Wikipedia: The most remarkable formula in the world/Talk

At least according to Richard Feynman, the most remarkable formula in the world is:

: e^iπ + 1 = 0

where e is the base of the natural logarithms, i is the square-root of -1 (see imaginary numbers), and π is the ratio of the circumferance of a circle to its diameter.

-- RaviDesai.

Is there a name for the formula? (see below)

What does it mean--what does it show--why is it remarkable? (Not sure what I'm asking here, but you probably do.)

Richard Feynman is a Nobel Prize winner in physics ([Quantum Electrodynamics]?, 1950s?). He found this formula funny because it links all the main constants a human being is exposed to in this world. Zero and unity arise kinda naturally: one is how one starts to count, and zero comes later... when one does not want to :). π is a constant related to our world being Euclidean (otherwise, the ratio of the length of a circumference to its diameter would not be a universal constant, i.e. the same for all circumferences). The e constant is related to the speed of change, or growth, or whatever like that, as the solution to the simplest growth equation

: dy / dx = y

: y = e^x

Finally, i is the concept introduced mathematically to have a nice property that all polynomials of degree n have exactly n roots in the complex plane. So, quite a lot of rather deep concepts are interrelated within this formula. Of course, there is a number of other ways to arrive to any of those numbers... which only underlines their fundamentality :).

Stas

The most remarkable formula in the world is an example of Euler's Theorem from Complex Analysis, which states that

: e^ib = cos b + i sin b

where b is a real number.

So, if b = π, then

: e^iπ = cos π + i * sin π.

Then, since cos(π) = -1 and sin(π) = 0,

: e^iπ = - 1

and

: e^iπ + 1 = 0

The proof of Euler's Theorem involves the definition of e, by a [Taylor's series expansion]? of e^z (where z is a complex number), De Moivre's formula, and the series expansion of the sine and cosine functions.

Despite this last remark, Euler's Theorem is considered a direct consequence of the extension of the definition of the function e^x over the complex numbers.

e is also known as the limit of (1 + 1/n)ⁿ as n increases indefinitely. With any pocket calculator one can experiment with this function for increasing values of n to see the convergence.

Does e have a name?

I think it's Eulers number.

: It's also called the Exponential constant. Or just plain e.

The rate of growth of e^x (see the growth function dy/dx = x above) is e^x itself:

: (e^x)' = e^x

There are two solutions to the equation x² + 1 = 0, but only one is known as the imaginary unit i. We should probably write "...where i is the imaginary unit, a complex number with the property i² = -1, see imaginary numbers."
de Moivre's formula is not needed to prove Euler's formula. It follows directly from the Taylor series descriptions of the functions e^z for complex z and sin(x) and cos(x) for real x.
Instead of e^ib, it is more common to use e^{i φ} because one can think of φ as an angle. Euler's formula can be interpreted as saying that the function e^{i φ} traces out the unit circle in the complex plane; φ is the angle with the positive real axis, measured counter clockwise and in radians.

FWIW, I totally agree. Like Larry always says, "be bold ..." :-) -- JanHidders

It's interesting to note that even though the previous page says that zero and 1 are elementary counting numbers, 0 was discovered/invented in math hundreds of years after π!

I find it weird that people would consider "The most remarkable formula in the world" to be an appropriate title for an encyclopedia article such as this. Surely there are a multitude of different opinions on the subject. Wouldn't this material be more appropriate in an article about Feynman, or perhaps as a small diversion in an article on mathematics?

: No - this is the actual name by which mathematicians refer to the formula. Ask any mathematician what is "The most remarkable formula..." and they will answer this one, whether they actually agree with the idea or not. Mathematicians are weird. Also, I would dispute whether or not Feynman was the first to use this description, I think it predates him by a substantial period. But I ain't got the facts to prove it right now.--MB

Although I am a grad student in applied maths, I have only come across across this twice or so, and even then it has been qualified as "was once voted as..." or "some people think...". That is why I objected to the title. But I suppose I cannot complain if the established mathematicians here agree.

Ah well, that explains it... you see, it's a pure maths thing. (I said they were weird)

I don't think the formula is the most remarkable one in the world, but I think that it is the only formula that has ever been consistently refered to as "The most remarkable formula in the world". So ideally, the title of this article should be in quotes, but I don't think that's possible at this point. But I'll fix it on the Mathematics page. --AxelBoldt

Amongst the math nerds and geeks at my college this formula is well-known... although we call it the Universal Theory of Everything. ~$0.02~ --KA