Thursday, May 7, 2009

A man you should know about

There once was a man name Leonhard Euler (pronounced "Oil-er"). No other man in the history of mathematics, with perhaps the only exception being Newton, has his named attached to so many ideas, postulates, proofs, and former NFL football teams than the Master Euler. Without a doubt, he was the most prolific writer of mathematics ever, his total works filling at least 100 volumes (and still counting). He averaged about 800 pages of new work each year during his long, productive life. It is said that it would take an entire volume itself just to publish the table of contents for his works.

He left his mark on virtual every branch of mathematics in such diverse fields as number theory, analysis, hydrodynamics and mechanics, topology, cartography, astronomy, and even dabbled in the seemingly unrelated fields of science, public affairs, philosophy, and even theology. Without his passion for discovery, the world would be much less advanced than it is today, and common mathematical symbols like i (the imaginary unit, equal to the square root of negative one), π (pi ~3.14159...), f(x), and e (called Euler's number, or the natural base ~2.71728...) would likely look more like numbers rather than those sneaky symbols that look like letters. Because of Mr. Euler, I can write my name like this:

Euler's most important contributions were so vast and numerous that if I referred to something as simple as "Euler's formula" or "Euler's theorem" people would either look at me like I was a dorky math nerd, or if they were a dorky math nerd themselves, would chuckle before incredulously saying, "which one?" (We mathematicians love the humor brought about by an elliptic statement, like the previous one, or hyperbolic statements, like, "Have you ever heard of Korpi's theorem??") Depending on the context in which you refer to the greatest mathematician of the 18th century, his "formula" or "theorem" can take on many, many different meanings. Just in mechanical physics alone, one has Euler angles--which specify the orientation of a rigid body, Euler's theorem--simply stating that every rotation has an axis, Euler's equations for motion of fluids, and the Euler-Lagrange equation--arising from the calculus involved.

Despite what you learned growing up that there are only two types of numbers in the whole-wide-world: the number Zero, and numbers that aren't zero, there are actually many types of numbers. In fact, when it comes to Euler, there are both Euler numbers and Eulerian numbers, and Euler's number (the number e already mentioned), and they are NOT the same thing. Some numbers resemble the three previous types, but are not those types. These numbers I have personally named Euleresque numbers. Euler's study of the Bridges of Königsberg can be seen as the beginning of combinatorial topology (which is why the Euler characteristic bears his name, but to make matters worse, the Euler characteristic is sometimes called the Euler number too!! I told you it depends on the context). The Bridge problem itself lends itself to defining what's called an Euler walk.

Can you cross all yellow bridges only once??

If you were to ask any advanced calculus student what the "Euler Formula" is, they might give you two different answers. On one hand, it can refer to the equation that defines the exponentials of imaginary numbers in terms of trigonometric functions. But there is another "Euler's formula" that (to use the modern terminology adopted long after Euler's death) gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers. Wow! That's a mouthful. My BC calculus students would tell you that Euler's Formula is something entirely different.

They would tell you that it's the equation that proves the existence of God! That's right, they have a good teacher. Sometimes called Euler's Identity or Euler's Equation, it can be derived from the first context of Euler's formula above evaluated at π, we can arrive at the following equation via infinite series:

Why does it proves God's existence? Well, it would fall under the Teleological argument of creation by design. The equation is too perfect and beautiful, not to mention TRUE, that it couldn't have existed unless it was created by a god, by THE God. The equation itself contains the three basic arithmetic operations occuring exactly once each: addition, multiplication, and exponentiation. The equation also links the five fundamental mathematical constants: 0, 1, π, i, and e itself. Like the likelihood of a finely-crafted, precision Swiss watch just coming together by a random arrangement of its parts, or the probability of the monkeys at typewriters producing the complete works of Shakespeare, the equation itself had to have had a creator, a divine one.The creator is God. Euler merely discovered it.

Are you still not impressed with this guy?

