Q3(iii):

Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (iii) $0.\overline{001}$

Step 1: Initial Setup & Variable Assignment

Let the given repeating decimal be assigned to a variable $x$. The vinculum (the bar over the digits) indicates the specific block of digits that repeats infinitely.

$x = 0.\overline{001}$

Expanding the repeating block, we can write this as an infinite decimal:

$x = 0.001001001... \quad \text{--- (Equation 1)}$

Step 2: Determining the Multiplier [Base-10 Shift]

To convert a recurring decimal into a rational fraction of the form $\frac{p}{q}$, we must eliminate the infinitely repeating fractional part. [Per the properties of repeating decimals], we achieve this by shifting the decimal point to the right by exactly the number of digits in the repeating block.

The repeating block is "001", which consists of exactly $3$ digits. Therefore, we multiply both sides of Equation 1 by $10^3$ (which is $1000$).

$1000 \cdot x = 1000 \cdot (0.001001001...)$

$1000x = 1.001001001... \quad \text{--- (Equation 2)}$

Step 3: Algebraic Subtraction & Elimination of the Infinite Series

We now have two equations where the fractional parts (the digits after the decimal point) are identical. By subtracting Equation 1 from Equation 2, the infinite repeating sequence will cancel out completely [By the Subtraction Property of Equality].

Mathematically, this is executed as:

$1000x - x = (1.001001001...) - (0.001001001...)$

$999x = 1$

Step 4: Isolation of the Variable

Divide both sides of the equation by $999$ to isolate $x$ [By the Division Property of Equality]:

$x = \frac{1}{999}$

Step 5: Verification of Rational Conditions

The problem requires the final answer to be in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

• $p = 1$, which is an integer ($1 \in \mathbb{Z}$).
• $q = 999$, which is an integer ($999 \in \mathbb{Z}$).
• The denominator $q = 999 \neq 0$.

All conditions for a rational number are strictly satisfied. Furthermore, the greatest common divisor $\text{GCD}(1, 999) = 1$, meaning the fraction is in its simplest form.

Final Solution: The repeating decimal $0.\overline{001}$ expressed in the form $\frac{p}{q}$ is $\frac{1}{999}$.