Two way to two...

I learned in Mr. Block's 9th grade geometry class that 1.999... = 2 . That is, "One point 9 repeating" is another way to write "two." Speaking of all this making sense but not agreeing stuff, and a recent comment by Anne on the same topic, I went in search for a proof. I found this discussion. Here are some clever attempts...

If x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.

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1.99.. does equal 2

1.99.. = 1.99..

1.99.. = 1+ .99..

1.99.. = 1 + 9 * .11..

1.99.. = 1 + 9 * 1/9

1.99.. = 1 + 1

1.99.. = 2

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Because .9999... means a decimal with an infinite number of nines after it, the only possible value for 1-.9999... would be .0000000....1 in other words, an infinite number of zeroes..... followed by a one. Clearly, this is absurd, and therefore 1-.999999... must be zero.

As fun as these are, as far as I can tell, they aren't based on real math. I think 1.999... = 2 by definition.