Numbers | Class 8 Mathematics Notes
By ConceptScroll Team · Published on 17 July 2026 · 3 min read
Numbers – this guide gives you a concise, exam-ready overview of Numbers from Class 8 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a ratio of two integers, meaning they cannot be written in the form p/q where p and q are integers and q ≠ 0. Their decimal expansions are non-terminating and non-repeating, which distinguishes them from rational numbers. Examples include √2, √3, π, and e. The section explains the historical discovery of irrational numbers, such as the proof that √2 is irrational, which was a significant milestone in mathematics. It elaborates on the fact that irrational numbers fill the gaps on the number line that rational numbers cannot, thus completing the real number system. The section also discusses how irrational numbers can be approximated by rational numbers to any degree of accuracy, which is essential in practical computations. Students are introduced to the concept of real numbers as the union of rational and irrational numbers, emphasizing the completeness of the number line. The section includes proofs, decimal expansions, and examples to help students understand the nature and importance of irrational numbers.
📊 Diagram: Diagrams include the number line showing irrational numbers like √2 placed between integers 1 and 2, and a geometric square illustrating the length √2.
🧪 Activity: Activity 3.3: Using the Pythagorean theorem to find the length of the diagonal of a square (√2) and locating it on the number line.
🔗 Connection: This section connects to the next, which discusses the representation of real numbers on the number line and the completeness property.
Frequently asked questions
1. Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions? 2. Form a base-2 place value system using ‘ukasar’ and ‘urapon’ as the digits. Compare this system with that of the Gumulgal’s. 3. Where in your daily lives, and in which professions, do the Hindu numerals, and 0, play an important role? How might our lives have been different if our number system and 0 hadn’t been invented or conceived of? 4. The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers, and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?
1. The Chinese alternated between the Zong and Heng symbols likely to avoid confusion and to clearly distinguish between different place values. If only the Zong symbols were used, the numeral 41 would be represented by four Zong symbols followed by one Zong symbol for the units place. Without significant spacing, this could be misinterpreted as 11 or 14, causing ambiguity.
2. Using ‘ukasar’ and ‘urapon’ as digits for base-2, 'ukasar' can represent 0 and 'urapon' can represent 1. Numbers would
Which of the following statements correctly describes a rational number?
A number that can be expressed as a ratio of two integers, where the denominator is not zero
What is the decimal expansion of the rational number $\frac{1}{3}$?
0.333... (repeating)
Which of the following decimal expansions represents an irrational number?
3.1415926535... (non-terminating, non-repeating)
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