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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 · 3 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.

Real Numbers

Real numbers comprise all rational and irrational numbers. This set includes every number that can be represented on the number line. The chapter explains that real numbers are used to measure continuous quantities and are fundamental in mathematics and science. It discusses the properties of real numbers, including closure under addition, subtraction, multiplication, and division (except by zero). The section also explains the representation of real numbers on the number line, emphasizing that the number line is a visual representation of the real number system. The chapter introduces the concept of decimal expansions of real numbers, which can be terminating, repeating, or non-terminating and non-repeating. It highlights the importance of real numbers in solving equations and representing measurements. The section also briefly mentions the irrationality of some famous constants like π and e, reinforcing the concept of real numbers as a complete set. The chapter concludes by summarizing the classification of numbers and their relationships within the real number system.

📊 Diagram: Number line showing all types of numbers—natural, whole, integers, rational, and irrational—highlighting that all are subsets of real numbers.

🧪 Activity: Activity: Students classify given numbers into natural, whole, integers, rational, irrational, and real numbers and plot them on the number line.

🔗 Connection: This section sets the foundation for understanding operations on real numbers and their applications in subsequent chapters.

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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