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Last updated date: 09th Dec 2023
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MVSAT Dec 2023

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

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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
Thus, on evaluating the above equation, we get
We can now divide both sides of this equation by 999, to get
So, now we have
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.