Can you think of other way(s) to find a rational number between any two rational numbers?

Question

Homi · Accepted Answer

Understand the Question

Between any two distinct rational numbers, there exist infinitely many rational numbers.
Several methods can be used to find them:
- Mean (Average) Method: Calculate the average $\dfrac{a + b}{2}$ of the two numbers.
- Common Denominator Method: Convert both fractions to equivalent fractions with a larger common denominator to find intermediate numerators.
- Decimal Expansion Method: Convert both fractions into decimals and choose any terminating or repeating decimal between them.

Step 1 · Mean Method

For any two rational numbers $a$ and $b$ , their average $\dfrac{a + b}{2}$ always lies strictly between them.

For example, to find a rational number between $a = \dfrac{1}{4}$ and $b = \dfrac{1}{2}$ :

Sum of the numbers

\begin{aligned} \dfrac{1}{4} + \dfrac{1}{2} &= \dfrac{1}{4} + \dfrac{2}{4} \\[0.6em] &= \dfrac{3}{4} \end{aligned}

Divide the sum by $2$

\begin{aligned} \dfrac{3}{4} \div 2 &= \dfrac{3}{4} \times \dfrac{1}{2} \\[0.6em] &= \dfrac{3}{8} \end{aligned}

Thus, $\dfrac{3}{8}$ lies between $\dfrac{1}{4}$ and $\dfrac{1}{2}$ .

Step 2 · Common Denominator Method

Make the denominators equal and scale them up using equivalent fractions.

For $\dfrac{1}{4}$ and $\dfrac{1}{2}$ , convert to a common denominator of $4$ $\dfrac{1}{4} = \dfrac{1}{4}$ $\dfrac{1}{2} = \dfrac{1 \times 2}{2 \times 2} = \dfrac{2}{4}$

Multiply numerator and denominator of both by $10$ to create space between numerators $\dfrac{1}{4} = \dfrac{1 \times 10}{4 \times 10} = \dfrac{10}{40}$ $\dfrac{2}{4} = \dfrac{2 \times 10}{4 \times 10} = \dfrac{20}{40}$

Any rational number with a numerator between $10$ and $20$ (such as $\dfrac{11}{40}$ ) lies between them.

Step 3 · Decimal Expansion Method

Convert the rational numbers into their decimal representations.

$\dfrac{1}{4} = 0.25$ $\dfrac{1}{2} = 0.5$

Pick any terminating decimal between $0.25$ and $0.5$ , such as $0.3$ , $0.4$ , or $0.45$ , and write it in $\dfrac{p}{q}$ form $0.3 = \dfrac{3}{10}$

Answer

Yes, other methods include:

Mean Method: Finding the average $\dfrac{a + b}{2}$ .
Common Denominator Method: Converting to equivalent fractions with a larger common denominator.
Decimal Expansion Method: Converting both numbers into decimals and picking a terminating or repeating decimal between them.

Common Mistakes

Arithmetic Error in Mean: Forgetting to divide the sum by $2$ after adding the two fractions.
Non-Equivalent Fractions: Multiplying only the numerator or only the denominator when creating equivalent fractions.
Choosing Irrationals in Decimal Form: Writing a non-terminating, non-repeating decimal instead of a terminating or repeating decimal (rational number).

Question 6

More questions in Exercise 3.4