# E_ the Unnatural Natural Number

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Todays number is e, aka Euler’s constant, aka the natural log base. e is a very odd number, but very fundamental
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Todays number is e , aka Euler’s constant, aka the natural log base. e  is a very odd number, but veryfundamental. It shows up constantly, in all sorts of strange places where you wouldn’t expect it. What is e? e  is a transcendental irrational number. It’s roughly 2.718281828459045. It’s also the base of the naturallogarithm. That means that by definition, if ln(x)=y, then e y =x. Given my highly warped sense of humor, andmy love of bad puns (especially bad geek  puns) , I like to call e the unnatural natural number  . (It’s natural inthe sense that it’s the base of the natural logarithm; but it’s not a natural number according to the usualdefinition of natural numbers. Hey, I warned you that it was a bad geek pun.)But that’s not a sufficient answer. We call it the natural  logarithm. Why is that bizarre irrational number just abit smaller than 2 3/4 natural  ?Take the curve y=1/x. The area under the curve from 1 to n is the natural log of n. e  is the point on the xaxis where the area under the curve from 1 is equal to one:It’s also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/1!+ 1/2! + 1/3! + 1/4! + …)It’s also what you get if you take the limit: lim n → ∞  (1 + 1/n) n .It’s also what you get if you work out this very strange looking series:2 + 1/(1+1/(2+2/(3+3/(4+..))))It’s also the base of a very strange equation: the derivative of e x  is… e x . And of course, as I mentioned in my post on i  , it’s the number that makes the most amazing equation in  mathematics work: e iπ =-1.Why does it come up so often? It’s really deeply fundamental. It’s tied to the fundamental structure of numbers. It really is a deeply natural  number; it’s tied into the shape of a circle, to the basic 1/x curve. Thereare dozens of different ways of defining it, because it’s so deeply embedded in the structure of  everything  .Wikipedia even points out that if you put \$1 into a bank account paying 100% interest compoundedcontinually, at the end of the year, you’ll have exactly e  dollars. (That’s not too surprising; it’s just another way of stating the integral definition of e , but it’s got a nice intuitiveness to it.) History e  has less history to it than the other strange numbers we’ve talked about. It’s a comparatively recentdiscovery.The first reference to it indirectly by William Oughtred in the 17th century. Oughtred is the guy whoinvented the slide rule, which works on logarithmic principles; the moment you start looking an logarithms,you’ll start seeing e . He didn’t actually name it, or even really work out its value; but he did  write the firsttable of the values of the natural logarithm.Not too much later, it showed up in the work of Leibniz – not too surprising, given that Liebniz was in theprocess of working out the basics of differential and integral calculus, and e  shows up all the time incalculus. But Leibniz didn’t call it e , he called it b.The first person to really try to calculate a value for e  was Bernoulli, who wasfor some reason obsessed with the limit equation above, and actually calculated it out.By the time Leibniz’s calculus was published, e  was well and truly entrenched, and we haven’t been able toavoid it since.Why the letter e ? We don’t really know. It was first used by Euler, but he didn’t say why he chose that.Probably as an abbreviation for “exponential”. Does e  have a meaning? This is a tricky question. Does e  mean anything? Or is it just an artifact – a number that’s just a result of theway that numbers work?That’s more a question for philosophers than mathematicians. But I’m inclined to say that the numbere  is anartifact; but the natural logarithm is deeply meaningful. The natural logarithm is full of amazing properties –it’s the only logarithm that can be written with a closed form series; it’s got that wonderful interval propertywith the 1/x curve; it really is a deeply natural thing that expresses very important properties of the basicconcepts of numbers. As a logarithm, some number had to be the base; it just happens that it works out tothe value e . But it’s the logarithm that’s really meaningful; and you can calculate thelogarithm without  knowing the value of e .

Jul 23, 2017

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Jul 23, 2017
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