February 2015 - mathfun@mailman.xmission.com

Re: [math-fun] another blot on Stanford
by Henry Baker 13 Feb '15

13 Feb '15

Realistically, how long does it really take to master the multiplication tables? I grew up in a large family, and my siblings drilled these tables to one another; it became a bit of a game, and didn't really take very long. If someone had asked us to memorize the sqrt table, we probably would have done that, as well. I also recall doing a prime sieve by hand shortly after learning how to count, because I was fascinated by how random the results were. We now know that the capacity of human memory is easily in the Gigabytes, so there's no worry about filling up a limited capacity. At 12:37 AM 2/13/2015, Bill Gosper wrote: >http://dailycaller.com/2015/02/11/stanford-professor-says-memorizing-multip…

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[math-fun] Does zero divide zero?
by James Propp 13 Feb '15

13 Feb '15

Mathematica refuses to answer the question, and I've seen textbooks that duck the issue as well ("Suppose *a* and *b* are integers, with *a* nonzero. We say *a* divides *b* if and only if ..."). In fact, Mathematica also refuses to answer the question "Does 0 divide 1?" Is it a standard convention that *a*-divides-*b* is a relation on (*Z*\{0}) x *Z*, so that asking whether 0 divides 1 is no more sensible than asking whether pi divides the square root of 2? I would've been naively inclined to the view that 0 divides 0 is TRUE, while 0 divides 1 is FALSE. I'm afraid to post this question to MathOverflow, lest I be reprimanded for asking such an inappropriate question ("Is this a homework problem that your professor assigned you?"). You guys are nicer, and more to the point, you all know me. Jim Propp

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Re: [math-fun] multiplication in school
by Warren D Smith 13 Feb '15

13 Feb '15

I agree with whoever said that understanding WHY arithmetic works, is helpful, not just memorizing how to do it. I did memorize the mul-table, and while I do not think my memory is very good, I also do not think I had problems with memorizing this table. Obviously the mul-table is easier to memorize than a same-size table of random numbers because if you forget an entry, you can re-obtain it from an adjacent one. So it is like self-sealing rubber, gaps in your memory self-repair in this application. IF you understand that. Which I did, although I do not think any teacher ever told me this. Now the real reason schoolboy multiplication works is the distributive law, which again I do not think any teacher ever told us. But I started to appreciate that about age 10, I think (?), which for me was something of an "aha!" moment. Understanding what was going on. Versus being a robot. Now I do not think I ever, or anyhow not for a long time, learned how to do long division, but I always managed nevertheless because I used some sort of crude iterative correction technique instead to get the right answer. (I haven't long-divided by hand in years, can I even still do it?) Actually, the way schoolboy long-division works IS an iterative correction technique, when you break it down, depending again on distributive law, and I think I was unable to do long division until I came to understand that, which was when I was a pretty old schoolboy. My father apparently got taught in school to take square roots via a paper & pencil procedure, but I was never taught that in school but again was able to use iterative correction techniques when I needed that. If all we do is churn out children who can do arithmetic robotically without understanding why it works, I do not think that is a worthwhile educational accomplishment in the modern era. Understanding WHY procedures work, is worthwhile and also will cause your roboticism to be better. I also think one of my school teachers thought a/b + c/d = (a+b)/(c+d) and taught this to the class, which was not helpful...

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Re: [math-fun] another blot on Stanford
by Henry Baker 13 Feb '15

13 Feb '15

Everyone who makes it through college has a good memory; each such person has a vocabulary of at least 50,000 words. https://en.wikipedia.org/wiki/Mental_lexicon The biggest problem is presenting facts in such a way that they are easier to remember. All mammals seem to be quite good at navigating in complex physical environments; they utilize the hardware of the visual cortex to organize information. The "Memory Palace"/"Method of Loci" technique appears to be quite useful in taking advantage of this hardware. https://en.wikipedia.org/wiki/Method_of_loci At 08:12 AM 2/13/2015, Charles Greathouse wrote: >Not everyone is gifted with a good memory.

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[math-fun] digits of e, one at the time,
by Simon Plouffe 13 Feb '15

13 Feb '15

Hello, speaking of a short program, here is one from 1990 I worked on : at the time I was a BASIC fan, the compiler I had was Zbasic, a fantastic engine on a mac, capable of computing digits up to 54 digits! and compiled, very fast. that simple routine, now called a spigot algorithm can extract digits of exp(1) one column at the time but of course, the program takes memory, 5 years later this is where I could do it for Pi but in base 2, or in base 10 but very slowly. program 'decde' digits of e, August 5 1990. DEFDBL INT A-Z 'OPEN"O",1,"essai" B=10000 D=0 DIM R(10000) S=0:A=B DO N=1 DO Q=A/N R(N)=A MOD N S=S+Q A=R(N+1)*B+Q N=N+1 UNTIL A=0 AND R(N+1)=0 D=D+1 PRINT D*4; S=0 IF INKEY$="s" THEN STOP UNTIL N>9900 STOP best regards and have a nice day. Simon Plouffe

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[math-fun] another blot on Stanford
by Bill Gosper 13 Feb '15

13 Feb '15

http://dailycaller.com/2015/02/11/stanford-professor-says-memorizing-multip… --rwg

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Re: [math-fun] Short programs for e
by Steve Witham 13 Feb '15

