Infinite numbers are strange
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I missed out all the boilerplate phrases along the lines of "As you increase the number of digits 7*0.14285714... tends towards 1 so, in the limit, is assumed to be 1...", etc.
Kornfeld Eliyahu Peter wrote:
0.1428573 * 7 = 1.0000011 0.14285731428573 * 7 = 1.00000120000011 --- 0.142857 * 7 = 0.999999 0.142857142857 * 7 = 0.999999999999
You've misread the first number (second in my message). It's... ........2857142857142857142857142857143 or ........2857142857142857142857142857142857142857142857142857142857143 That is, an integer with an infinite number of repetitions of ...285714... followed by 3. Although there's a notation for recurring decimals, I don't know of a shorthand for a p-adic number (which is what this is). The point is that I (and the video) didn't suggest stopping the calculation at any point. Obviously, if you do they aren't the same and, in the case of the 'infinite' integer, you don't actually have a result! It's only if you project to the theoretical limit that the results are equivalent. Many years ago, I researched for a PhD in Nuclear Structure Physics and studied some High-Energy (particle) Physics so am aware of renormalization to get rid of infinites in theories, the meaning (or lack of it) of anything divided by zero, etc. I'd just never encountered p-adic numbers[^] before I watched that video[^].
Finally I had time to watch the video... Now I understand you better... However I have to say that, that video cuts corners in a very horrible way IMHO... It seems that its simplifications and inclusions are chosen to server a specific end result, but not clear and whole or precise...
"If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization." ― Gerald Weinberg
