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Solution 1.1:7c

From Förberedande kurs i matematik 1

Revision as of 14:00, 14 September 2008 by Ian (Talk | contribs)
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If we look more closely at this number, we see that the combination 001 is repeated from the second decimal place onwards,

02 001 001 001...

and this reveals that the number is rational.

By multiplying a certain number of times by 10 we can move the decimal place step by step to the right.

x=02 001 001 001...
10x=2001 001 001 1...
100x=2001 001 001 1...
1000x=2001 001 001 1...
10000x=2001001 001 1...

In this list, we see that 10x and 10000x have the same decimal expansion, which means that

10000x−10x=2001001 001 001...−2001 001 001...
=1999.(decimal parts cancel)

As 10000x−10x=9990x then

9990x=1999x=99901999.
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