Yesterday we discussed the limit of the process of compounding interest. As we move through even shorter periods, from semi-annual to quarterly to monthly compounding and so forth, and as we approach an infinity of infinitesimal periods, i,e. continuous compounding, we find the number 2.71828 arises to stare us in the face as a limit.
This was Jacob Bernoulli's result. So why is the number called Euler's number, and sometimes just e? Did Euler name it for himself? And, if so, isn't that an injustice? Shouldn't it be called b?
Well, Bernoulli found 2.71828... but he neglected to name it. Gottfried Leibniz, the man recognized as THE founder of calculus in the German speaking world, seems to have called it "b" in letters he wrote in 1690 and 1691. Perhaps this was a recognition of Bernoulli's role.
More important, though, Bernoulli neglected to explore the properties of this number outside of banking. It was Euler (pictured above) who defined it as the base of the natural logarithm. Euler connected it to exponential functions and proved it was irrational
Euler seems not to have had his own initial in mind at the time. He had used a, b, c, and d for other constants and variables, and (if that is not reason enough) he may have had in mind the word "exponential".
I'll mention just one more Eulerian contribution here. He said that since the limit identified by Bernoulli, taken to a power defined by the square root of negative 1 (called i) multiplied by pi, equals -1. Further, he expressed that in what is known as Euler's identity.
People rhapsodize over the beauty of that equation. It combines the three key letter symbols into one expression and connects them to the binary pair 1/0. Two mathematicians writing in 1940 said that it "appeals equally to the mystic, the scientist, the philosopher, the mathematician."
And perhaps alchemists.
"Let's take something imaginary and combine it with something irrational!"
"Great, then let us call THAT an exponent of the base of ... something else irrational. But newer!"
"Ooooh, good. Then add it to unity, and get...?"
"Nothingness!"
"Aaaaah."
As to practical applications of all this.....beyond banking of course: I am told that circuits of AC [alternating current] are modelled using sinusoidal functions, and that Euler's identity helps with this work. Please don't press me too hard on what that means.
The fact that Euler started to use "e" for the expression of 2.71828..., along with his importance in developing the number's importance, and the beauty of his identity, have all made and kept it "Euler's number," and e for short.
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