Classify the following numbers as rational or irrational: (i) (ii) (iii) (iv) (v) (Notice the pattern: Is it repeating a single block?) (vi) Find the explicit fractions in case they are rational.

The World of Numbers | Exercise 3.5

Question 3

Classify the following numbers as rational or irrational:

(i) $\sqrt{81}$

(ii) $\sqrt{12}$

(iii) $0.33333 \dots$

(iv) $0.123451234512345 \dots$

(v) $1.01001000100001 \dots$ (Notice the pattern: Is it repeating a single block?)

(vi) $23.560185612239874790120$

Find the explicit fractions in case they are rational.

Solution

Understand the Question

A number is rational if it can be written as a fraction $\dfrac{p}{q}$ of two integers ( $q \neq 0$ ). Its decimal expansion is either terminating or non-terminating repeating (recurring).
A number is irrational if its decimal expansion is non-terminating and non-repeating (such as square roots of non-perfect squares or patterned non-repeating decimals).
For rational numbers, we find their explicit fraction form $\dfrac{p}{q}$ .

(i) Classify $\sqrt{81}$ as rational or irrational and find its explicit fraction if rational.

Step 1 · Simplify and Classify $\sqrt{81}$

Diagram 1

\begin{aligned} \sqrt{81} &= \sqrt{9 \times 9} \\ &= \sqrt{9^2} \\ &= 9 \end{aligned}

Expressing as a fraction of integers $9 = \dfrac{9}{1}$

Since it can be written in the form $\dfrac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$ , $\sqrt{81}$ is a rational number.

Answer

(i) Rational, $\dfrac{9}{1}$

(ii) Classify $\sqrt{12}$ as rational or irrational and find its explicit fraction if rational.

Step 1 · Simplify and Classify $\sqrt{12}$

Simplifying $\sqrt{12}$

\begin{aligned} \sqrt{12} &= \sqrt{4 \times 3} \\ &= \sqrt{2^2 \times 3} \\ &= 2\sqrt{3} \end{aligned}

Since $3$ is not a perfect square, $\sqrt{3}$ is an irrational number with a non-terminating, non-repeating decimal expansion.

The product of a non-zero rational number ( $2$ ) and an irrational number ( $\sqrt{3}$ ) is irrational.

Therefore, $\sqrt{12}$ is an irrational number.

Answer

(ii) Irrational

(iii) Classify $0.33333 \dots$ as rational or irrational and find its explicit fraction if rational.

Step 1 · Convert Repeating Decimal to a Fraction

Let $x = 0.33333 \dots \quad \dots (1)$

Multiply by $10$ $10x = 3.33333 \dots \quad \dots (2)$

Subtracting $(1)$ from $(2)$

\begin{aligned} 10x - x &= 3.33333 \dots - 0.33333 \dots \\[0.6em] 9x &= 3 \\[0.6em] x &= \dfrac{3}{9} = \dfrac{1}{3} \end{aligned}

Since it is a repeating decimal and can be expressed as $\dfrac{1}{3}$ , it is a rational number.

Answer

(iii) Rational, $\dfrac{1}{3}$

(iv) Classify $0.123451234512345 \dots$ as rational or irrational and find its explicit fraction if rational.

Step 1 · Convert Repeating Block Decimal to a Fraction

The repeating block has $5$ digits ( $12345$ ).

Let $x = 0.1234512345 \dots \quad \dots (1)$

Multiply by $10^5 = 100000$ $100000x = 12345.1234512345 \dots \quad \dots (2)$

Subtracting $(1)$ from $(2)$

\begin{aligned} 100000x - x &= 12345.12345 \dots - 0.12345 \dots \\[0.6em] 99999x &= 12345 \\[0.6em] x &= \dfrac{12345}{99999} \end{aligned}

Simplifying by dividing numerator and denominator by $3$ $x = \dfrac{12345 \div 3}{99999 \div 3} = \dfrac{4115}{33333}$

Since it can be written as a fraction of integers, it is a rational number.

Answer

(iv) Rational, $\dfrac{4115}{33333}$

(v) Classify $1.01001000100001 \dots$ as rational or irrational and find its explicit fraction if rational.

Step 1 · Analyze Pattern and Classify

In $1.01001000100001 \dots$ , the number of zeros between successive ones increases each time ( $1$ zero, then $2$ zeros, $3$ zeros, etc.).

Because no fixed block of digits repeats periodically, the decimal expansion is non-terminating and non-repeating.

Therefore, $1.01001000100001 \dots$ is an irrational number.

Answer

(v) Irrational

(vi) Classify $23.560185612239874790120$ as rational or irrational and find its explicit fraction if rational.

Step 1 · Convert Terminating Decimal to a Fraction

The number $23.560185612239874790120$ terminates after $21$ digits following the decimal point.

Every terminating decimal is rational and can be converted into a fraction with a power of $10$ in the denominator $23.560185612239874790120 = \dfrac{23560185612239874790120}{1000000000000000000000}$

Since it is of the form $\dfrac{p}{q}$ , it is a rational number.

Answer

(vi) Rational, $\dfrac{23560185612239874790120}{1000000000000000000000}$

Common Mistakes

Pattern vs. Periodicity: Confusing patterned decimals like $1.010010001\dots$ with repeating decimals. A decimal is only rational if a fixed block of digits repeats periodically, not when the gap between digits grows.
Terminating Length: Thinking very long decimals are irrational simply because they have many decimal places. Any decimal that terminates is strictly rational.

Question 3

More questions in Exercise 3.5