Wikipedia:Reference desk/Archives/Mathematics/2011 October 13

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October 13

1959 Textbook with Chapter by Kernighan

In the early eighties I bought my first book on C programming language by Kernighan and Ritchie. At the time I remembered I had a College text book with a chapter written by Kernighan . The chapter was on symbolic logic and the book's title was something like "Concepts in Modern Mathematics" It was a small brown book. I forget when I lost ownership of it . The thing is I was born in 1939 and Kernighan and Ritchie were both born in 1942. You mean I was busting my head on a book written by someone two years younger than me??? Can anyone help me? I wrote my first program in machine language at about that time, and got paid for it in London by 1963. My memory has a right to be faulty, but help me out here. Oh. and I became aware of this because Ritchie has just passed. The pioneers are getting old now. --sesquepedalia== — Preceding unsigned comment added by Sesquepedalia (talk • contribs) 16:22, 13 October 2011 (UTC)

Teaching math: paper against computer

Should math teachers make their students (maybe the younger, at least) do their homework on paper or could the students just do everything on the computer? Quest09 (talk) 00:24, 13 October 2011 (UTC)

They should make their students write up their homework in TeX. It's the only reasonable answer! --COVIZAPIBETEFOKY (talk) 03:51, 13 October 2011 (UTC)

Do mean have students write up the homework on paper instead of a word processor or do you mean don't allow students to use calculators, symbolic algebra programs, AI proof generators, ... ? For either it depends on the level of math. Word processors can do basic math notation but you need something like TeX to do more complicated or advanced notation, and diagrams are almost always easier to do on paper. An experienced teacher can usually create problems that don't require a lot of tedious calculation, so having electronic aids shouldn't make that much of a difference. When I was teaching I told my students that if I allowed calculators etc. then I would give problems where the answers didn't come out nicely, sort of like the problem you might get in the real world. The students said they'd rather have the answers come out nicely.--RDBury (talk) 04:36, 13 October 2011 (UTC)

I meant paper vs. word processor/TeX. Quest09 (talk) 13:29, 13 October 2011 (UTC)

American students at all levels write mathematics at a semiliterate level. I'm sure many of these university students are able to write very well on sophisticated topics, but give them a basic math problem, and they might as well be using crayons and fingerpaint. It's as though everyone suffers from selective dysgraphia. Sławomir Biały (talk) 10:36, 13 October 2011 (UTC)

And what would be better for those semiliterate students? Learn to write math on paper or learn TeX? I see an advantage of paper: it doesn't let you choose from a set of symbols, so it forces you to choose more (and maybe think more). Quest09 (talk) 13:23, 13 October 2011 (UTC)

Technology can be used well and used poorly. It seems to me that it is often used as a crutch, though. The calculator/computer obsession has definitely contributed to the decline of mathematics education in my opinion. When a university calculus student is working a problem, sees the fraction 27/9, and whips out a calculator to evaluate it, that's definitely a problem. I think there is too much emphasis on results and not enough on process (standardized testing is no doubt largely to blame for this). Students get to higher education with no idea how to *think* or *write* about solutions to mathematics problems. Anyway, I'm not in principle opposed to using computers. Actually, I think they are necessary. But they should be used in conjunction with brains, not as a substitute. Sławomir Biały (talk) 14:24, 13 October 2011 (UTC)

I would think learning it on paper first, then moving on to a word processor, would be best. I still find it painful to do even basic math with a word processor. Students also need to learn that:

6/2/12 could mean either 6/2   or     6  
                          12        2/12

Perhaps practice on a word processor could teach them that difference. StuRat (talk) 13:46, 13 October 2011 (UTC)

What do you mean by that StuRat? It looks kind of a broken layout...Quest09 (talk) 14:21, 13 October 2011 (UTC)

I believe that there is a fixed order of computation. Certain computations are done before others. If I recall, it's division, then multiplication, then either addition or subtraction. So 2/3*4 = (2/3)*4 = 8/3; moreover 6/2/12 = (6/2)/12 = 1/4. — Fly by Night (talk) 19:18, 13 October 2011 (UTC)

There is no rule that says "division must come before multiplication". They commute, so order is immaterial. Repeated divisions are normally evaluated left to right, as are repeated subtractions. Dbfirs 11:56, 14 October 2011 (UTC)

Maybe I didn't explain myself very well. There is an order in which a computer (or calculator) deals with an arithmetic computation. I believe it does all the divisions first, then the multiplications and then the additions and subtractions (not sure on the order of these last two). The key point is that there is an accepted hierarchy of operation when doing arithmetic calculations. Obviously, one operation has to come first, another second, another third, etc. I simply meant that a computer will do division then multiplication. That is not to say that we would get a different answer if we did multiplication before division, or any other operation before another; although we might. Sorry for not making myself clear. — Fly by Night (talk) 17:54, 14 October 2011 (UTC)

