This equation basically tells you that if you rotate a vector in a 2D space by an angle of 180° then you will get the same vector but opposite direction. I do not see any relation with "Human Soul" here.

Simen, I fully agree with you. The fact that this equation includes pi, e, 1 and 0 makes it not only extremely beautiful but also fascinating as it seem to make a link between distinct branches of mathematics. On the other hand, I do not see the relation with "Human Soul" as suggested in the original question.

Both e and pi are very special irrational numbers. e governs the time evolution because exponential function is the eigenvector of the derivative map. pi is essentially the geometrical structure of our local universe. i is a generalization of the well known real numbers and is correlated to rotations around one axis. So this simple equation combines the geometry with the evolution of the universe due to simple rotation!

Sometimes, one refers to this equation as the "Euler's pearl". This is due to the fact that this equation establish a relationship between the 5 most used matematical constants (0, 1, e, pi and i), using basic operators (addition, multiplication, power). Really a bright pearl!

Besides, if you take a closer look to the first term, you have a real, positive and irrational number elevated to an imaginary and irrational number, an this gives a negative integer number. AWESOME!

If this equation had inside the phi number (the golden section) then it would contain all! (evolution, geometry, beauty and counting).

Let's eliminate 'e': exp(i*pi)=-1 => Ln(exp(i*pi))=Ln(-1)=>i*pi=Ln(-1)=>pi^4=(Ln(-1))^4. Now we have only pi, -1 and 4.

It is not quite the jewel that people think it is, because there is no such thing as taking a real number to an imaginary power. Euler DEFINED e^i*theta as cos theta + i sin theta. This is a notational convention. The right hand side is a complex function of a real number, and hence it makes sense. When you differentiate it, multiply it, etc., it behaves LIKE an exponential, so that is why the artificial definition. If you choose theta = pi, then you get e^i pi = cos pi + i sin pi = -1 + i(0) = -1. Thus, e^i pi+ 1 = 0. Sorry to bust everyone's bubble, but it's not that exciting. It's true because cos pi = -1 and sin pi = 0, but you all knew that in middle school geometry.

Euler's identity uses each of 0,1,pi,e,i only once.

0 and 1 are the two most imperative whole numbers.

pi and e are the two most popular and conventional irrational numbers.

i is imaginary.

An identity which uses each of these 5 numbers exactly once...I think that's beautiful.

Demetris, Ln(-1) ??

The principal branch of logarithmic function is denoted by Ln(z).

But "Ln" is not defined on negative real half-axis, where the number -1 is located.

Marek, your point strictly is accurate. But look here:

http://en.wikipedia.org/wiki/Complex_logarithm

Log(-1)=i*pi

Also every program computes ln(-1):

->R:

> log(1i^2)

[1] 0+3.141593i

-->Octave:

octave:1> log(-1)

ans = 0.00000 + 3.14159i

-->Maple:

ln(-1)=i*Pi

-->Mathematica:

In[2]:= Log[-1]

Out[2]= \[ImaginaryI] \[Pi]

... and so on.

Probably they take care about only the discontinous nature at the non-positive real axis.

It is an important issue. What do you think?

Well, I can only regret that mathematical packages contain some flaws, no matter that they all produce the same wrong answers. Worse yet: some of such nonexistent results are later introduced without any comment into quite prestigious books. This is my second practical encounter with logarithm of negative number. Maybe I am already allergic for it. My only hope is I'm not alone.

You are certainly not alone Marek!

Marek, can you provide us a link with a book that does not define the complex logarithmic function at negative real axis? Because from a search I realized that the only point that Log(z) is not defined is the z=0 and it is not continuous at negative real axis, or strictly at {z\inC:z+|z|=0.}

Math programs cause ln(negative numbers) to make sense as a consequence of Euler's identity, but that's just carrying a notational convention too far. Read my post above about Euler's identity being a notational convention, not a mathematical truism.

For example look here:

http://books.google.gr/books?hl=el&id=Nilzh5adHPgC&q=logarithmic#v=snippet&q=logarithmic&f=false

Demetris, I know that Wikipedia is not the authoritative source, but look at:

http://en.wikipedia.org/wiki/Complex_logarithm

You will not find here the values of Log on negative part of real axis. This function is continuous on the *open* set \mathbb{C}-\mathbb{R}_{\le 0}, as they correctly write. The results you cite are the only possible according to your example book, but the truth is that their negatives may be considered equally good. Two different values for one argument? If so, then Log doesn't deserve the name of a function, even if it has many branches. No wonder the complex analysis folks dislike this situation and prefer not to define Log for x+i0, when real x

Marek, it is supposed that the remedy for the problem of log(-1)=+i*pi,-i*pi is the proper definition of principal argument Arg(z)\in(-pi,pi], so log(-1)=+i*pi, only. Ofcourse I agree that if we do not define Log(z) at x+i0, when real x

I think it is easy to understand that:

1) Log is the inverse function of exp

2) if we can define an expression of power of imaginary number, then we can define its inverse function too

3) if exp of an imaginary number is a negative number, then the log of a negative number can be defined as an imaginary number.

That is it!

Note that exp(s*t) where s can be a complex number is the solution to a differential equation that describes a real dynamic phenomenon.

I think that the “mystery” of the formula stems from the fact that it is highly symbolic and that the meaning of the symbols has changed in the course of the evolution of the corresponding concepts.

For a positive real number a originally a^n has been a symbol for n-fold multiplication, e.g. a^3=aaa. This implies a^(n+m)=(a^n)(a^m) for all positive integers. Then it turned out that if you write 1/a=a^(-1) then a^(n+m)=(a^n)(a^m) remains valid for all integers m,n . If you write the squareroot of a as a^(1/2) then you also have a= a^(1/2) a^(1/2). More generally you can use the symbol a^r for each rational number r in a natural way. But in the meantime the original interpretation as n-fold multiplication has been lost. Then by continuity you can extend this to the function x -> a^x on the real line. This turns out to be differentiable and the simplest formula for the derivative occurs for a=e. The next step is the observation that the exponential function e^x has a power series expansion e^x = 1 + x + x^2/2! + x^3/3! + ... .This power series has nothing to do with the original meaning of exponentiation. Then you can compare this power series with that of cos(x) and sin(x) and you observe that the series expansion of e^(ix) is the same as that of cos(x)+isin(x) if i is a symbol satisfying i^2=-1. This eventually leads you to introduce the complex exponential function e^z for a complex number z=x+iy as the power series e^z = 1 + z + z^2/2! + z^3/3! + ... and to your formula e^(i pi)+1=0.

Thus this formula can either be considered as a rather simple result about power series or as a highly condensed code of the evolution of mathematical analysis.

Johan, your short lecture is simply fantastic! I wish all the teachers could be so concise but informative and crystal clear at the same time. Excellent job!

http://www.bbc.co.uk/news/science-environment-26151062

This is an interesting article!