Date: Oct 3, 2017 7:47 AM
Author: bursejan@gmail.com
Subject: Re: reason why 1=.99999....99(10/9) and why 1/3 = .333333......33(1/3),<br> never that nonsense 1=.999.... 1/3 = .3333.....

So they are not the reals, and the only reason
Newton might have used them, is explained by

Newton himself on page 162:

The method of fluxions and infinite series
https://archive.org/stream/methodoffluxions00newt#page/162/mode/2up

Its a nummeric tool, to perform r/s for two
rational numbers, as 1/s * r, and if we want
to see a few decimal numbers of the result.
There is a certain advantage in using "compleat

Quotients", one can trade a big number division
by two small number division and "aubstitution".
Substitution is Newtons trick to get larger
and larger number of digits.

Lets do what he did, we saw already:

1/29 = 0.03448(8/29)

8/29 = 0.27586(6/29)

But note (8/29) is the tail of the first number,
so we can "substitute" it, Newton writes "I
substitute this instead of the Fraction in
the first equation, and I shall have:"

1/29 = 0.0344827586(6/29)

So he needs to do the division only once
for a small number of digits, and then can
reuse the result by multiplication and
a small modulo division.

Isn't this amazing?

Am Dienstag, 3. Oktober 2017 13:31:38 UTC+2 schrieb burs...@gmail.com:
> The proof that they are rational numbers only,
> hence not the reals, is very easy.
>
> Lets first do it by example. To get the
> following result:
>
> 1/29 = 0.03448(8/29)
>
> One did compute the remainder of the
> following modular problem:
>
> 100000 = 3448*29 + 8
>
> If we divide both sides first by 100000:
>
> 100000/100000 = 3448/100000*29 + 8/100000
>
> 1 = 0.03448*29 + 0.00001*8
>
> And then by 29 we get:
>
> 1/29 = 0.03448 + 0.00001*8/29
>
> 1/29 = 0.03448(8/29)
>
> Now assume we have an arbitrary representation
> as "compleat Quotient":
>
> d0.d1 ... dn (p/q)
>
> We can easily reverse the process, we first see
> that it is:
>
> d0.d1 ... dn + 1/10^n*(p/q)
>
> Which is:
>
> d0 d1 ... dn/10^n + 1/10^n*(p/q)
>
> Lets only write m for the mantissa:
>
> m/10^n + 1/10^n*(p/q)
>
> Now do the usual rational number addition:
>
> m*q/(10^n*q) + p/(10^n*q)
>
> And in the end we have the following rational number:
>
> (m*q+p)/(10^n*q)
>
> A special case would be when:
>
> 10^n*r = m*q+p
>
> Then the origial rational number had a much smaller
> representation, namely:
>
> r/q
>
> But this doesn't happen necessarely, for example,
> note the change of 8 to 9:
>
> 0.03449(8/29)
>
> Is only:
>
> 100029
> -------
> 2900000
>
> No further factors to cancel.
>
> Am Dienstag, 3. Oktober 2017 04:31:44 UTC+2 schrieb burs...@gmail.com:

> > Unfortunately this is nothing new, and was already
> > used by Newton. He calls it "compleat Quotient".
> >
> > See for yourself here:
> >
> > The method of fluxions and infinite series
> > https://archive.org/stream/methodoffluxions00newt#page/162/mode/2up
> >
> > You find on page 162:
> >
> > 1/29 = 0.03448(8/29)
> >
> > Then he goes on (multiply by 8):
> >
> > 8/29 = 0.27584(64/29)
> >
> > Then he says, he uses "or rather":
> >
> > 8/29 = 0.27586(6/29)
> >
> > And guess what, its the same as calculating with
> > rational numbers (hence the name "compleat Quotient).
> >
> > Be sure you can figure it out that this representation
> > is the same as rational numbers. So noting to do with
> >
> > infinite decimal representation.
> >
> > Am Dienstag, 3. Oktober 2017 04:21:37 UTC+2 schrieb Archimedes Plutonium:

