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=3.8 required=5.0 tests=BAYES_00,INVALID_MSGID, RATWARE_MS_HASH,RATWARE_OUTLOOK_NONAME 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: 109fba,baaf5f793d03d420 X-Google-Attributes: gid109fba,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,public From: "Dann Corbit" Subject: Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?] Date: 1996/09/27 Message-ID: <01bbaca9$4218e980$2bac399d@v-cnadc1>#1/1 X-Deja-AN: 185720148 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> <1996Sep24.133312.9745@ocsystems.com> content-type: text/plain; charset=ISO-8859-1 organization: Microsoft Corporation mime-version: 1.0 newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada Date: 1996-09-27T00:00:00+00:00 List-Id: Joel VanLaven wrote in article <1996Sep24.133312.9745@ocsystems.com>... > 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. There are far simpler examples of functions that are not integrable. Consder f(x) = 1/x over [-1,1] (It is not continuous) Or consider the integral of e^x over [0, infinity] There is no finite value to the integral of this smooth, continuous function. There is however, for e^(-x*x) over the same interval being sqrt(pi)/2. > 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. Often, infinte integrals can be converted to simple finite integrals with a change of variables. "Numerical Recipies in C" discusses several solutions to "improper" integrals. > -- A Mathematician > is correct > > If the the most REAL :) definition of integration (as I was taught in > advanced calculus) is: > -- > -- Joel VanLaven > -- "I speak for myself and all of the lawyers of the world" If I say something dumb, then they will have to sue themselves.