Rational Numbers: Representation & Decimal Expansions

Rational numbers are numbers that can be expressed in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not equal to zero. They include integers, fractions, and terminating and recurring decimals.

Representation on the Number Line: Rational numbers are represented on a number line by dividing the unit interval between two integers into equal parts based on the denominator of the fraction. For example, to plot $\frac{3}{4}$, divide the interval between 0 and 1 into four equal parts, and mark the point corresponding to the third division from 0.

Decimal Expansions: Every rational number has a decimal expansion that either terminates (ends) or recurs (repeats). Terminating decimals have a finite number of digits after the decimal point. Recurring decimals have an infinite number of digits after the decimal point, with a repeating pattern of digits.

To convert a fraction to a decimal, divide the numerator by the denominator. If the division ends (leaves a remainder of 0), the decimal terminates. If the division continues with a repeating pattern of remainders, the decimal recurs.

Formulae

There aren’t specific “formulae” for rational numbers themselves. However, the following is useful:

To convert a fraction $\frac{p}{q}$ to a decimal, perform the division $p \div q$.
To convert a terminating decimal to a fraction, write the decimal as a fraction with a denominator that is a power of 10 (e.g., 10, 100, 1000) and simplify.
To convert a recurring decimal to a fraction, use algebraic manipulation (explained in the Examples section).

Examples

Example-1: Representing Fractions on the Number Line

Represent the following rational numbers on a number line: $-\frac{1}{2}$, $\frac{3}{4}$, and $2\frac{1}{3}$.

For $-\frac{1}{2}$: Divide the interval between -1 and 0 into two equal parts. Mark the point that corresponds to one division to the left of zero.
For $\frac{3}{4}$: Divide the interval between 0 and 1 into four equal parts. Mark the point that corresponds to three divisions from zero.
For $2\frac{1}{3}$: This is equivalent to $\frac{7}{3}$. Divide the interval between 2 and 3 into three equal parts. Mark the point corresponding to one division from 2 (or $\frac{1}{3}$ of the way to 3).

Example-2: Converting Recurring Decimals to Fractions

Convert the recurring decimal $0.333…$ (written as $0.\overline{3}$) to a fraction.

Let $x = 0.\overline{3}$
Multiply both sides by 10: $10x = 3.\overline{3}$
Subtract the first equation from the second: $10x – x = 3.\overline{3} – 0.\overline{3}$
Simplify: $9x = 3$
Solve for $x$: $x = \frac{3}{9} = \frac{1}{3}$

Common Mistakes by Students

Incorrectly dividing fractions to decimals: Forgetting to include the decimal point when performing long division.
Misinterpreting recurring decimals: Not understanding how to properly represent them (e.g., writing $0.333$ instead of $0.\overline{3}$) or the process of conversion.
Misunderstanding the number line representation: Incorrectly placing fractions on the number line, especially negative fractions.
Not simplifying fractions: Failing to reduce fractions to their simplest form after converting from decimals.

Real Life Application

Rational numbers are used extensively in everyday life:

Cooking and Baking: Recipes often use fractions (e.g., $\frac{1}{2}$ cup of flour).
Measuring: Lengths, weights, and volumes are frequently represented as rational numbers (e.g., 2.5 meters, 1.75 kg).
Finance: Calculating interest rates, discounts, and percentages involves rational numbers.
Construction: Building and designing often involve precise measurements and fractions.

Fun Fact

The term “rational” comes from the word “ratio,” because rational numbers can be expressed as a ratio of two integers. Irrational numbers, which cannot be written as a ratio of integers, are the opposite!