From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-1.3 required=5.0 tests=BAYES_00,INVALID_MSGID autolearn=no autolearn_force=no version=3.4.4 X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: fc89c,97188312486d4578 X-Google-Attributes: gidfc89c,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public X-Google-Thread: 109fba,baaf5f793d03d420 X-Google-Attributes: gid109fba,public From: jvl@ocsystems.com (Joel VanLaven) Subject: Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?] Date: 1996/09/24 Message-ID: <1996Sep24.133312.9745@ocsystems.com>#1/1 X-Deja-AN: 185038668 references: <01bb8df1$2e19d420$87ee6fce@timpent.airshields.com> <4vcac4$gm6@zeus.orl.mmc.com> <01bb8f19$9a89d820$32ee6fce@timhome2> <841797763snz@genesis.demon.co.uk> <322f864d.42836625@news.demon.co.uk> <01bb9bf9$61e9e0e0$87ee6fce@timpent.airshields.com> <50sj6q$aci@mtinsc01-mgt.ops.worldnet.att.net> <01bb9d25$9cb3cb00$32ee6fcf@timhome2> <50v6k3$soo@mtinsc01-mgt.ops.worldnet.att.net> <01bb9ded$cd0fdf00$32ee6fcf@timhome2> <5136on$7qj@goanna.cs.rmit.edu.au> <01bb9f26$36c870e0$87ee6fce@timpent.a-sis.com> <51knhg$j61@dub-news-svc-8.compuserve.com> organization: OC Systems, Inc. followup-to: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada Date: 1996-09-24T00:00:00+00:00 List-Id: George (grs@liyorkrd.li.co.uk) wrote: : jsa@alexandria (Jon S Anthony) wrote: : > In article <01bb9f26$36c870e0$87ee6fce@timpent.a-sis.com> "Tim Behrendsen" writes: : > > It is similiar to the difference between summation and integration; : > > one consists of individual sums, the other of an infinite number : > > of sums. However, both are fundamentally adding. : > Well, that is one option. But as "everyone" knows, the FTC allows you : > to compute definite integrals without taking the limits of sums or : > using summations at all. Incidentally, none of the standard : > definitions (Riemann Sum or something) uses "an infinite number of : > sums". Can't - infinity is not part of the real numbers... : Surely the definition of integration contains the phrase "...tends to : infinity", i.e. it's _as if_ there was an infinite number of sums. : G. Actually, not all functions are integrable. The most complete and irrefutable definition of the integral of a function to my knowledge is : Given a partition P of [a,b] (P is a finite subset of [a,b] including a and b) P={x0,x1,x2,...xn} such that a=x0, b=xn, and x(j+1)>xj call Mj the lub(f([x(j-1),xj]) (least upper bound) call mj the glb(f([x(j-1),xj]) (greatest lower bound) The number Uf(P)=SUM(Mj(xj-x(j-1))) 1<=j<=n is called the P upper sum for f The number Lf(P)=SUM(mj(xj-x(j-1))) 1<=j<=n is called the P lower sum for f The unique number I that satisfies the inequality Lf(P)<=I<=Uf(P) for all possible P of [a,b] is called the definite integral of f from a to b. So, you could talk about an infinite (aleph 2)! number of sums, but it is ridiculus. We are talking about more uncountable that uncountable. No one integrates this way. It is a mathematical abstraction. This is the definition that we then prove is equivalent to some simpler method for certain "nice" functions. Riemann sums only work for uniformly continuous functions. Even then we don't actually USE the Riemann sums! We use them only as ways to let us use even better methods. In the mathematics there are REAL infinities but a good mathematician NEVER actually calculates an infinite anything, as that is impossible! they PROVE that thier answer is correct through techniques that remove infiniteness like induction or certain properties of the real numbers like completeness. While a hand-wavy infinite sums explanation usually satisfies the non-mathematicians (like say engineers), the truth is at once more complicated, simple, and beautiful. -- A Mathematician is correct If the the most REAL :) definition of integration (as I was taught in advanced calculus) is: -- -- Joel VanLaven