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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.

Integers

Integers extend whole numbers by including negative numbers. The set of integers includes ..., -3, -2, -1, 0, 1, 2, 3, ... and is denoted by Z. Integers are important because they allow representation of quantities that can be less than zero, such as temperatures below freezing or debts in finance. The chapter explains the properties of integers, including closure under addition, subtraction, and multiplication, but not under division. It introduces the concept of additive inverse, where for every integer 'a', there exists an integer '-a' such that a + (-a) = 0. The number line is used to represent integers, showing negative numbers to the left of zero and positive numbers to the right. The section also discusses the ordering of integers and how to compare them. For example, -3 < -1 < 0 < 2 < 5. The operations on integers are explained with examples, including addition and subtraction using the number line method. The section emphasizes the importance of understanding integers for solving real-world problems involving gains and losses, elevations above and below sea level, and temperature changes.

📊 Diagram: Number line showing integers with negative numbers to the left of zero and positive numbers to the right, with arrows indicating addition and subtraction movements.

🧪 Activity: Activity: Using a number line, students perform addition and subtraction of integers by moving right for positive and left for negative numbers.

🔗 Connection: Understanding integers leads to the study of rational numbers, which include fractions and decimals, expanding the number system further.

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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