Mathematics

Real Numbers

In Class 10 Mathematics, the chapter Real Numbers develops number concepts through Euclid’s Division Lemma, prime factorisation, and the Fundamental Theorem of Arithmetic. Students use prime factorization to find HCF and LCM, examine rational and irrational numbers, and understand proofs of the irrationality of √2, √3, and √5. The chapter also explains how the prime factors of a denominator determine whether a rational number has a terminating or recurring decimal expansion.

Euclid’s Division Lemma

In Class 10 Mathematics, under the chapter Real Numbers, Euclid’s Division Lemma introduces the relationship a = bq + r, where a and b are positive integers, q is the quotient, and the remainder r satisfies 0 ≤ r < b. Students learn how repeated division forms Euclid’s division algorithm and use it to find the highest common factor (HCF) of two numbers. The topic also strengthens understanding of divisibility, quotients, remainders, and the logical steps used in number-theory proofs.

650questions

Fundamental Theorem of Arithmetic

In this Class 10 Mathematics topic from the chapter Real Numbers, students learn that every integer greater than 1 can be expressed as a product of prime numbers, and that this prime factorisation is unique apart from the order of the factors. They practise finding prime factors and use the theorem to understand and determine the HCF and LCM of numbers. The topic builds clear reasoning about the structure of whole numbers and supports later work with divisibility and number relationships.

549questions

Prime Factorisation

In Class 10 Mathematics, under the Real Numbers chapter, Prime Factorisation teaches students to express a composite number as a product of prime numbers. Students practise identifying prime factors and writing factorisations using multiplication and exponents. This foundational skill supports the Fundamental Theorem of Arithmetic and later work with HCF and LCM.

750questions

HCF and LCM using prime factorisation

In this Class 10 Mathematics topic from Real Numbers, students learn to find the HCF and LCM of two or more numbers through prime factorisation. They break each number into prime factors, compare the resulting powers, and select the appropriate common factors: the lowest powers for HCF and the highest powers for LCM. The topic also develops accuracy in writing factor trees, simplifies comparison between methods, and helps students apply these concepts to numerical and everyday problem-solving situations.

504questions

Irrational numbers

In Class 10 Mathematics, this topic from the Real Numbers chapter introduces irrational numbers as numbers that cannot be expressed as a ratio of two integers. Students learn to identify them through non-terminating, non-repeating decimal expansions, distinguish them from rational numbers, and locate them on the number line. The topic also develops understanding of examples such as √2 and how irrational numbers fit into the wider system of real numbers.

571questions

Proof of irrationality of √2, √3, √5

In this Class 10 Mathematics topic from the Real Numbers chapter, students learn how to prove that √2, √3 and √5 are irrational numbers. The proof begins by assuming that a square root can be written as a fraction p/q in lowest terms, then uses prime divisibility and the resulting contradiction to reject that assumption. Students also strengthen their understanding of rational and irrational numbers, prime factorisation, and the logical structure used in mathematical proofs.

600questions

Decimal expansion of rational numbers

In Class 10 Mathematics, this topic from the Real Numbers chapter explains how rational numbers are represented in decimal form. Students learn to distinguish terminating decimals from non-terminating recurring decimals and determine the type of expansion by reducing a fraction to its simplest form and examining the prime factors of its denominator. The topic also develops accuracy in long division, fraction-to-decimal conversion, and understanding the relationship between rational numbers and their decimal representations.

609questions

7: Decimal expansion of rationa numbers

In this Class 10 Mathematics topic from the Real Numbers chapter, students explore how rational numbers are represented in decimal form. They learn to distinguish terminating decimals from non-terminating recurring decimals, convert fractions into decimals using division, and determine the type of expansion by examining the prime factors of a fraction’s denominator after reduction. The topic also connects decimal patterns with rational numbers and strengthens accuracy in calculations and mathematical reasoning.

1questions