion to its radius. I've seen this equation for in 2d. It's an exponential growth spiral "staircase" I want.

But as to the euler equation, does it look like anything in particular, or should i wait and be surprised?

---------- Post added 04-19-2010 at 04:28 PM ----------

Twirlip;153992 wrote:

Series might have to be used in the rigorous justification of this hand-waving argument, but I hope that that that doesn't take away all of its charm! The full original thread in sci.math is here:
sci.math | Google Groups


Thanks! Sounds you like you have some skills. I've got very little practice tackling such problems. Lack of experience and obsession with foundations. What do you think of this? I've read that the maxima of the x root of x is equal to e. or shall we write it it "x^(1/x)" it pleases me for some reason that the answer is e, as e strikes me as some sort of fulcrum. of course it is its own derivative.
it's as if 0 is additive identity, 1 is the mult identity, and e is the exponential identity. but i'm still working out what i mean by exponential identity. the fact that it's the maxima of the x root of x is pretty damn fascinating. does this touch on its essence?

also it appears in probability. i note that its infinite series is built on the natural series of factorials. i know that probability is exponentially calculated in many cases. so this is fascinating too. if we put all ten digits in a hat, and pick them blindly, removing them one at a time..how many possibilities of sequences do we have? there's something like that that tends to e as the numbers of digits (or whatever) increases toward infinity. thoughts?
 
longknowledge
 
Reply Mon 19 Apr, 2010 09:18 pm
@Reconstructo,
Reconstructo;154109 wrote:
That's sounds wild. I've never used 3-d graph paper. I've scribbled my approximation of an ascending and widening equiangular spiral. But I couldn't figure out how to write this spiral down in polar cylindrical coordinates. How does one avoid being drowned in possible solutions? I want a curve/spiral that both rises and widens in proportion to its radius. I've seen this equation for in 2d. It's an exponential growth spiral "staircase" I want.

But as to the euler equation, does it look like anything in particular, or should i wait and be surprised?

I'm not sure about the spiral, but an ascending and widening surface is what I do remember. The lower portion was like a round bottom vase or cup, but then as it rose it did funny bending. That's all I remember.

:flowers:
 
Reconstructo
 
Reply Tue 20 Apr, 2010 12:22 am
@Reconstructo,
Another charming equation is the solution to i to the power of i. Apparently there are many or an infinite number, but the most natural or simplest is: e^(-pi/2), which is real, and something like:
0.207879576
 
 

 
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