{"id":723,"date":"2008-12-27T08:20:46","date_gmt":"2008-12-27T08:20:46","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2008\/12\/27\/e-the-unnatural-natural-number-classic-repost\/"},"modified":"2014-05-22T10:38:38","modified_gmt":"2014-05-22T14:38:38","slug":"e-the-unnatural-natural-number-classic-repost","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2008\/12\/27\/e-the-unnatural-natural-number-classic-repost\/","title":{"rendered":"e: the Unnatural Natural Number (classic repost)"},"content":{"rendered":"

I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.<\/em><\/p>\n

Anyway. Todays number is e<\/em>, aka Euler’s constant, aka the natural log base. e<\/em> is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn’t expect it. <\/p>\n

<\/p>\n

What is e?<\/h3>\n

e<\/em> is a transcendental irrational number. It’s roughly 2.718281828459045. It’s also the base of the natural logarithm. That means that by definition, if ln(x)=y, then e<\/em>y<\/sup>=x. Given my highly warped sense of humor, and my love of bad puns (especially bad geek<\/em>puns) , I like to call e<\/em> theunnatural natural number<\/em>. (It’s natural in the sense that it’s the base of the natural logarithm; but it’s not a natural number according to the usual definition of natural numbers. Hey, I warned you that it was a bad geek pun.)<\/p>\n

But that’s not a sufficient answer. We call it the natural<\/em>logarithm. Why is that bizarre irrational number just a bit smaller than 2 3\/4 natural<\/em>?<\/p>\n

Take the curve y=1\/x. The area under the curve from 1 to n is the natural log of n. e<\/em> is the point on the x axis where the area under the curve from 1 is equal to one:<\/p>\n

\"ln.jpg\"<\/p>\n

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! + …) <\/p>\n

It’s also what you get if you take the limit: lim<\/em>n → ∞<\/sub> (1 + 1\/n)n<\/sup>.<\/p>\n

It’s also what you get if you work out this very strange looking series:<\/p>\n

2 + 1\/(1+1\/(2+2\/(3+3\/(4+..))))<\/p>\n

It’s also the base of a very strange equation: the derivative of e<\/em>x<\/sup> is… e<\/em>x<\/sup>. <\/p>\n

And of course, as I mentioned in my post on i<\/em>, it’s the number that makes the most amazing equation in mathematics work: e<\/em>iπ<\/sup>=-1.<\/p>\n

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 deeplynatural<\/em>number; it’s tied into the shape of a circle, to the basic 1\/x curve. There are dozens of different ways of defining it, because it’s so deeply embedded in the structure ofeverything<\/em>. <\/p>\n

Wikipedia even points out that if you put $1 into a bank account paying 100% interest compounded continually, at the end of the year, you’ll have exactly e<\/em> dollars. (That’s not too surprising; it’s just another way of stating the integral definition of e<\/em>, but it’s got a nice intuitiveness to it.)<\/p>\n

History<\/h3>\n

e<\/em> has less history to it than the other strange numbers we’ve talked about. It’s a comparatively recent discovery. <\/p>\n

The first reference to it indirectly by William Oughtred in the 17th century. Oughtred is the guy who invented the slide rule, which works on logarithmic principles; the moment you start looking an logarithms, you’ll start seeing e<\/em>. He didn’t actually name it, or even really work out its value; but hedid<\/em>write the first table of the values of the natural logarithm. <\/p>\n

Not too much later, it showed up in the work of Leibniz – not too surprising, given that Liebniz was in the process of working out the basics of differential and integral calculus, and e<\/em> shows up all the time in calculus. But Leibniz didn’t call it e<\/em>, he called it b. <\/p>\n

The first person to really try to calculate a value for e<\/em> was Bernoulli, who was for some reason obsessed with the limit equation above, and actually calculated it out.<\/p>\n

By the time Leibniz’s calculus was published, e<\/em> was well and truly entrenched, and we haven’t been able to avoid it since.<\/p>\n

Why the letter e<\/em>? 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”.<\/p>\n

Does e<\/em> have a meaning?<\/h3>\n

This is a tricky question. Does e<\/em> mean anything? Or is it just an artifact – a number that’s just a result of the way that numbers work? <\/p>\n

That’s more a question for philosophers than mathematicians. But I’m inclined to say that the number<\/em> e<\/em> is an artifact; but the natural logarithm<\/em>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 property with the 1\/x curve; it really is a deeply natural thing that expresses very important properties of the basic concepts of numbers. As a logarithm, some number had to be the base; it just happens that it works out to the value e<\/em>. But it’s the logarithm that’s really meaningful; and you can calculate the logarithm without<\/em>knowing the value of e<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"

I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised. Anyway. Todays number […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[13,43],"tags":[],"class_list":["post-723","post","type-post","status-publish","format-standard","hentry","category-classics","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-bF","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/723","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=723"}],"version-history":[{"count":2,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/723\/revisions"}],"predecessor-version":[{"id":2975,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/723\/revisions\/2975"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=723"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=723"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}