# NCERT Class 10 Mathematics Chapter 1 Real Numbers: Summary, Concepts, and Revision Notes | SwaVid This chapter builds upon your understanding of numbers from previous classes, specifically focusing on Real Numbers. You will revisit properties of inte... Canonical: https://swavid.com/maths/class/10/chapter/real-numbers Source: https://swavid.com/maths/class/10/chapter/real-numbers # Real Numbers ## Concepts in this chapter ## The ideas this chapter keeps returning to ## Where this chapter usually trips people up ## Find the gaps before this chapter finds them for you. ## Related chapters ### Euclid's Division Lemma Chapter 1 · Class 10 Mathematics This chapter builds upon your understanding of numbers from previous classes, specifically focusing on Real Numbers. You will revisit properties of integers and rational numbers, and delve deeper into irrational numbers. The chapter introduces fundamental concepts like Euclid's Division Lemma and the Fundamental Theorem of Arithmetic, which are crucial for understanding number theory and have practical applications. Your study route Key topics Euclid's Division Lemma states that given any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 <= r < b. Here, 'a' is the dividend, 'b' is the divisor, 'q' is the quotient, and 'r' is the remainder. This lemma is a fundamental statement from which other results are derived. Example Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. (NCERT Example 1, modified for clarity) Watch out Confusing the lemma with the algorithm. The lemma is a statement, while the algorithm is a procedure or a series of well-defined steps. Euclid's Division Lemma states that given any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 <= r < b. Here, 'a' is the dividend, 'b' is the divisor, 'q' is the quotient, and 'r' is the remainder. This lemma is a fundamental statement from which other results are derived. Tap the card for an example Example Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. (NCERT Example 1, modified for clarity) Why it matters This lemma is the basis for the Euclid's Division Algorithm, which is used to find the HCF of two numbers. It also helps in proving various properties of integers, such as the forms of even/odd numbers or squares/cubes of integers. Watch out Confusing the lemma with the algorithm. The lemma is a statement, while the algorithm is a procedure or a series of well-defined steps. Ask at home Ask your child to explain the statement a = bq + r and what each variable represents. Provide two small numbers (e.g., 17 and 5) and ask them to find q and r using the lemma. Chapter 1, Real Numbers, introduces Euclid's Division Lemma, a foundational statement used to derive Euclid's Division Algorithm for finding the Highest Common Factor (HCF) of two positive integers. It then presents the Fundamental Theorem of Arithmetic, stating that every composite number can be uniquely expressed as a product of primes, irrespective of the order. This theorem is applied to find HCF and Least Common Multiple (LCM) and to prove the irrationality of numbers like sqrt(2) and sqrt(3) using proof by contradiction. Finally, the chapter revisits rational numbers and their decimal expansions, establishing the conditions under which a rational number will have a terminating or non-terminating repeating decimal expansion based on the prime factors of its denominator. This chapter solidifies the understanding of properties and classifications of real numbers. Chapter summary Chapter 1, Real Numbers, introduces Euclid's Division Lemma, a foundational statement used to derive Euclid's Division Algorithm for finding the Highest Common Factor (HCF) of two positive integers. It then presents the Fundamental Theorem of Arithmetic, stating that every composite number can be uniquely expressed as a product of primes, irrespective of the order. This theorem is applied to find HCF and Least Common Multiple (LCM) and to prove the irrationality of numbers like sqrt(2) and sqrt(3) using proof by contradiction. Finally, the chapter revisits rational numbers and their decimal expansions, establishing the conditions under which a rational number will have a terminating or non-terminating repeating decimal expansion based on the prime factors of its denominator. This chapter solidifies the understanding of properties and classifications of real numbers. What you should learn Keep these close Real numbers include both rational and irrational numbers. Euclid's Division Lemma: a = bq + r, where 0 <= r < b, for positive integers a and b. Euclid's Division Algorithm is a series of steps based on the lemma to find the HCF of two positive integers. The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of primes. To find HCF by prime factorization, take the product of the smallest power of each common prime factor. To find LCM by prime factorization, take the product of the highest