Understanding Repeating Decimals and Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, a numerator $p$ and a non-zero denominator $q$. Repeating decimals, like 0.232323..., are a type of rational number.

To convert a repeating decimal into its rational form, we can use an algebraic method. Let's apply this method to the given number 0.232323...

Converting 0.232323... to a Fraction

Let the given repeating decimal be represented by the variable $x$.

• Step 1: Set the decimal equal to $x$.
  $x = 0.232323...$ (Equation 1)
• Step 2: Identify the repeating block of digits. In this case, the repeating block is '23'.
• Step 3: Count the number of digits in the repeating block. There are 2 digits ('2' and '3').
• Step 4: Multiply Equation 1 by $10^n$, where $n$ is the number of digits in the repeating block. Since there are 2 repeating digits, we multiply by $10^2 = 100$.
  $100x = 100 \times 0.232323...$
  $100x = 23.232323...$ (Equation 2)
• Step 5: Subtract Equation 1 from Equation 2. This step is crucial because it eliminates the repeating part of the decimal.
  $(100x) - (x) = (23.232323...) - (0.232323...)$
  $99x = 23$
• Step 6: Solve for $x$ by dividing both sides by 99.
  $x = \frac{23}{99}$

Thus, the rational form of 0.232323... is $\frac{23}{99}$.

Verification of the Rational Form

To verify our answer, we can divide 23 by 99:

The division confirms that $\frac{23}{99}$ is indeed equivalent to the repeating decimal 0.232323....

Comparing with Given Options

Let's compare our result with the provided options:

• Option 1: $\frac{23}{999}$
• Option 2: $\frac{23}{99}$
• Option 3: $\frac{23}{990}$
• Option 4: $\frac{23}{9}$

Our calculated rational form, $\frac{23}{99}$, matches Option 2.

Revision Table: Repeating Decimal Conversion

Additional Information: Rational Numbers

Rational numbers include all integers ($\dots, -2, -1, 0, 1, 2, \dots$), fractions (like $\frac{1}{2}$, $-\frac{3}{4}$), and all terminating and repeating decimals. They can be plotted on a number line. The set of rational numbers is denoted by $\mathbb{Q}$. Numbers that cannot be expressed as a simple fraction $\frac{p}{q}$ are called irrational numbers, such as $\sqrt{2}$ or $\pi$.