April 2006 - mathfun@mailman.xmission.com

[math-fun] equality of floating point numbers.
by Ray Tayek 19 Apr '06

19 Apr '06

hi, reading junit recipies by martin fowler. talking about equality as an equivalence relation on objects. if one of the fields of the objects that is in included in the equals method has a floating point number, then one uses some idea of "close enough" (i.e. within some epsilon). this is clearly reflexive and symmetric, but not transitive. he conjectures that this idea of "close enough" is the *only* case (in the space of equals methods for object) of having a equals relation that is symmetric and reflexive and not transitive. does anyone have a counter example? thanks --- vice-chair http://ocjug.org/

1 0

[math-fun] cubic thirds
by Schroeppel, Richard 19 Apr '06

19 Apr '06

There are some interesting possibilities for dissecting a cube into three congruent pieces. Select a triagonal, and split the cube in half with a plane perp-bisecting the triagonal. The two pieces have regular triangle symmetry, and each can be cut into thirds using 120deg wedges with the triagonal as the spine. The starting wedge angular position is arbitrary, so the two halves of the cube can have different cutting positions. Wedges from the two halves can be glued for a rather peculiar looking congruent trisection. I believe one of Gardner's columns was about dissections, and said the square-into-5-congruent- pieces problem had only the trivial solution, and that a proof existed. Rich

2 2

Re: [math-fun] Square thirds
by dasimov＠earthlink.net 19 Apr '06

19 Apr '06

Michael Kleber wrote: << There are clearly lots of choices for what restrictions you put on the pieces. Personally, I'd be happy to learn anything: a proof that the obvious trisection is the only one possible with polygonal pieces would be great, as would an abstract, AC-dependent construction that created three congruent point-sets whose union is the square. >> My hunch is that under modest restrictions on the kind of pieces, the following line of attack will help: Keep your eye on where the 4 corners and the 8 one-third and two-thirds points on each edge go among the 3 pieces. Also, the problem should be defined loosely enough that boundary points of the pieces can be ignored, and care must be taken to avoid showing that no such decomposition is possible! A RELATED PROBLEM that some people could've predicted I'd ask is: In what ways can the square torus be decomposed into n congruent pieces for any positive n (other than the obvious n parallel annuli decomposition [whose slopes, btw, can be any fixed rational number]) ? --Dan

2 1

[math-fun] What use are primes?
by Stan E. Isaacs 19 Apr '06

19 Apr '06

I had a student in my computer class ask me a mathematics question: she asked what was the use of prime numbers? She knew there were lots of work done on primes, but what she wanted to know was, what *good* are they, besides being a fascination to math people. It caught me unawares, I mumbled something about that there are theorems where it is easier to prove them for primes, and then extend the proof to all numbers, but I couldn't think of an example. I looked around a little, but most of the things I see, off-hand, talk about all the things you can prove about primes themselves, not what use they might be elsewhere. I could probably mention mods, since you need mod p to be prime, to allow division. But what I really want is some examples where they might be of use in "everyday life", or at least in some mathematics that would apply to more everyday things. Anyone have ideas of what to tell her? Or references of places I could look? Thanks, -- Stan -- Stan Isaacs 210 East Meadow Drive Palo Alto, CA 94306 stan(a)isaacs.com

9 11

[math-fun] Question on Fourier transform for positive probability densities
by dasimov＠earthlink.net 19 Apr '06

19 Apr '06

Suppose p: R -> R is a continuous mapping with p(x) > 0 everywhere, and integral{R} p(x) dx = 1. Question: Can the set of Fourier transforms P(t) := (1/sqrt(2pi)) integral{R} p(x) exp(ixt) dx of such p's be characterized explicitly? --Dan

2 1

Re: [math-fun] Square thirds
by dasimov＠earthlink.net 19 Apr '06

19 Apr '06

Franklin wrote: << I've never seen this problem before, but I'm sure the answer is no. Since there are four corners and only 3 pieces, one of the pieces must have two of the corners. It must then have the edge connecting them . . .. . . . >> How do you see this last assertion (without assuming convexity)? --Dan << -----Original Message----- From: Michael Kleber <michael.kleber(a)gmail.com> . . . Is it possible to dissect a square into 3 congruent pieces in any way other than the trivial one? Surely the answer is no, but I can't think of a proof. >>

2 1

[math-fun] Square thirds
by Michael Kleber 19 Apr '06

19 Apr '06

Surely everyone other than me already knows this, but when my 7-year-old son asked me this over dinner tonight, I couldn't give him a satisfying answer. Is it possible to dissect a square into 3 congruent pieces in any way other than the trivial one? Surely the answer is no, but I can't think of a proof. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.

3 2

Re: [math-fun] What use are primes?
by Henry Baker 18 Apr '06

18 Apr '06

One of my intuitions re numbers is that analysts worry about the high-order bits of numbers, while number theorists worry about the low-order bits. For example, some quick compositeness tests look only at the low-order digits of numbers. The usual GCD algorithm worries about reducing the length of the numbers, but the so-called "binary algorithm" (see Knuth if you need to know more about this) worries about reducing the number of non-zero bits in the numbers. If you really want to blow your student's mind, introduce her to p-adic numbers, which dramatically change the definition of "closeness".

1 0

Re: [math-fun] Little number theory puzzle
by dasimov＠earthlink.net 15 Apr '06

15 Apr '06

Yes. Like many truths, it contains a sliver of humor. --Dan --------------------- Franklin writes: << Umm, this is a joke, right? -----Original Message----- From: dasimov(a)earthlink.net Mike Stay inquires: << How do you select two integers independently at random? To pick one integer at random, just consider a dense subgroup of the circle C = R/Z -- say G = {n*sqrt(2) mod 1 : n is in Z} -- and consider the cosets C/G. Let W be a subset of C containing exactly one representative of each coset. Then C is the disjoint union of the countably many translates of W of the form T_k = k + W, for k in Z. Finally, pick a point at random from C. It belongs to exactly one T_k, and then k is your random integer. Pick another point independently from C in the same way to get two independent random integers. >> >>

1 0

Re: [math-fun] Little number theory puzzle
by dasimov＠earthlink.net 15 Apr '06

15 Apr '06

Mike Stay inquires: << How do you select two integers independently at random? To pick one integer at random, just consider a dense subgroup of the circle C = R/Z -- say G = {n*sqrt(2) mod 1 : n is in Z} -- and consider the cosets C/G. Let W be a subset of C containing exactly one representative of each coset. Then C is the disjoint union of the countably many translates of W of the form T_k = k + W, for k in Z. Finally, pick a point at random from C. It belongs to exactly one T_k, and then k is your random integer. Pick another point independently from C in the same way to get two independent random integers. --Dan

2 1