Problems I have with 1.999...=2

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.
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