At 08:04 PM 5/15/01 +0100, you wrote: >From: "Jean-Louis Couturier" > > > Robert Shaw wrote: > > > > >When I was a student we spent one rainy afternoon working > > >out what numbers we could get from our calculators using > > >only the unitary buttons, starting fom 0 and never touching > > >another number afterwards. > > > > >It took a dozen maths students twenty minutes to work > > >out how to generate all the integers, then we went after > > >the rationals. > > > > > > Now THAT sounds like fun! > > > > When you say never touching another number, does that include > > zero? > >Yes. > >When you switch the calculator on you get a zero, but after that >you never touch another number key. These were scientific >calculators so we had ^2,^3,^-1,cos, sin, tan, cosh, sinh, tanh, >ln, log and their inverses plus factorial. For anyone who is interested, this is what I was playing with last night: MirCalc, 9x, 06 Nov 98, 31K, Multi-digit calculator, up to 10,000 digits. <http://www.winsite.com/edu/calculator/page10.html> It has all the above functions. For some reason when I was closing my e-mail because I really needed to finish grading finals and go to bed, I saw the MirCalc icon staring at me and started playing with it, and noticed the interesting behavior of cos( cos( cos( ... > > If so, 3 is going to take some work. > >Actually, there's a trivial way of doing it. By composing >the allowed functions you can produce the function x+1. >Repeat indefinitely and you can produce all integers. Which is what you do when developing the nonnegative integers in a course on set theory or other fundamentals: starting with 0 and using a "successor of" function, e.g., "ss0 + ss0 = ssss0" is the way of writing "2 + 2 = 4." By the time you get that far, though, you're near the end of the book and the semester is almost over. >Similarly ln(sqrt(exp(x)))=x/2 so once you've got all >integers you can easily get all halves, quarters, eights, etc. > >That means you can get arbitarily close to any number. > >Pi is ln(square(square(exp(arctan(exp(0)))))) >though there may be quicker ways to get it. > > > I have been a mth student in the past. I can imagine that this > > might sound boring to most of you. Or that you and I (and apparently others here) are the kind of folk who find this kind of thing a fun distraction when we are bored. > Kind of like when I tried to > > solve Fermat's theorem... > > >-- >Robert Of course, the thing I found interesting was that it was oscillating back and forth but fairly rapidly converging to lim (cos^n (0)) = 0.73908 51332 15160 64165 53120 87673 87340 40134 11758 90075 74649 65680 63577 ... I guess the next step is to try to express the exact value of the above in closed form. One starting place might be cos^n (0) = (2^-n) Sum C(n,j) (summation over j | j = 0, 1, ..., n) -- Ronn! :)
