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Nitella [24]
3 years ago
8

How do i turn reapeating decimals into fractions?

Mathematics
1 answer:
inysia [295]3 years ago
6 0
Easy example that shows 0.999\ldots=1:

Let x=0.999\ldots=0.\overline9.

Then 10x=9.999\ldots=9.\overline9.

So 10x-x=9.\overline9-0.\overline9, or 9x=9, and so x=1.

The basic idea is to find the period of the repeating decimal, move the n digits belonging to one period over to the left of the decimal point by multiplying by 10^n, then subtract the original repeating decimal from this new number, and finally divide by 10^n-1.

A slightly more complicated example:

Let x=0.142857142857142857\ldots=0.\overline{142857}.

Then 10^6x=142857.\overline{142857}.

Then 10^6x-x=142857.\overline{142857}-0.\overline{142857}, or

999999x=142857\implies x=\dfrac{142857}{999999}=\dfrac17
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