Euler's Beautiful Equation on His 300th

Today is the 300th birthday of the great mathematician Leonhard Euler. Julie J. Rehmeyer discusses his beautiful equation in Science News:

"Gentlemen, that is surely true, 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." —Benjamin Pierce, a Harvard mathematician, after proving Euler's equation, e^[i(pi)] = –1, in a 19th-century lecture.

Sunday, April 15, is the 300th birthday of Leonhard Euler (pronounced "oiler"), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math.

One of the many discoveries for which he is famous is the equation eip = –1 . In a 1988 poll, readers of the journal Mathematical Intelligencer chose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers, e, p, i, and –1, in an astonishingly simple way.

But what does eip = –1 really mean?

Comments

[Daryl McCullough]

What do you mean, what does eip=-1 really mean? Do you mean what does it mean, mathematically, or what does it mean, philosophically, or what?

I can explain what it means mathematically, but it's a little "deep", in that it involves lots of different mathematical concepts combined together in a misleadingly simple result.

In "eip = -1", e is Euler's constant, which is about 2.7.
i is the "imaginary" number whose square is -1. i*i=-1.
p is pi, which is the circumference of a circle whose diameter is 1.

eip = -1 means that when e is raised to the power of (i*pi), the result is -1. What does it mean to raise one number to the power of x? Well, if x is a whole number, then it just means e multiplied by itself x times: e raised to the power of 2 means e*e.

To be able to understand what e raised to the power of x when x is not a whole number, you can use power series: You can prove that e^x = 1 + x + x^2/2 + ...
Now, once you have this form, you can let x be anything, including i*pi. Summing the infinite series, and showing that it equals -1 is a little work, but Euler did it.

[Pat]

Daryl,
The whole paragraph is a quote form the article where they go on explaining what it means ...

Maybe it would have been less confusing to have the "read more" link after the quoted paragraph.

[Daryl McCullough]

Sorry, I was confused by the quotations within quotations. I thought that that line was a comment by Robin.

[Winfield J. Abbe]

If you look in the book "Biographical Encyclopedia of Scientists", Facts on File, 1981, the birthdates of Baron Augustin Louis Cauchy was August 21, 1789 (Paris, France) and Karl Friedrich Gauss April 30, 1777 (Brunswick, now West Germany), they were respectively 82 and 70 years after the genius Leonhard Euler on April 15, 1707 at Basel, Switzerland. In a sense Euler discovered analytic continuation before Cauchy sort of by hit and miss, like Feynman used to do with the "Delta Function" and all his fancy "divergent" integrals. The beauty of Cauchy's Theorem is that integral representations can be used to "extend" or analytically continue power series like the one presented for the exponential function to almost anywhere in the complex plane. Fortunately for Euler, his method worked because this series has an infinite radius of convergence. However, if one uses the series expansion for the function 1/(1-x) for example, this series converges only inside the circle mod x = 1. But the answer is valid for a much large domain of x. This is the beauty of analytic continuation. As the above reference notes about Gauss: "At the age of ten he astonished his school teacher by discovering for himself the formula for the sum of an arithmetical progression..." All three, Euler, Cauchy and Gauss were mathematical physics geniuses. Every day we should give thanks for their births.

[P. Taborsky]

One way to look at it, is that it asserts that there is (or is really the result of) a simple relationship between the exponential function (e^x) and the trigonometric functions. The exponential function is the inverse of the natural logarithm, which is connected with the area under the hyperbola. The trigonometric functions allow one to calculate the area of circles. As both are conic sections, one might guess that there would have to be some simple connection.