13 Feb '15

I said, > You can speed it up by a factor of four by converting to base 10**4. I think that's true as long as you're taking better advantage of what a machine can do with a single word in one cycle, but > ...for that matter, why not generate all the digits in one pass? base = 10 ** 35 # < 32! c = 0 for j in range(32, 1, -1): # 32 down to 2. c = (c + base) / j # flooring int division. print "2." + str(c) This reminds me of Minsky's two-register Turing machine...but it's a typical way to evaluate polynomials. In Python it looks simpler than the nested loop with factorial notation, but that's possible because Python switches to bignums automatically. For 35 digits, the short loop runs faster than the one-digit-at-a-time factorial-notation version. 35 digits, 32 terms: calc_e_fact_nota 0.00173 sec calc_e_one_pass 0.00005 sec That's unfair, though, racing the bignum math written in C against the inner loop in Python. But, as the number of digits goes up, even the Python factorial-notation version is faster than the bignum math. For instance: 100000 digits, 25206 terms: calc_e_fact_nota 1.04 sec calc_e_one_pass 19.09 sec --Steve

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[math-fun] Fwd: Tadashi Tokieda's 2.6 minute Valentine video
by Bill Gosper 12 Feb '15

12 Feb '15

---------- Forwarded message ---------- From: Nancy Blachman <nancy(a)googleguide.com> Date: Thu, Feb 12, 2015 at 2:53 PM Subject: Tadashi Tokieda's 2.6 minute Valentine video To: A Valentine from Möbius || Radcliffe Institute <https://www.youtube.com/watch?v=5xLFf_SwaK4&authuser=0> A Valentine from Möbius || Radcliffe Institute <https://www.youtube.com/watch?v=5xLFf_SwaK4&authuser=0>

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[math-fun] Does zero divide zero ...
by Bill Gosper 12 Feb '15

12 Feb '15

Guy Haworth wrote This is a deeper question than it appears ... and there are of course some well-informed books on the history of zero. A colleague of mine created an algebra J = {(a, b) | a, b in the Reals R} ... with (a, b) being analogous to a/b in R. Thus, addition, inverses, subtraction and multiplication can be defined naturally ... and division by (a, b) is defined as multiplication by (b, a) without any exceptions. There is also an equivalence relationship: (ca, cb) is equivalent to (a, b). Thus, he has elements in the algebra like (a, 0), (-a, 0) and (0, 0) with a > 0. He gives these the names 'infinity', '-infinity' and 'nullity'. Anything equivalent to nullity is also 'nullity'. Nullity is what I (and maybe others) would call an 'annihilator' ... if it is combined with anything, the result is 'nullity'. Let us add 0/0 as a notional 'number' to the set of numbers a/b in the Reals R. Let this algebra be R'. rwg> For a very informal treatment, www.tweedledum.com/rwg/rectarith12.pdf p7/14 <rwg Thus, there is a 1-1 mapping from R into J. Any expression not featuring 0/0 in R' maps in an obvious way to J ... and expressions featuring 0/0 in R' map to nullity in J. So J is equivalent to a number system with 0/0 being treated as a number rather than an exception. However, it does not follow that '2', i.e. 2/1, equals '1', i.e. 1/1, in this algebra. "All expressions have a value" might be a neat mantra ... but we haven't really created - IMHO - a number system which is different from what we have now. The history of new numbers - minus 1, zero, famously sqrt(2) and presumably the unfortunately named complex numbers ... is that they are not universally received with peace of mind. They had a troubled birth. I have not been able to persuade my colleague 'J' that he has not created anything new. One cannot simply ignore the fact that a number is zero if it is effectively being divided into something. Many mathematical methods have 'if zero' get-out clauses. The standard formula for the roots of a quadratic does not work if the coefficient of x^2 is zero. Maybe there is a neat proof that this concept of 'nullity' does not bring anything to the party: I sure wish there was ... and then I could save my colleague's time writing conference papers and being convinced that he is onto something. I believe his interest in 0/0 stemmed from an interest in Project Geometry, and then in Interval Arithmetic ... where he was trying to tackle the problem of Instability in computations. Guy

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Re: [math-fun] Factorial base
by Bill Gosper 12 Feb '15

12 Feb '15

On 2015-02-11 20:14, Bill Ackerman wrote: >> ---------- Forwarded message ---------- >> From: <rcs(a)xmission.com> >> Date: Wed, Feb 11, 2015 at 2:57 PM >> Subject: Re: [math-fun] Factorial base >> To: math-fun(a)mailman.xmission.com >> Cc: rcs(a)xmission.com >> >> >> The only theorem I know of is that a rational number always has a >> finite representation in factorial base; therefore e is irrational. >> >> Back in the day, I was looking through the file of paper tapes in >> the PDP1 room (this served as the software library), and I came >> across one labeled simply "e". The handwriting might have been either >> Eric Jensen or Bill Ackerman. It was only a couple of folds, a few >> hundred characters. Curious, I printed it out, and then ran the >> assembler on it. Indeed, it promptly printed out a few thousand >> digits of e. I examined the code more closely, and discovered it >> created a factorial base representation of (the fractional part of) e, >> one digit per machine word, and then did a simple decimal conversion, >> multiplying the factorial base representation by decimal 10, and >> printing the integer "carry" that falls off the front end; rinse, repeat. >> The program, in assembly language, easily fit on one page. A gem. >> >> Rich >> > The program was most likely written by the late Wow. Details? > Richard P. Howell, one > of the noteworthy PDP-1 folks in the late 60's. I seem to recall he > had some extremely clever program for calculating e. I don't remember > ever examining it. > > Bill It certainly is a pleasure. Dick put the endblocks on the glider gun. I was preparing to search big rectangles of four or eight shuttles. At some point, I would have thought of eaters. I also seem to remember he wrote a too-clever Life simulator that ran for weeks with an undetected bug, then miscomputed a pattern. He was one of the few hackers who didn't dodge the Viet Nam draft. (Peace with honor.) --rwg

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