Yesterday I paid tribute here to a great mathematician. Today I offer a quote from him. "To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as there are usually believed to be." The significance of that observation may not be obvious outside of some consideration of the history of math, the development of calculus in particular. But once we do a little grappling with that history, we see that Euler is making a point here that takes us back to ancient Greece. Back to Zeno, Achilles and tortoise. If Achilles is to catch up with the tortoise here must be a moment at which the difference between them is zero. One way of looking at the problem is to ask what is the next lower number -- one really really close to zero but still a positive number! Euler here is saying "That is the wrong way of looking at it." Or, "don't create mys...
Leonhard Euler (1707-1783) was surely one of the most prolific of great mathematicians. Among his contributions, we need to mention two, each of which comes down to us as a single letter: the letter e and the letter i . If we were literally to "pay"tribute, we might do so in increments of $2.72, rounding up a bit the value of irrational e. We'll get to that. First: biography. Euler was born in Basel, Switzerland, so his life and work might fittingly be considered a riposte to the old anti-Swiss jibe (originally from The Third Man ) that Switzerland has produced nothing for all its years of peace and democracy, nothing more than the humble cuckoo clock. Since Euler’s day and because of his work, i has stood for the simplest of the numbers that Descartes had called “imaginary.” This i refers to the square root of -1. We don’t need to bother ourselves further with the question “ what is the square root of -1?” It is i , by stipulation. Also since Eul...