Class 10 — Mathematics (NCERT)

Real Numbers

Class 10

✓By the end of this lesson students will be able to state and apply Euclid's Division Lemma to find the HCF of two positive integers.
✓By the end of this lesson students will be able to state the Fundamental Theorem of Arithmetic and use it to find the HCF and LCM of positive integers.
✓By the end of this lesson students will be able to prove the irrationality of numbers like √2, √3, and √5.
✓By the end of this lesson students will be able to distinguish between rational and irrational numbers based on their decimal expansions.

Key concepts

Euclid's Division Lemma

For any two given positive integers 'a' and 'b', there exist unique whole numbers 'q' (quotient) and 'r' (remainder) such that a = bq + r, where the remainder 'r' satisfies the condition 0 ≤ r < b. This lemma is a statement used for proving other statements. The algorithm based on this lemma is called Euclid's Division Algorithm, which is used to find the HCF of two positive integers.

a = bq + r, where 0 ≤ r < b

Fundamental Theorem of Arithmetic

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. This theorem is also known as the Unique Factorisation Theorem. It implies that for any composite number, there is only one way to write it as a product of prime numbers, if the order of the primes is ignored.

N = p₁⁺¹ × p₂⁺² × ... × pₙ⁺ₙ (where N is a composite number, p₁, p₂, ..., pₙ are prime numbers, and a₁, a₂, ..., aₙ are positive integers)

Irrational Numbers

A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0. The decimal expansion of an irrational number is non-terminating and non-recurring. Examples include √2, √3, √5, π, etc. We will learn to prove that certain numbers are irrational using the method of proof by contradiction.

Key facts to remember

1Real Numbers comprise all rational and irrational numbers.
2Euclid's Division Lemma states that for positive integers a and b, a = bq + r, where 0 ≤ r < b.
3The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of primes, ignoring the order of factors.
4For any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b.
5Rational numbers have decimal expansions that are either terminating or non-terminating recurring.
6Irrational numbers have decimal expansions that are non-terminating and non-recurring.
7A prime number has exactly two factors: 1 and the number itself.
8A composite number has more than two factors.

Worked examples

Example 1

Use Euclid's Division Algorithm to find the HCF of 135 and 225.

ISince 225 > 135, we apply the division lemma to 225 and 135.

II225 = 135 × 1 + 90

IIISince the remainder 90 ≠ 0, we apply the division lemma to the divisor 135 and remainder 90.

IV135 = 90 × 1 + 45

VSince the remainder 45 ≠ 0, we apply the division lemma to the divisor 90 and remainder 45.

VI90 = 45 × 2 + 0

VIIThe remainder has now become zero. The divisor at this stage is 45.

Answer

The HCF of 135 and 225 is 45.

Euclid's Division Algorithm is a repetitive process of applying Euclid's Division Lemma until the remainder becomes zero. The last non-zero divisor is the HCF.

Example 2

Find the HCF and LCM of 6 and 20 by the prime factorisation method.

IFirst, find the prime factorisation of each number.

II6 = 2 × 3

III20 = 2 × 2 × 5 = 2² × 5

IVTo find HCF, we take the product of the smallest power of each common prime factor in the numbers.

VCommon prime factor is 2. Smallest power of 2 is 2¹.

VIHCF(6, 20) = 2¹ = 2

VIITo find LCM, we take the product of the greatest power of each prime factor involved in the numbers.

VIIIPrime factors involved are 2, 3, 5.

9Greatest power of 2 is 2².

10Greatest power of 3 is 3¹.

11Greatest power of 5 is 5¹.

12LCM(6, 20) = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

Answer

HCF(6, 20) = 2, LCM(6, 20) = 60.

Verify the relationship: HCF(a,b) × LCM(a,b) = a × b. Here, 2 × 60 = 120 and 6 × 20 = 120. L.H.S. = R.H.S.

Example 3

Prove that √2 is irrational.

ILet us assume, to the contrary, that √2 is rational.

IISo, we can write √2 = a/b, where a and b are coprime integers and b ≠ 0. (Coprime means they have no common factors other than 1).

IIISquaring both sides, we get 2 = a²/b², which implies a² = 2b². (Equation 1)

IVThis means a² is divisible by 2.

VBy Theorem 1.3 (from NCERT Class 10 textbook: If p is a prime number and p divides a², then p divides a), if 2 divides a², then 2 must divide a.

VISo, we can write a = 2c for some integer c.

VIISubstituting a = 2c in Equation 1, we get (2c)² = 2b².

VIII4c² = 2b²

9b² = 2c²

10This means b² is divisible by 2.

11Again, by Theorem 1.3, if 2 divides b², then 2 must divide b.

12So, we have shown that 2 divides both a and b.

13But this contradicts our initial assumption that a and b are coprime (i.e., they have no common factors other than 1).

14This contradiction arose because our initial assumption that √2 is rational was false.

Answer

Hence, √2 is irrational.

This method is called proof by contradiction. The key is to assume the opposite of what you want to prove and then show that this assumption leads to a contradiction.

Common mistakes

✗Confusing Euclid's Division Lemma (a statement) with Euclid's Division Algorithm (a procedure).
✗Not ensuring that 'a' and 'b' are coprime in the proof of irrationality.
✗Incorrectly applying the Fundamental Theorem of Arithmetic, especially when finding HCF and LCM (e.g., using greatest powers for HCF or smallest powers for LCM).
✗Forgetting the condition 0 ≤ r < b in Euclid's Division Lemma.
✗Making calculation errors during prime factorisation.

Exam tips

★Practice the steps of Euclid's Division Algorithm thoroughly to avoid errors in finding HCF.
★Clearly state the Fundamental Theorem of Arithmetic when using it in problems.
★Master the proof by contradiction for irrational numbers (e.g., √2, √3, √5) as it is a frequently asked question.
★Always show all intermediate steps clearly and logically in your solutions, especially for proofs.
★Remember the relationship HCF(a,b) × LCM(a,b) = a × b, as it can be used to verify answers or solve related problems.

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