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 Question from Asia, a student: Does 1= 0.9999....? There seems to be different opinions on this.

Yes, it does. A nonterminating decimal has the value that its sequence of truncations converges to; thus pi is the limit of

3, 3.1, 3.14, 3.141, 3.1415, ....

and 0.9999... is the limit of

0.9, 0.99, 0.999,...

which is 1.

Decimal fractions (numbers of the form A/10^n) can be expressed in all the following ways as decimals (for simplicity I will use the example 123/100 = 1.23:

terminating, without trailing zeros: 1.23
terminating, with finitely many trailing zeros: 1.23000
(infinitely many choices!)
with infinitely many trailing zeros: 1.23000...
with infinitely many trailing nines: 1.22999...

All other real numbers, rational or not, have a unique decimal expansion, which never terminates.

Good Hunting!
RD

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