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MathematicsClass 9The World of Numbers

The World of Numbers | Class 9 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 4 min read

The World of Numbers – this guide gives you a concise, exam-ready overview of The World of Numbers from Class 9 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

Rational Numbers

Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. This set includes integers, fractions, and decimals that either terminate or repeat. Rational numbers are important because they represent quantities that are not whole numbers, such as parts of a whole or ratios. The chapter explains how rational numbers can be positive or negative. It also discusses the decimal representation of rational numbers, highlighting that they either terminate (e.g., 0.75) or repeat periodically (e.g., 0.333...). The section explains the properties of rational numbers, including closure under addition, subtraction, multiplication, and division (except division by zero). It also introduces the concept of equivalent rational numbers, where different fractions represent the same rational number (e.g., 1/2 = 2/4). The section includes methods to convert decimals to rational numbers and vice versa. The number line representation of rational numbers is also discussed, showing that rational numbers are dense, meaning between any two rational numbers, there exists another rational number. This property is crucial for understanding the continuity of the number line.

📊 Diagram: Number line showing rational numbers including fractions and decimals, illustrating density by marking points between two rational numbers.

🧪 Activity: Activity: Students convert given decimals into rational numbers and plot them on the number line to observe density.

🔗 Connection: This section prepares for the introduction of irrational numbers, which are numbers that cannot be expressed as rational numbers.

Frequently asked questions

1. Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: \(\frac{7}{20}, \frac{4}{15}\) and \(\frac{13}{250}\). Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

To determine whether the decimal expansion of a rational number \(\frac{p}{q}\) is terminating or repeating, factorize the denominator \(q\) into primes. If the denominator (in simplest form) has only 2 and/or 5 as prime factors, the decimal is terminating; otherwise, it is repeating.

1) \(\frac{7}{20}\): Denominator 20 = 2^2 * 5. Only 2 and 5 as prime factors, so decimal terminates. Long division: 7 ÷ 20 = 0.35 (terminating)

2) \(\frac{4}{15}\): Denominator 15 = 3 * 5. Since 3 is a prime fact

2. Perform the long division for \(\frac{1}{13}\) . Identify the repeating block of digits. Does it show cyclic properties if you evaluate \(\frac{2}{13}\) ? Now compute \(\frac{3}{13}, \frac{4}{13}\) , etc. What do you notice?

Performing long division for \(\frac{1}{13}\):

1 ÷ 13 = 0.076923076923... The repeating block is '076923' (6 digits).

For \(\frac{2}{13}\): 2 ÷ 13 = 0.153846153846... Repeating block '153846'.

Similarly: \(\frac{3}{13} = 0.230769230769...\) repeating '230769' \(\frac{4}{13} = 0.307692307692...\) repeating '307692' \(\frac{5}{13} = 0.384615384615...\) repeating '384615' \(\frac{6}{13} = 0.461538461538...\) repeating '461538'

Notice that the repeating blocks are cyclic permutations of the digi

3. Classify the following numbers as rational or irrational: (i) \(\sqrt{81}\) (ii) \(\sqrt{12}\) (iii) 0.33333 ... (iv) 0.123451234512345 ... (v) 1.01001000100001 ... (Notice the pattern: Is it repeating a single block?) (vi) 23.560185612239874790120 Find the explicit fractions in case they are rational.

(i) \(\sqrt{81} = 9\), which is a rational number (integer).

(ii) \(\sqrt{12} = 2\sqrt{3}\), irrational because \(\sqrt{3}\) is irrational.

(iii) 0.33333 ... is a repeating decimal (0.\overline{3}), rational. Explicit fraction: \(\frac{1}{3}\).

(iv) 0.123451234512345 ... is a repeating decimal with block '12345', rational. Explicit fraction: Let x = 0.1234512345..., multiply by 10^5 = 100000: 100000x = 12345.1234512345... Subtracting x: 100000x - x = 12345.1234512345... - 0.1234512345... = 12

4. The number \(0.9\) (which means \(0.99999 \ldots\) ) is a rational number. Using algebra (let \(x = 0.9\) , multiply by 10, and subtract), explain why \(0.9\) is exactly equal to 1.

Let \(x = 0.99999...\)

Multiply both sides by 10: \(10x = 9.99999...\)

Subtract the original equation from this: \(10x - x = 9.99999... - 0.99999...\) \(9x = 9\)

Divide both sides by 9: \(x = 1\)

Since \(x = 0.99999...\), it follows that \(0.99999... = 1\).

Hence, the decimal 0.9 (repeating) is exactly equal to 1.

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