A definition of rational numbers
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From: fishfry on 9 Aug 2008 12:39 In article <dd5cb308-ef26-4bdb-804c-f4b1854a83e9(a)c58g2000hsc.googlegroups.com>, translogi <wilemien(a)googlemail.com> wrote: > On Aug 9, 9:59�am, smn <smnewber...(a)comcast.net> wrote: > > On Aug 9, 1:33�am, kunzmilan <kunzmi...(a)atlas.cz> wrote:> Rational numbers > > have either a finite number of valid digits, > > > as 1/2 = 0.5, 50/100 = 0.50, > > > or if their length is infinite, as 1/3 = 0.33333..., > > > 1/7 = 0.142857142857..., they contain a repeating sequence. > > > > �Any change in such infinite sequences, as 0.33133... gives an > > irrational > > > > NO .33133... =.331+(.333.../1000) is still rational.You have to be > > a;;owed to start the repeated finite sequences of digits after a > > finite number of arbitrarily chosen digits to characterize the > > rationals. > > Also .5=.5000... =.499999... so you needent mention "finite number of > > valid digits" smn > > > > > > > > > an irrational number. It is not possible to name > > > two finite natural numbers giving the changed sequence, > > > since normal infinite natural numbers can not have infinite number of > > > digits.- Hide quoted text - > > > > - Show quoted text - > > A rational number is a number that can be described as the quocient of > two natural numbers (except the dividor must be unequal to o) > Still imprecise. Are 1/2 and 2/4 different rational numbers, or the same rational number?
From: Dave on 9 Aug 2008 12:55 On Aug 9, 11:39 am, fishfry <BLOCKSPAMfish...(a)your-mailbox.com> wrote: > Still imprecise. Are 1/2 and 2/4 different rational numbers, or the same > rational number? The definition doesn't speak to that. It just asserts that if 2*x = 1 and 4*y = 2, then x and y are rational numbers, which we can represent as ordered pairs, (1, 2) and (2, 4), or more commonly, as fractions, 1/2 and 2/4. That 1/2 and 2/4 are members of an equivalence class (that we can call the rational number 1/2) is a deduction that follows from a definition of equality of fractions: two fractions a/b and c/d are equal if and only if a*d = b*c. Dave
From: porky_pig_jr on 9 Aug 2008 13:00 On Aug 9, 4:33 am, kunzmilan <kunzmi...(a)atlas.cz> wrote: > Rational numbers have either a finite number of valid digits, > as 1/2 = 0.5, 50/100 = 0.50, > or if their length is infinite, as 1/3 = 0.33333..., > 1/7 = 0.142857142857..., they contain a repeating sequence. > Any change in such infinite sequences, as 0.33133... gives > an irrational number. It is not possible to name > two finite natural numbers giving the changed sequence, > since normal infinite natural numbers can not have infinite number of > digits. > kunzmilan so you define 1/2 = 0.5, without defining 0.5 or explaining why 1/2 = 0.5. The standard definition of rational numbers as an ordered pair (a,b) where a is either natural number or 0 and b is natural number, with appropriate rules for addition and multiplication, works just fine for everyone. The issue of finiteness of decimal expansion of rational numbers comes as a consequence of their definition, not as *their* definition. Please before you start reinventing the wheel, at least check what already has been done.
From: Virgil on 9 Aug 2008 14:45
In article <e379f14a-8947-4017-a3bf-e4512057f43f(a)m45g2000hsb.googlegroups.com>, kunzmilan <kunzmilan(a)atlas.cz> wrote: > Rational numbers have either a finite number of valid digits, > as 1/2 = 0.5, 50/100 = 0.50, > or if their length is infinite, as 1/3 = 0.33333..., > 1/7 = 0.142857142857..., they contain a repeating sequence. > Any change in such infinite sequences, as 0.33133... gives > an irrational number. It is not possible to name > two finite natural numbers giving the changed sequence, > since normal infinite natural numbers can not have infinite number of > digits. > kunzmilan A real number r is rational if and only if there is some non-zero integer s such that r times s is integral. |