{"id":97,"date":"2006-08-02T11:17:44","date_gmt":"2006-08-02T11:17:44","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/08\/02\/e-the-unnatural-natural-number\/"},"modified":"2006-08-02T11:17:44","modified_gmt":"2006-08-02T11:17:44","slug":"e-the-unnatural-natural-number","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/08\/02\/e-the-unnatural-natural-number\/","title":{"rendered":"e – the Unnatural Natural Number"},"content":{"rendered":"

Looks like I’ve accidentally created a series of articles here about fundamental numbers. I didn’t intend to; originally, I was just trying to write the zero article I’d promised back during the donorschoose drive.
\nAnyway. Todays number is *e*, aka Euler’s constant, aka the natural log base. *e* is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn’t expect it.
\nWhat is e?
\n————
\n*e* 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*y<\/sup>=x. Given my highly warped sense of humor, and my love of bad puns (especially bad *geek* puns) , I like to call *e* the *unnatural natural number*. (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.)
\nBut that’s not a sufficient answer. We call it the *natural* logarithm. Why is that bizzare irrational number just a bit smaller than 2 3\/4 *natural*?
\nTake 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 x axis where the area under the curve from 1 is equal to one:
\n\"ln.jpg\"
\nIt’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! + …)
\nIt’s also what you get if you take the limit: *lim*n → ∞<\/sub> (1 + 1\/n)n<\/sup>.
\nIt’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*x<\/sup> is… *e*x<\/sup>.
\nAnd of course, as I mentioned yesterday, it’s the number that makes the most amazing equation in mathematics work: *e*iπ<\/sup>=-1.
\nWhy 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. There are dozens of different ways of defining it, because it’s so deeply embedded in the structure of *everything*.
\nWikipedia 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* dollars. (That’s not too suprising; it’s just another way of stating the integral definition of *e*, but it’s got a nice intuitiveness to it.)
\nHistory<\/p>\n","protected":false},"excerpt":{"rendered":"

Looks like I’ve accidentally created a series of articles here about fundamental numbers. I didn’t intend to; originally, I was just trying to write the zero article I’d promised back during the donorschoose drive. Anyway. Todays number is *e*, aka Euler’s constant, aka the natural log base. *e* is a very odd number, but very […]<\/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":[43],"tags":[],"class_list":["post-97","post","type-post","status-publish","format-standard","hentry","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-1z","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/97","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=97"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/97\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=97"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=97"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}