NOTICE: This article originally appeared in the September, '88 issue ofMichigan Atari Magazine and may be freely distributed or reprinted innon-profit User Group publications as long as the article's author andMichigan Atari Magazine are credited AND this notice is reprinted with thearticle. All other publications must obtain written permission from UnicornPublications, 3487 Braeburn Circle, Ann Arbor, MI 48108, Phone: (313) 973-8825before using this article.Binary Maybe's, And the IF..THEN..Who Cares Maxim.by Clinton Pierce (GAG)In the last installment, I stressed the importance of logical thinking andthinking in general. Now, I hope to give you the tools needed to do this.First of all, walk (do not run and trip over your computer cables) to yourbookshelf and pull down an algebra textbook (or calculus, geometry etc..).Theorems and postulates are not just made up at the whim of the publisher.They are derived from logical steps, which themselves are derived. (From What?you say.)Let's make an assumption: the Truth is true. Ok, not bad. (For a proof, seeDescartes.) Let's use the equals instead of the word "is." P=P is true. Wantsome more? P<>_P (P is not, not P), still true. The list of logicaloperators is extensive and I'm not going to derive each one of them. Youshould know what each of these does: AND, OR, _(NOT), XOR (Exclusive Or), =,<>, TRUE, FALSE, -> (Implies) and IFF (If and Only If).You should note that there isn't a "looks like" or "should" function.Computers understand is, was, shall be, and nothing else. They don't workwell with implications, indecision, morality, and justice. They do what iscorrect.Let's look at some other things of interest. Numbers. (And for the sake ofclarity, integers.) Oh sure, you know all about them. You learned those inthe first grade, did you? No. You learned that 5 preceeds 6 and that 2+3 is5. You didn't learn numbers.Every number greater than 1 is a unique product of primes. [Don't believe me?If N is an integer, then N must be equals to some axb (where a AND b < N) If Aor B are nonprime, then they each become their own N and the process startsover.] What good is all of this symbolic garbage? Plenty. Take a simpleproblem: Find the primes from 1 to 70. Well, we cound take each number anddivide it by every number lower than that. That'll only take 2280repetitions. If we were smart, we would only go up to halfway-1 of eachnumber testing for divisibility (knowing that every number cannot be dividedby another evenly, when the second>First/2). We're down to 1061 repetitions.And if we start with 3 (because we KNOW that 2 is prime) and only count odds,we're down to 619. Using Euclid's Lemma, we can only check the odds up to theSquare Root of the number to be checked to see if it is prime. We are nowdown to 135 repetitions. If we stop checking a number as soon as it isnonprime, then we only need 89 repetitions.Knowing all of that symbolic garbage saved us 2191 repetitions. Findingprimes was simple, so what? What if you were trying to find a number in atelephone directory of 1,000,000 people? Or trying to sort that listalphabetically? Time and energy used can be saved exponentially.In the next installment, these techniques and others will be used for more"practical" applications.
