Re: Problems I have with 1.999...=2 - variation on a well-known theme

UNKNOWN VARIATION ON A WELL-KNOWN THEME 

Kirby Cook wrote:

Problems I have with 1.999...=2

Commonly, the first "proof" or demonstration offered involves setting .999... =x, multiplying .999... by 10, subtracting x from 10x, and asserting that 9x=9. Taking that step by step, if x=.9 and 10x=9, 9x=8.1. When x=.99 and 10x=9.9, 9x=8.91. Lining up the first few examples,
10x 9 9.9 9.99 9.999
-x -.9 -.99 -.999 -.9999
9x 8.1 8.91 8.991 8.9991

So, even if you claim leave to ignore the convention that multiplying a number by ten always adds a zero to the end of the number by arguing that there is no "end" to .999... (a dubious dodge in itself, IMO), if you let those two numbers, x and 10x march out to "without limit" in lockstep, the upper 9 will always be one step to the left of the lower, and 10x-x will never equal 9.

The flaw in the oft-cited second "proof", or demostration, which tries to show that there is no number between 1.999... and 2, may be found in the law of the reciprocal, by analogy and inference. Say, for instance, that there is no number between the infinite sum (difference) 1-.999... and zero.Taking it step by step, 1-.9 has a reciprocal of 1/(1-.9) or 10; 1-.99 has a recirpocal of 100, and so on. So, if you assert that there is no number between 1-.999... and zero, then you are also asserting that there is no number between 1000... and a point often termed the point at infinity. And how many of you will buy that?

It should be obvious, I hope, that I am directing my remarks not to those who carefully assert that 1.999... =2 is merely sloppy shorthand for the idea that 1.999... *approaches* 2. No, my remarks are for those hopelessly superstitious minds that claim that "equals means equals" in this case.

I use the words "hopelessly superstitious" advisedly, as it seems to me that the common invocation of the words "without limit" and all similar such are used with no more understanding than a twelve year old's fervent incantation to charm warts. The frank translation of "without limit" seems to be "the magic happens here", the magic being, most often, that something becomes nothing. The fact that we can trace this impenetrably ignorant superstition back to such a luminary as Leonhard Euler, who said in one of his most quoted dissertations, "There is no doubt that any quantity can be diminished until it vanishes and is transformed into nothing", is really no excuse for its perpetuation. At any rate, *I* certainly doubt his assertion! That Euler was a brilliant mathematician is true, but that he was something of a crackpot is also true, as evidenced by the above quote and the accompanying exposition.
Consider: If *at any point* applying division to a finite quantity results in zero, that point is as much where you are as, well, as this point is where I am. So if I can say "Alakazam!" and make a small bit vanish, why can't I say it (maybe louder?) and make a big bit vanish in exactly the same way?

Or, if you persist in asserting that Euler was right, what *exactly* is it about the words and concept, "without limit", that makes the magic happen, that transports us from the familiar world of 2+2=4 to a world where four divided by something equals ("Alakazam!") nothing? Just as though that four never was.

I would much like to know exactly what part of which work of Euler you are quoting. Please do not forget that Cauchy's theory of sequences, series and limits found its way among mathematicians only after the death of Euler. Therefore all earlier attempts at catching the concepts of limit and continuity seem clumsy and even crackpot-like to some of us today.

As regards the specific issue of 1.999... = 2, I repeat a sci.math posting I did twice previously. I do this at the risk of causing a new avelanche of debates on matters infinitesimal.

===========================================================
As long as I am surfing the world-wide web and the newsgroups for mathematical stuff I have been intrigued by the phenomenon "Is 0.999999.... = 1?". I got the impression that it is a recurring phenomenon. It seems to coincide more or less with the start of the education season.

In my opinion 0.999999..... is the most elusive representation of 1, even more so than 1 =
exp(2.pi.i). Let me explain.

0.9999... is a fabricated number. It cannot be obtained in any way as the outcome of a long
division carried out in the usual manner. As far as I know nobody before noticed this. This may be the reason why it pops up time and again in newsgroups and similar forums.

But you can perform long division in a different way as follows:

Consider long division A/B as a process of exhaustion in the style of Archimedes, i.e. take away from the dividend A as many times of B as possible. If nothing is left then you are done; otherwise proceed with taking away from the remainder as many parts B/10 as possible, etc.

Now look at A/B = 1/1, but consider A as 10 * 0.1 instead of 1 * 1, and take away nine parts B/10 = 0.1 instead of all ten parts. Write down 0.9 in the quotient field to record that you took away nine parts of size 0.1
Shift down a factor of 10 and treat the remainder like you treated the original dividend
previously. The quotient becomes 0.99 and the new remainder is 0.01; etc.

Only two things really matter in the process of long division:

(1) maintaining the relation A = Qn * B + Rn where Qn and Rn are the quotient and the remainder at the n-th stage of the process;

(2) getting Rn eventually equal to zero, or at least getting Rn arbitrarily close to zero.

I hope that it is clear from the above that 0.9999... has a sensible meaning and can be equal only to unity.

I have no illusion that this note will stop newsgroup postings on the phenomenon
"0.999999...... = 1?".
===========================================================

Johan E. Mebius
.

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