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Express $0.\overline{001}$ in the simplest form.

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Last updated date: 15th Jul 2024
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Answer
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Hint: We must first check how many digits are being repeated in the given number, and if the number of such digits is n, we must multiply the number by ${{10}^{n}}$. By subtracting this new number with the original given number, we can get an equation free from recurring parts. Thus, we can solve it further to convert it into a fraction in its simplest form.

Complete step by step answer:
We know that a specific type of decimal numbers, in which the digits after the decimal point keep on repeating endlessly, are called non terminating, repeating decimals. We represent these repeating decimals with a bar above the repeating number.
So, $0.\overline{001}$ simply means that the digits 001 are being repeated endlessly. Thus, we can also represent $0.\overline{001}$ as 0.001001001001…
We know that a division by 10 results in the shifting of the decimal point on the left side by one place, and a multiplication by 10 results in the shifting of the decimal point on the right side by one place.
So, we can say that a multiplication by 1000 will result in the shifting of the decimal point to the right, by one place.
So, let us assume $x=0.\overline{001}...\left( i \right)$
Let us multiply equation (i) by 1000 on both sides. Thus, we get
$1000x=1000\times 0.\overline{001}$
By using the concept explained above, we can write
$1000x=1.\overline{001}...\left( ii \right)$
Now, subtracting (i) from (ii), we get
$1000x-x=1.\overline{001}-0.\overline{001}$
Thus, on evaluating the above equation, we get
$999x=1$
We can now divide both sides of this equation by 999, to get
$\dfrac{999x}{999}=\dfrac{1}{999}$
So, now we have
$x=\dfrac{1}{999}$
Here, we can see that the fraction is in its simplest form.
Hence, we can say that the non-terminating repeating decimal $0.\overline{001}$ is equal to the fraction $\dfrac{1}{999}$.

Note: Here, in this question, we must note that in $0.\overline{001}$, three digits, namely 001, is being repeated, and so, the multiplication is done by ${{10}^{3}}=1000$. If the number of repeating digits would have been n, then instead of multiplying by 1000, we would have multiplied by ${{10}^{n}}$. We should also know that repeating decimals are also called recurring decimals.