I was actually asking due to the layout (not due to the order of operations), the fractions looks confusing for me. With LaTex is would be OK, equally is the case on paper. At least, I can draw the conclusion that no one should use a common word processor. Quest09 (talk) 23:39, 13 October 2011 (UTC)

Computers have the added benefit of audio and video feedback. Some (not all) students may benefit from that. There is the immediate turn-around benefit also. Instead of waiting (possibly a full day) to get a response, the computer can immediately tell you if you are right or wrong. Finally, paper is not necessary throughout life. I've had the same notepad on my desk for 8 years (I know because it has a calendar on the front from 8 years ago). I've written on about 10 pages - and that was mostly doodling for my kids when they visit me in the office. Therefore, doing math on paper is not important to me. Doing it in latex is far more important. -- kainaw™ 13:50, 13 October 2011 (UTC)

We should stop teaching writing using paper and pencil. From Kindergarten onwards we should let children use only computers, mobile phones and other electronic devices. The ability to write using paper and pencil requires some brain capacity which isn't used later in life. But because after the age of about 8, the brain becomes more hard wired, that brain capacity reserved for writing cannot be freed up to do other things. Count Iblis (talk) 15:36, 14 October 2011 (UTC)

Quest09, you might find this resource to be helpful.

• Math Forum: Teacher2Teacher (T2T) ("Welcome to Teacher2Teacher, a resource for teachers and parents who have questions about teaching mathematics.")

—Wavelength (talk) 18:43, 14 October 2011 (UTC)

Logical implications of "I propose a construction"

Hello,

my question is in fact both on mathematics and on the English language. Suppose I want to construct something of type B, and I can prove that an object of type A gives me an object of type B. However, I cannot prove that an object of type A exists.

Would it be alright to write "I propose a construction of an object of type B"? Or would that imply that I in fact definitely can construct an object of type B?

Many thanks, Evilbu (talk) 16:35, 13 October 2011 (UTC)

I'd say something like 'Suppose A is a right angled equilateral triangle, then...' Dmcq (talk) 17:10, 13 October 2011 (UTC)

It's perfectly fine to assume anything you like, provided you make sure to follow the rules of logic from that point on. In fact, it's quite a handy method in mathematical proofs. For example, there is a method known as proof by contradiction where one assumes the opposite of that which one would like to prove, and then logically derives a contradiction; meaning that the original assumption is false (and so the thing that one originally wanted to prove is indeed true). This method of proof is used to show that the square root of any non square number is irrational. — Fly by Night (talk) 19:03, 13 October 2011 (UTC)

I think you mean "... any non-square integer". The square root of, for example, (4/81) is rational; and (4/81) is a non-square number. Nimur (talk) 19:37, 13 October 2011 (UTC)

Well, yes, but that was a tacit assumption; otherwise every (complex) number would be a square number, and so there would be no non-square numbers. Just like when you say prime number, you tacitly assume them to be integers, and their factors to be integers. Otherwise there would be no prime numbers: x = 2 × (x/2). Square numbers, odd numbers, even numbers, triangle numbers, prime numbers; they are all assumed to be whole numbers. — Fly by Night (talk) 18:16, 14 October 2011 (UTC)

Nobody has really addressed your question on English phrasing of mathematics, and it is somewhat subjective. Here's my two cents: Phrases similar to "a proposed construction of X" do appear in titles and abstracts, especially in older works. I've always read this phrase to mean that the object is constructed, or at minimum proven to be constructable. You clearly state that you cannot do that in this case. As others point out, there is value to methods like the one you describe, but I wouldn't call it a construction if you cannot rigorously establish the existence of each intermediate object. SemanticMantis (talk) 19:59, 13 October 2011 (UTC)

Thanks for your replies. The problem is that I need to summarize what I did in just one sentence, so it has to be something like "I suggest a construction for an object of type B" or "I propose a possible construction for an object of type B". So you don't think that this way of writing makes it look like a claim that you can certainly prove existence of objects of type B?213.118.182.209 (talk) —Preceding undated comment added 20:01, 13 October 2011 (UTC).

Unless it's horribly inconvenient, I would specifically mention the existence of type A. Something like "We show that the existence of an object of type A allows a construction of an object of type B." This sort of thing is not too uncommon, and many important results are of this form. A key step in the proof of Fermat's last theorem was the observation (which came many years before the eventual proof) that a solution to Fermat's equation would imply a construction of a prohibitively weird elliptic curve. Staecker (talk) 21:37, 13 October 2011 (UTC)