Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106274] Re: [mg106250] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Thu, 7 Jan 2010 02:29:12 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

> Bobby, you're looking at how machine reals are *encoded*, and a > *subset* of them is encoded as if they were precise rationals, as you > say. But what matters is how they *behave*. If RootApproximant@RandomReal[] succeeds, that random number is algebraic, according to Mathematica... yes or no? RootApproximant@RandomReal[] Root[3 - #1 - #1^3 - #1^4 - 4 #1^5 + 4 #1^6 - 2 #1^7 + 2 #1^8 + #1^9 + #1^10 - 2 #1^11 - 2 #1^12 + #1^14 &, 1] (It succeeds every time, as far as I can tell.) Does that constitute behaving like an algebraic number? Bobby On Wed, 06 Jan 2010 04:59:52 -0600, Noqsi <jpd at noqsi.com> wrote: > On Jan 4, 11:46 pm, DrMajorBob <btre... at austin.rr.com> wrote: >> Computer reals are precisely equal to, > > No. Computer reals are imprecise. That's the source of their utility. > And their difficulties. > >> and in one-to-one correspondence >> with, a miniscule subset of the rationals. Every one of them has a >> finite >> binary expansion. > > This proves only that they are contained within a finite set. > > But it makes much more sense to consider the correspondence as one-to- > uncountably-many: each individual machine real represents every real > number within an interval determined by its precision. Except for > those that don't represent numbers at all. > >> >> x = RandomReal[] >> digitForm = RealDigits@x; >> Length@First@digitForm >> rationalForm = FromDigits@digitForm >> {n, d} = Through[{Numerator, Denominator}@rationalForm] >> d x == n >> >> 0.217694 >> >> 16 >> >> 1088471616079187/5000000000000000 >> >> {1088471616079187, 5000000000000000} >> >> True > > Proves nothing. > > 0.1 == 1/10 > True > > Now, 1/10 is *not* a member of the "miniscule subset of rationals" you > referred to above, and the machine number 0.1 does not precisely > represent it: > > RealDigits[0.1,2] > {{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0},-3} > > Note that the representation is finite and the last two bits reflect > rounding. > > Nevertheless, the machine manages to create a representation of 0.1, > and even reports it to be equal to 1/10. Now, if machine reals behaved > as if they were "precisely equal to ... a miniscule subset of > rationals" neither of these achievements would be possible. Their > imprecision makes this possible. > >> >> A number can't get more rational or algebraic (solving a FIRST degree >> polynomial with integer coefficients) than that. > > All you've done is to find a number with that property equal to x > within machine precision. But since there is, in fact, an infinite set > of such numbers, finding one isn't remarkable. > >> >> If computer reals are THE reals, > > I never said they were. Go back and read what I wrote. > >> why is it that RandomReal[{3,4}] can >> never return Pi, Sqrt[11], or ANY irrational? > > Well, I'm not patient enough for RandomReal[{3,4}]. But, > > Map[Pi==#&,RandomReal[{3.14159265358970,3.14159265358980},10]] > {False,True,False,False,False,True,False,True,False,False} > > Seems we can easily get Pi from RandomReal. And no, I'm not cheating: > the numbers that yielded True here are equal to Pi in the only sense > that matters with an imprecise number: they are equal to Pi within > their precision. > > Bobby, you're looking at how machine reals are *encoded*, and a > *subset* of them is encoded as if they were precise rationals, as you > say. But what matters is how they *behave*. They do not generally > behave like precise rationals: they behave imprecisely. And what > rational number does a NaN encode? > > Rational addition and multiplication are associative, but machine real > arithmetic isn't. > > Rational arithmetic is distributive, but machine real arithmetic > isn't. > > etc. > > Machine reals are not the reals or the rationals: they are themselves, > with their own special properties. Those who reason as if machine > reals are either real or rational often suffer adverse consequences. > Much of the art of numerical analysis depends upon understanding these > special properties and their consequences. > -- DrMajorBob at yahoo.com