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. > 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. 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. Kind of like when I tried to > solve Fermat's theorem... > -- Robert