> > > Newsgroups: sci.math
> > > Date: Mon, 2 Oct 2017 08:32:50 -0700 (PDT)
> > >
> > > Subject: Why is there a difference between fractions 1/3 and .333 repeating
> > > multiplied by 3
> > > From: Archimedes Plutonium <plutonium....@gmail.com>
> > > Injection-Date: Mon, 02 Oct 2017 15:32:50 +0000
> > >
> > >
> > > Because you cannot mix fraction with a decimal unless the decimal ends in 0's
> > >
> > > For example 10 divided by 3 is 10/3 or written as 3+1/3
> > >
> > > But completely wrong when writing it as 3.333.... Because the dots mean nothing, unless you write it as 3.3333...33(1/3) so you include the carryover at infinity
> > >
> > > This means that .99999.... is not 1, unless you included the carryover at infinity border as this
> > >
> > > .99999....99(10/9) which in fact is 1
> > >
> > > For you divide 10 by 9 carry the 1 leaving behind 0, 1 added to 9 is 10, carryover the 1, leaving behind another 0, finishing off with
> > >
> > > .9999....99(10/9) = 1.0000.....
> > >
> > > You see Old Math was too dumb and lazy to define how those dots ......... Interfaces with fractions, too dumb too lazy
> > >
> > > AP
> > >
> > > Newsgroups: sci.math
> > > Date: Mon, 2 Oct 2017 19:10:39 -0700 (PDT)
> > >
> > > Subject: the reason why 1=.99999....99(10/9) and why 1/3 = .333333......33(1/3)
> > > From: Archimedes Plutonium <plutonium....@gmail.com>
> > > Injection-Date: Tue, 03 Oct 2017 02:10:39 +0000
> > >
> > > the reason why 1=.99999....99(10/9) and why 1/3 = .333333......33(1/3)
> > >
> > > On Monday, October 2, 2017 at 10:33:05 AM UTC-5, Archimedes Plutonium wrote:

> > > > Because you cannot mix fraction with a decimal unless the decimal ends in 0's
> > > >
> > > > For example 10 divided by 3 is 10/3 or written as 3+1/3
> > > >
> > > > But completely wrong when writing it as 3.333.... Because the dots mean nothing, unless you write it as 3.3333...33(1/3) so you include the carryover at infinity
> > > >
> > > > This means that .99999.... is not 1, unless you included the carryover at infinity border as this
> > > >
> > > > .99999....99(10/9) which in fact is 1
> > > >
> > > > For you divide 10 by 9 carry the 1 leaving behind 0, 1 added to 9 is 10, carryover the 1, leaving behind another 0, finishing off with
> > > >
> > > > .9999....99(10/9) = 1.0000.....
> > > >
> > > > You see Old Math was too dumb and lazy to define how those dots ......... Interfaces with fractions, too dumb too lazy
> > > >
> > > >

> > >
> > > Alright, let me expand and expound on the ideas above, of undefined and ignorant dots .............
> > >
> > > The snobs, and slobs of Old Math who think that 1/3 = .333333..... and that 1 = .999999.....
> > >
> > > Well, ask those slobs and snobs, ask them to add
> > >
> > > 0.9999999.....
> > > +.9999999.......
> > >
> > > watch the goon squad try to get rid of this 1.99999.....9998
> > >
> > > ask the goon squad how they got rid of the "8"
> > >
> > > But, on the other hand, when you well define the ellipsis, for those series of dots ...... is called Ellipsis
> > >
> > > If you well define the ellipsis with an infinity border and where you include all REMAINDERS in division.
> > > Then you have a correct and proper mathematics.
> > >
> > > The example that must always be followed in division is the remainder
> > >    ________
> > > 3| 1000       = 333+1/3
> > >
> > > We never imagine that the correct final answer is without that fraction 1/3 added on
> > >
> > > Thus, when we have
> > >
> > >    ________
> > > 3| 1.000...       = .333....(+1/3)
> > >
> > > for we have a ending fraction of (+1/3)
> > >
> > > Now, the slobs and snobs never realized how important that (+1/3) was for they boneheadedly did this
> > >
> > > .3333333333.......... plus .3333333...........  = .666666666........ and thought everything was A-okay
> > >
> > > But look what the true addition is like:
> > >
> > > .333333.....33(+1/3) + .333333.....33(+1/3) = .66666666........66(+2/3)
> > >
> > > You see how unmessy that is, because, well look at this by the boneheads:
> > >
> > > .99999999..... + .99999999..... = 1.999999......998
> > >
> > > Whereas true math has
> > >
> > > .9999999......99(+10/9) + .9999999.....99(+10/9) = 1.999999.....98(+20/9) = 2.00000....(+0)
> > >
> > > So, you see how clean that all is, rather than what the slobs, snobs and boneheads dish out in their fantasies of math.
> > >
> > > They are failures, regular failures of math for they refuse to define infinity with a borderline and then they make up this crap that .9999.... is the same as 1, or that .33333.... is the same as 1/3 when they forgot about the remainder.
> > >
> > > AP