0.99999999999 divided by 1 = 1 / 9 = 0.11111111111111
1 divided by 9 = 1 / 9 = 0.111111111111
So, does that make 1 = 0.99999999999999
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their distance is as tiny as you want so... – Vicfred Oct 04 '13 at 14:32
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For what you have written, no. – peterwhy Oct 04 '13 at 14:32
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But $0.999\cdots = 1$ exactly – peterwhy Oct 04 '13 at 14:33
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$0.999 \dotsc$ divided by 1 is $0.999 \dotsc$. – tylerc0816 Oct 04 '13 at 14:33
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3But $0.99999999999999=1-0.00000000000001<1$. SNCR – Michael Hoppe Oct 04 '13 at 14:35
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1 divided by 9 is $0.111\dots$ (with an infinite number of $1$s), not $0.111111111111$. – ShreevatsaR Oct 04 '13 at 14:40
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@Vicfred, if the OP meant the infinite periodic decimal expansion $;0.99\dot{9};$ , then this number is exactly $;1;$....which means, of course, that it is "as close as desired" to one. – DonAntonio Oct 04 '13 at 14:58
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Let's assume that: $$ 0.a_1a_2....=\frac{a_1}{10}+\frac{a_2}{10^2}+.... $$ Then it can be seen that: $$ 0.9999....=\frac{9}{10}+\frac{9}{10^2}+....=\frac{9}{10}\frac{1}{1-\frac{1}{10}}=1. $$ Another way to see that is:
$$ 0.9999....=9(\frac{1}{10}+\frac{1}{10^2}+....)=9\times 0.1111... $$ But then $0.1111...=\frac{1}{9}$ which means that $9\times 0.1111...=1$.
This means that the convention of representing real numbers in decimals has the difficulty of having non-unique representation. Therefore $0.99999999$ and $1$ both represents the identity element of multiplication for real numbers.
Arash
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