Which Euler Equation?

One of Leonhard Euler's equations is below, but it's far from the only one (how many Euler equations are there anyway?). However, for reasons explained at the end, many people believe this was his most "beautiful" equation, and one of the most beautiful equations of all-time. Here's a brief history of Euler's life and achievements:

The Countless Achievements of a Math Master, by David Brown Washington Post: ...In 1988, the journal Mathematical Intelligencer asked its readers to list the most beautiful equations in mathematics. Of the top five, [Leonhard] Euler, who was born in Basel, Switzerland, 300 years ago next Sunday, discovered three of them, including No. 1:

e^iπ + 1 = 0.

...In 2004, Physics World put the same question to its readers. Of the top 20 equations, Euler had two. The one listed above, known as "Euler's equation," was second only to James Clerk Maxwell's equations describing electromagnetism...

Some have called Euler the "Mozart of Mathematics," not only because of his genius but because of his prodigious output. Before his death at 76, he had written more than 800 papers and books on pure and applied mathematics. In 1775, he composed about one paper a week, ranging in length from 10 to 50 pages. (Twenty papers is considered a good lifetime output for modern mathematicians.) His collected works fill 25,000 pages in 79 volumes, including five of correspondence to the leading thinkers of his day.

Amazingly, that's not all of it. More letters and a dozen notebooks will be published over the next decade. If the past is a guide, they are likely to contain work that in some sense is original even today.

Three centuries after his birth, Euler is far from a household name... He didn't jump out of the bathtub and run naked through the streets, like Archimedes. His head didn't get hit by an apple, like Newton's. He didn't figure out, before age 10, how to add every number from 1 to 100 in less than a minute, like Gauss. Nevertheless, he's right there with them. ...

William Dunham, a professor of mathematics at Muhlenberg College in Pennsylvania, added that Euler is "an amazingly seminal figure in physics, as well. He wrote about optics, classical mechanics, fluid mechanics and astronomy -- in those days it was all sort of one big subject."

Euler (pronounced "oiler") was the first child of a pastor and his wife. His father had a talent for mathematics and instructed Leonhard, who enrolled in the University of Basel at age 13.

There, he studied under Johann Bernoulli, one of Europe's eminent mathematicians, and met Bernoulli's sons, Nicholas and Daniel, who were to become famous scientists themselves. Daniel was to be Euler's best friend.

The younger Bernoullis went to St. Petersburg to join the Russian Academy of Sciences. Soon after arriving, they persuaded Catherine I, Peter the Great's widow, to invite Euler, too. He arrived in 1727, at age 20. Euler spent about 30 years in Russia in two long stints, interrupted by about 25 years in Berlin...

As one would expect, Euler was good at all kinds of things. His first language was German. He wrote principally in Latin, with many papers in French ... and a few in German. He spoke Russian. A few letters to London's Royal Society in English survive. As a young man, he studied Greek and Hebrew.

Euler contributed to essentially every field of mathematics -- calculus, geometry, number theory and the vast realm of applied mathematics. ... Euler's greatest achievements may lie in what became mathematical analysis, which includes calculus and differential equations.

Although Newton and Gottfried Leibniz discovered calculus, Euler systematized it, made hundreds of discoveries and invented differential equations, which he successfully applied to mechanics and astronomy, transforming them from geometry-based disciplines to fully calculus-based ones. He almost single-handedly invented the calculus of variations...

Euler also recognized the importance of the number e, which he named... It is ... the base of natural logarithms and essential to the calculation of such things as compound interest. Like π, it is also a value that pops up in all sorts of unexpected places -- one of the universe's favored numbers.

Euler's achievements were all the more remarkable because he lived a life that was both relatively normal and quite difficult.

He married twice and fathered 13 children. Only five of them survived into adolescence. He played the clavier and composed a small corpus of music based on mathematical equations. ... He was a masterful chess player. ...

In his early 30s, Euler lost most of the sight in his right eye. He developed a cataract in the other and was legally blind for the last dozen years of his life. As his sight failed, he took to writing on a huge slate on a round table, dictating his papers to a Swiss secretary.

He worked incessantly even after his eyesight failed, and was, it appears, a happy man. On the day he died in St. Petersburg, Sept. 18, 1783, his slate reportedly contained a calculation of the height to which a hot-air balloon could rise. News of the first balloon ascent, in Paris the previous June, had recently arrived. Says Dunham: "You could hardly argue that he wasted a day of his life."

The equation above is actually a special case of Euler's equation. More generally:

e^ix = cos x  + i sin x

and, when x = π, you get the equation above. The reason many people think the case where x = π is special is because:

The special case gives the beautiful identity, e^iπ + 1 = 0, an equation connecting the fundamental numbers i, π, e, 1, and 0, the fundamental operations +, x, and exponentiation, the most important relation =, and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician...

Comments

[Jonathan Lundell]

Bit of a typo there, in the general form of the equation.

[Mark Thoma]

Darn cut and paste - thanks, I'll fix that [had e ^ {i pi} rather than e ^ {i x}].

[John H. Morrison]

> how to add every number from 1 to 100 in less than a
> minute

Well, a couple seconds is less than a minute. And if you do it this way: 1 + 2 + 3 + ... + 99 + 100 = 100(101)/2 = 5050, you can do it in a couple seconds as well.

[billy]

"Mozart of Mathematics,"? Surely, the author was trying to play with words. The correct address would be "Bach of Mathematics", for reasons of both aesthetics and the mathematics of Bach's works.

[Patrick]

Billy said:

"Mozart of Mathematics,"? Surely, the author was trying to play with words. The correct address would be "Bach of Mathematics", for reasons of both aesthetics and the mathematics of Bach's works.

I don't think you get the metaphor. Bach was not well known for simplistic elegance, for being extremely proliferous, or for being a child prodigy.

[Eric H]

Actually, I think Billy is right. Bach was also a child prodigy and was far more prolific than Mozart. I vaguely recall something my sister told me about how you could listen to Mozart's entire output played continuously in about 24 hours, but it would take a week to listen to Bach. Besides of course the mathematics of Euler's music and the music of Bach's mathematics.

[dave]

I think Beethoven. Beethoven brought romance to music. Euler did it for maths.

Although, I'm with Mark and Patrick -the Mozart metaphor does fit better. Mozart is more noted for being the child prodigy. His music is simple/elegant (classical). Bach's was complex (baroque).