Date: Oct 3, 2017 7:31 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..... 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