Berkeley CSUA MOTD:Entry 26792
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2025/05/24 [General] UID:1000 Activity:popular
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2002/12/11-12 [Politics/Domestic/California] UID:26792 Activity:high
12/10   How does one explain proofs by induction to an idiot?
        \_ Lead him through it step by step.  He should be able to arrive
           at the conclusion.  -John
           \- sing "the sun will come out tomorrow, tomorrow, you can
                   bet your bottom dollar" --psb
           \_ No that's how you explain it to an intelligent person/adolescent
              who has never been taught induction. I don't think induction is
              for idiots.
                \_ It was supposed to be a joke.  -John
        \_ Here, hold this wire...
        \_ A journey of a thousand miles begins with one step
        \_ ask them to use an indirect proof to prove that induction doesn't
           work.  walk them through a simple proof, like how summing increasing
           odd numbers generates squares.
           \_ http://mathforum.org/library/drmath/view/56866.html
        \_ On the same note, does anyone have a good online reference to
           explain induction to a 16 year old brother?
           \_ see the above mathforum link.
           \_ is he an idiot?
           \_ Schools haven't taught this to 11th graders?
              \_ Go back to russia, jew.
                 \_ It was taught in the 10th year in Britain.
                 \_ I was taught induction in 8th grade (Cupertino CA)
                    I'm poor at geography and history but I'm pretty
                    sure that Cupertino is not has never been a part
                    of russia (or the USSR).
                    sure that Cupertino has never been a part of
                    russia (or the USSR).
                    \_ This one is a master of double-think.
                        \_ RAdm Poindexter is watching you. Don't
                           be surprised if the Ministry of Truth
                           pays you a visit.
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mathforum.org/library/drmath/view/56866.html
Math Sum of n Odd Numbers Date: 7/11/96 at 22:5:34 From: Thomas Mucha Subject: Sum of n Odd Numbers Dear Dr. I was thinking about patterns in numbers one day and I came up with a peculiar law. Also, on a different note - How do you find math in college? I am considering careers, and math is one of my strong points. Thomas Mucha Queensbury, NY Date: 7/12/96 at 9:54:28 From: Doctor Luis Subject: Re: Sum of n Odd Numbers You're very perceptive! Well, 999 is of the form 2(500)-1, so n, in this case, is 500, so the sum of all odd numbers (from 1) up to 999 is 500^2 or 250,000. The above theorem can be proven quite easily by a method called induction, which is a very powerful technique used in mathematics to prove statements about the natural numbers. Since by now I probably have you interested, I'll explain a tad more about induction, and prove a basic relation involving, again, the natural numbers: Induction (a variation of it, at least) involves three steps in proving a statement. The first one is called the Basis Step, and it just involves the assertion that the given statement is true for the natural number 1. The second step involves the assumption that the statement is true for all natural numbers less than or equal to some integer k, and, not surprisingly, is called the Hypothesis Step. The third step is the Inductive Step, and it involves proving that: if the statement is true for the integer k, then it is true for the integer k+1. This step usually comprises the bulk of inductive proofs. The reason induction works is because it satisfies the Peano axioms, which construct the set of natural numbers. You can find a proof that induction works in any good book on Elementary Set Theory or Elementary Number Theory. So the statement there is to prove (I'll label it 'a') is that S = n(n+1)/2 Now that I know what is it I'm trying to prove, I'll follow the steps I told you about above. Here, you need to prove that if the statement in the hypothesis is true for the integer k then it is true for the integer k+1. I'll proceed as follows: By hypothesis, S = k(k+1)/2 Now, I'll add (k+1) to both sides, and obtain S + (k+1) = k(k+1)/2 + (k+1) = k(k+1)/2 + 2(k+1)/2 = k(k+1) + 2(k+1) /2 = (k+1)(k+2) /2 (factor a (k+1) inside the brackets) Ok. So far, we have S + (k+1) = (k+1)(k+2)/2 Remember that S is, by definition, the sum of all integers from 1 to n. Thus, S(k+1) = (k+1)(k+2)/2 which is what you obtain if you substitute n by (k+1) in statement ! What I have just proven is the Inductive Step, and this completes the proof. QED (Quod Erat Demonstratum) (Latin for "Which was to be proven") I have tried my best to explain the proof thoroughly so that you may be able to prove for yourself the interesting pattern you discovered. I'm not exactly going to college (I'm still in high school) but I can tell you that mathematics is a very creative field - a field which requires its foremost individuals to possess a vast imagination and an almost superhuman intuition. Bear in mind, however, that these qualities can be acquired through experience and much toil. It does seem ironic that many people believe that mathematics is a rigid discipline, with no room for creativity or imagination, when in reality it is the utmost creation of the human mind.