It was said of him that he "calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind." But don't think that an industrious brainiac was all seriousness. He also had quite a sense of humor too. For instance, late in life he went blind in one eye, and when asked how it would affect his math studies, he quipped, "Now I shall be less distracted." Talk about making lemonade out of lemons. In fact, he soon thereafter became totally blind in both eyes for the last 17, and MOST PRODUCTIVE, years of his life. This gives entirely new meaning to being "in the dark" when it pertains to mathematics.

Euler's powers of memory and concentration were incredible and quite legendary. He could recite the entire Aeneid, all 12 books!! word-for-word. I have trouble just remembering how to spell Aeneaid. A guy who spent so much of his waking hours in his study discovering and writing mathematics still had time to father 13 children with two different wives. He was not troubled, however, by all the distractions, interruptions, and little league baseball games. In fact, he did most of his work with his children playing at his feet, never kicking them away. His mind was a human calculator, capable of doing prodigious calculations in his head, which he increasingly relied on after he went blind. Apocryphal evidence tells the story of two of Euler's students who had independently summed seventeen terms of a complicated infinite series, only to disagree in the fiftieth decimal place; Euler settled the dispute by recomputing the sum in his head.

Genius like Euler comes once a century, if that, and although he and I are so very different, we do have one thing in common besides our sense of humor . . . we're both Yankee fans.

Come on, Ump. A blind ma . . . I could have made that call.

1 comment:

Sam Gilbert said...

Dear readers,

I am a huge fan of Euler! What an amazing mathematician he was.

Here is an excerpt from my new book. Much of it is based on Euler's work. A print of Euler graces the front cover.

The Riemann Hypothesis & the Roots of the Riemann Zeta Function

by Samuel W. Gilbert

available from

© U. S. Copyrights - 2009, 2008, 2005

This book is concerned with the geometric convergence of the Dirichlet series representation of the Riemann zeta function at its roots in the critical strip. The objectives are to understand why non-trivial roots occur in the Riemann zeta function, to define the roots mathematically, and to resolve the Riemann hypothesis.

The Dirichlet infinite series parts of the Riemann zeta function diverge everywhere in the critical strip. Therefore, it has always been assumed that the Dirichlet series representation of the zeta function is useless for characterization of the roots in the critical strip. In this work, it is shown that this assumption is completely wrong.

The Dirichlet series representation of the Riemann zeta function diverges algebraically everywhere in the critical strip. However, the Dirichlet series representation does, in fact, converge at the roots in the critical strip ̵and only at the roots in the critical strip in a special geometric sense. Although the Dirichlet series parts of the zeta function diverge both algebraically and geometrically everywhere in the critical strip, at the roots of the zeta function, the parts are geometrically equivalent and their geometric difference is identically zero.

At the roots of the Riemann zeta function, the two Dirichlet infinite series parts are coincidently divergent and are geometrically equivalent. The roots of the zeta function are the only points in the critical strip where infinite summation and infinite integration of the terms of the Dirichlet series parts are geometrically equivalent. Similarly, the roots of the zeta function with the real part of the argument reflected in the critical strip are the only points where infinite summation and infinite integration of the terms of the Dirichlet series parts with reflected argument are geometrically equivalent.

Reduced, or simplified, asymptotic expansions for the terms of the Riemann zeta function series parts at the roots, equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument at the roots, constrain the values of the real parts of both arguments to the critical line. Hence, the Riemann hypothesis is correct.

At the roots of the zeta function in the critical strip, the real part of the argument is the exponent, and the real and imaginary parts combine to constitute the coefficients of proportionality in geometrical constraints of the discrete partial sums of the series terms by a common, divergent envelope.

Values of the imaginary parts of the first 50 roots of the Riemann zeta function are calculated using derived formulae with 80 correct significant figures using a laptop computer. The first five imaginary parts of the roots are:


It is further demonstrated that the derived formulae yield calculated values of the imaginary parts of the roots of the Riemann zeta function with more than 330 correct significant figures.