I am a retired senior mathematician who worked at RAND in Santa Monica. I taught at the RAND Frederick S. Pardee Graduate School during my 29 year career there.
Now that I am retired, I want to spend time helping students get over a fear of mathematics
. I have also been a volunteer math
tutor at Palisades High School. I really enjoyed my time spent with students there, but not the administrative details. I have taught in the classroom, but find the one-on-one interaction with a student to be more rewarding.
My goal as a tutor is to help the student learn the requisite mathematics of course, but also to give the student a feel for the context in which that mathematics is embedded. I was asked once what is so amazing about the equation that e to the power (i times pi) is equal to -1.
Here is what I responded:
Pi and e are quantities/ideas that exist, that are in the ether. They have been there since time began, waiting to be found by someone who was curious. Pi is the ratio of the circumference of a circle to its diameter and is the same for every circle. The naturalness of e is more difficult to describe, but it arises “naturally” in calculus
and its naturalness can be argued in an amazing variety of ways.
That said, both pi and e are very un-simple quantities. They are not integers, they are not fractions
(with repeating decimal expansions like 1/3), and they are not irrational numbers (like the square root of 2). It took the field of mathematics a long time to figure out that they are in a separate category of number that they named “transcendental” numbers. Transcendental numbers are defined by what they are not, so it is very difficult to demonstrate that a given number is transcendental (and pi and e are among the few numbers proven to be transcendental).
i is its own kind of strange quantity. Perhaps the best way to think about it is to say that it does the same thing for square roots that negative numbers do for positive numbers. If you add all of the negative numbers to the positive numbers you get the “real” numbers that go off to infinity in both directions on a number line. Adding i into the mix allows you to take the square root of any real number. (This notion of “expanding the space of mathematics” is a common theme in the history
of mathematics. In a way it’s a search for completeness and – though it often turns out to be useful – it is usually pursued for its own sake).
So pi, e and i are strange quantities. Each has at least one book dedicated to it alone (“The Joy of Pi”, “e: The Story of a Number”, and “An Imaginary Talk: The Story of i”). And each is useful in its own right. Pi is very useful in trigonometry
, for example, and e is very useful in calculus. The three of them (alone and together) show up powerfully in engineering, physics
, cosmology, economics
, etc., etc. They all contribute in practical ways to much of the modern world.
Now I’m ready to describe the beauty of the equation. The most important thing to understand is that it is not the kind of equation that one would go looking for. I seriously doubt that any mathematician ever thought “I wonder if I could take pi, e, and i and manipulate them in some way so that they would turn into something very simple.” I don’t know the actual progression of steps that revealed the professor’s equation, but it began with working in the “complex plane” with the real numbers as one axis and the purely imaginary numbers (i, 2i, -4.7i, ei, etc.) as the other. Euler (in the 1700s) was the first man to write the equation down in the form that we know it, and while he is one of the truly monumental minds in the history of mathematics, his work built on the work of the hundreds of mathematical discoveries that preceded him.
In some real sense, the equation is an afterthought of the work that Euler was doing. It’s not quite like Einstein’s famous equation, it’s really only a summary of the things that Euler was finding out about working in the complex plane. But to stand back and look at it is to notice that three very exotic and powerful concepts/quantities from different strands of mathematical thought combine in an astonishing way to produce something very simple. It is every bit as beautiful to mathematicians as the “golden ratio” (another fascinating and precise mathematical entity) was to Leonardo da Vinci.