From mboxrd@z Thu Jan 1 00:00:00 1970 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Michael Heerdegen Newsgroups: gmane.emacs.help Subject: Re: The next exercise Date: Fri, 09 Dec 2022 21:38:25 +0100 Message-ID: <87y1rgs472.fsf@web.de> References: <87tu24tmv6.fsf@web.de> Mime-Version: 1.0 Content-Type: text/plain Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="32033"; mail-complaints-to="usenet@ciao.gmane.io" User-Agent: Gnus/5.13 (Gnus v5.13) To: help-gnu-emacs@gnu.org Cancel-Lock: sha1:3i0mw/MZIsR8KkO3tUwQJoUzOVg= Original-X-From: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane-mx.org@gnu.org Fri Dec 09 21:39:13 2022 Return-path: Envelope-to: geh-help-gnu-emacs@m.gmane-mx.org Original-Received: from lists.gnu.org ([209.51.188.17]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1p3k9I-00084Q-KE for geh-help-gnu-emacs@m.gmane-mx.org; Fri, 09 Dec 2022 21:39:12 +0100 Original-Received: from localhost ([::1] helo=lists1p.gnu.org) by lists.gnu.org with esmtp (Exim 4.90_1) (envelope-from ) id 1p3k8t-0001Ng-BP; Fri, 09 Dec 2022 15:38:48 -0500 Original-Received: from eggs.gnu.org ([2001:470:142:3::10]) by lists.gnu.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.90_1) (envelope-from ) id 1p3k8m-0001NL-G1 for help-gnu-emacs@gnu.org; Fri, 09 Dec 2022 15:38:40 -0500 Original-Received: from ciao.gmane.io ([116.202.254.214]) by eggs.gnu.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.90_1) (envelope-from ) id 1p3k8j-00087H-Iy for help-gnu-emacs@gnu.org; Fri, 09 Dec 2022 15:38:38 -0500 Original-Received: from list by ciao.gmane.io with local (Exim 4.92) (envelope-from ) id 1p3k8e-00079V-BQ for help-gnu-emacs@gnu.org; Fri, 09 Dec 2022 21:38:32 +0100 X-Injected-Via-Gmane: http://gmane.org/ Received-SPF: pass client-ip=116.202.254.214; envelope-from=geh-help-gnu-emacs@m.gmane-mx.org; helo=ciao.gmane.io X-Spam_score_int: -13 X-Spam_score: -1.4 X-Spam_bar: - X-Spam_report: (-1.4 / 5.0 requ) BAYES_00=-1.9, FREEMAIL_FORGED_FROMDOMAIN=0.248, FREEMAIL_FROM=0.001, HEADER_FROM_DIFFERENT_DOMAINS=0.25, SPF_HELO_NONE=0.001, SPF_PASS=-0.001 autolearn=no autolearn_force=no X-Spam_action: no action X-BeenThere: help-gnu-emacs@gnu.org X-Mailman-Version: 2.1.29 Precedence: list List-Id: Users list for the GNU Emacs text editor List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Errors-To: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane-mx.org@gnu.org Original-Sender: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane-mx.org@gnu.org Xref: news.gmane.io gmane.emacs.help:141529 Archived-At: writes: > My choice was school maths. But I cheated, because I > studied physics later, so those maths were kept warm > for some longer while ;-) Ok, I owe the mathematical background, at least a bit. The general background is that /n + 1\ n |- - -| --> e \ n / and if you set n := 1000 you get that 1001^1000 is approximately e*1000^1000 (*). Regards, Michael. (*) But is this enough to be sure that the first three digits are those of e? To be really sure you either check with some sort of calculator which can calculate such large numbers - there are several ways in Emacs as we saw! Or you remember how e can be introduced: with / 1\ n / 1\ n+1 a_n := |1 + -| and b_n := |1 + -| \ n/ \ n/ you got that a_n is strictly monotonously increasing, b_n strictly monotonously decreasing, and b_n - a_n --> 0, so they both converge and the limit is called e. From a_n < e < b_n you then also see that e b_n n+1 b_n b_n n+1 --- < --- = --- and --- < --- = --- a_n a_n n e a_n n and from that you can even conclude how good the approximation is. For n=1000 the exact value must fulfill 1 ----- *e*1000^1000 < 1001^1000 < 1.001*e*1000^1000 1.001 which is good enough for three digits 271 from e. This is a quite surprising thing, I mean, here are no functions and derivatives, no fractions, no combinatorics etc involved. Still, (1+10^n)^(10^n), a simple product of natural numbers, always starts with the first n digits of e for every n. More and more the larger n gets. Ok - I really apologize for the little mathematical excursion. Have a look at M-x calc (it's built in) if you haven't yet. It's worth it!