3 ms·

If you are inclined to believe that 0.3333... = (1/3), can you explain why 0.9999... = 1 is a big leap for you? Whether the case is 0.3333... or 0.9999..., each

by rskar 15y ago

If you are inclined to believe that 0.3333... = (1/3), can you explain why 0.9999... = 1 is a big leap for you? Whether the case is 0.3333... or 0.9999..., each is the result of a convergent series. ( http://en.wikipedia.org/wiki/Convergent_series http://en.wikipedia.org/wiki/Convergent_series ) On the word "fraction," do you take it to mean the result of dividing an integer by another integer, and that it must be less than 1? An "improper fraction" is still a fraction, nonetheless; hence 1 is also a fraction, namely (1/1), among other representations. ( http://en.wikipedia.org/wiki/Fraction_(mathematics) http://en.wikipedia.org/wiki/Fraction_(mathematics) ) Fractions aside, 0.9999... is a legitimate, albeit strange, way to say "one." ---- The grade-school approach (which can be used for 0.3333... too): (1) y = 0.9999... (2) 10y = 9.9999... (3) 10y - y = 9.9999... - 0.9999... = 9y (4) 9y = 9 (5) y = 1

rskar 15y ago
I wonder if this Xamuel guy is a bit off his nut. He invents a word, "pseudo-real," to classify 0.9999... and other such representations. He eventually castigates the reasons behind declaring 0.9999...=1, and then asks a profound question: "The real question is not why 0.9999...=1, but rather, what on earth are real numbers in the first place?" It's OK to ask, of course, but he's asking it in a profoundly silly way. For starters, 0.9999... is no less a real number than 0.9999 (i.e. 9999/10000). If we wish to continue on his line of thought, then we should all concede that any scrawling used to represent a quantity - any quantity, real number or no - is a pseudo-quantity. A scrawling is just not the real thing, whether "quantity" is in fact something real or simply the product of human imagination. Hence, "1" is also a pseudo-real, along with "I", "a", "0.9999...", "cos 0", etc. It's anyone's guess if there is such a thing as a unit of anything at all.

by rskar 15y ago

rskar 15y ago

I wonder if this Xamuel guy is a bit off his nut. He invents a word, "pseudo-real," to classify 0.9999... and other such representations. He eventually castigates the reasons behind declaring 0.9999...=1, and then asks a profound question: "The real question is not why 0.9999...=1, but rather, what on earth are real numbers in the first place?" It's OK to ask, of course, but he's asking it in a profoundly silly way. For starters, 0.9999... is no less a real number than 0.9999 (i.e. 9999/10000). If we wish to continue on his line of thought, then we should all concede that any scrawling used to represent a quantity - any quantity, real number or no - is a pseudo-quantity. A scrawling is just not the real thing, whether "quantity" is in fact something real or simply the product of human imagination. Hence, "1" is also a pseudo-real, along with "I", "a", "0.9999...", "cos 0", etc. It's anyone's guess if there is such a thing as a unit of anything at all.