power of each prime factor involved. For any two positive integers a and b, HCF(a, b) * LCM(a, b) = a * b. Proof by contradiction is a key technique used to prove the irrationality of numbers like sqrt(2), sqrt(3), etc. A rational number p/q (in simplest form) has a terminating decimal expansion if the prime factorization of q is of the form 2^n * 5^m, where n, m are non-negative integers. A rational number p/q (in simplest form) has a non-terminating repeating decimal expansion if the prime factorization of q is not of the form 2^n * 5^m. Irrational numbers are real numbers that cannot be expressed in the form p/q, where p and q are integers and q != 0. Examples of irrational numbers include sqrt(2), sqrt(3), pi, and numbers with non-terminating, non-repeating decimal expansions. Common confusions It is easy to think Euclid's Division Lemma and Euclid's Division Algorithm are the same thing. The clearer idea The Lemma is a fundamental statement (a = bq + r). The Algorithm is a step-by-step procedure that uses the Lemma repeatedly to achieve a specific goal, such as finding the HCF. It is easy to think The product of HCF and LCM of any set of numbers is equal to the product of the numbers. The clearer idea The relation HCF(a, b) * LCM(a, b) = a * b is strictly true only for two positive integers. It does not generally hold for three or more integers. It is easy to think All non-terminating decimal expansions are irrational. The clearer idea Non-terminating *repeating* decimal expansions represent rational numbers (e.g., 1/3 = 0.333...). Only non-terminating *non-repeating* decimal expansions represent irrational numbers (e.g., sqrt(2) = 1.4142135...). It is easy to think When checking for terminating decimals, only the denominator's prime factors matter, even if the fraction is not in simplest form. The clearer idea It is crucial to first simplify the rational number p/q to its simplest form (where p and q are coprime) before examining the prime factors of the denominator q. Otherwise, a common factor might mask the true nature of the decimal expansion. Before you push ahead Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns. Keep exploring Class 10 Source - Understand and apply Euclid's Division Lemma to solve problems related to properties of integers. - Use Euclid's Division Algorithm effectively to find the HCF of two positive integers. - State and apply the Fundamental Theorem of Arithmetic for prime factorization of composite numbers. - Determine HCF and LCM of positive integers using the prime factorization method. - Prove the irrationality of numbers such as sqrt(2), sqrt(3), and sqrt(5). - Distinguish between terminating and non-terminating repeating decimal expansions of rational numbers without actual division. - Understand and apply Euclid's Division Lemma to solve problems related to properties of integers. - Use Euclid's Division Algorithm effectively to find the HCF of two positive integers. - State and apply the Fundamental Theorem of Arithmetic for prime factorization of composite numbers. - Determine HCF and LCM of positive integers using the prime factorization method. - Prove the irrationality of numbers such as sqrt(2), sqrt(3), and sqrt(5). - Distinguish between terminating and non-terminating repeating decimal expansions of rational numbers without actual division. - NCERT Class 10 Mathematics textbook: Mathematics : Chapter 1: Real Numbers ## Key Links - [NCERT Mathematics](https://swavid.com/maths) - [Class 10](https://swavid.com/maths/class/10) - [Start with concepts](https://swavid.com/maths/class/10/chapter/real-numbers) - [Check a prerequisite](https://swavid.com/learning-debt-identifier) - [Concepts](https://swavid.com/maths/class/10/chapter/real-numbers) - [Practice](https://swavid.com/maths/class/10/chapter/real-numbers/practice-questions) - [NCERT Solutions](https://swavid.com/maths/class/10/chapter/real-numbers/ncert-solutions) - [Find the gaps before this chapter](https://swavid.com/learning-debt-identifier) - [Learn this chapter adapted to you](https://swavid.com/learning-style-test) - [Next chapter 2 . Polynomials](https://swavid.com/maths/class/10/chapter/polynomials) - [All Class 10 chapters](https://swavid.com/maths/class/10) - [2 Polynomials Introduction to Polynomials and their Degree · Geometrical Meaning of the Zeroes of a Polynomial Open chapter](https://swavid.com/maths/class/10/chapter/polynomials) - [3 Pair of Linear Equations in Two Variables Pair of Linear Equations in Two Variables · Graphical Method of Solving a Pair of Linear Equations Open chapter](https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables) - [4 Quadratic Equations Introduction to Quadratic Equations · Solving Quadratic Equations by Factorisation Open chapter](https://swavid.com/maths/class/10/chapter/quadratic-equations) - [NCERT Class 10 Mathematics textbook: Mathematics](https://ncert.nic.in/textbook.php)