CBSE Class 10 Maths Chapter 1 Real Numbers Notes

Key Concepts & Formulas

• Euclid's Division Lemma (EDL): For any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b.
• Euclid's Division Algorithm (EDA): A technique to compute the Highest Common Factor (HCF) of two given positive integers by repeatedly applying Euclid's Division Lemma.
• Fundamental Theorem of Arithmetic (FTA): 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.
• HCF and LCM using FTA: For two positive integers 'a' and 'b', HCF(a, b) × LCM(a, b) = a × b. Prime factorization is used to find HCF (common minimum powers) and LCM (all prime factors maximum powers).
• 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. Examples: √2, √3, π.
• Proof by Contradiction: The standard method to prove a number is irrational (e.g., √2). Assume it's rational, derive a contradiction, thus proving the assumption was false.
• Decimal Expansion of Rational Numbers: A rational number p/q (q ≠ 0, p and q coprime) has a terminating decimal expansion if the prime factorization of 'q' is of the form 2^m × 5^n, where m and n are non-negative integers. Otherwise, it has a non-terminating repeating decimal expansion.

Essential Definitions

Real Numbers

The set of all rational and irrational numbers. They can represent any point on the number line.

Rational Numbers

Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

Irrational Numbers

Real numbers that cannot be expressed as a simple fraction p/q. Their decimal expansions are non-terminating and non-repeating.

Prime Number

A natural number greater than 1 that has no positive divisors other than 1 and itself.

Composite Number

A natural number greater than 1 that is not a prime number (i.e., it has more than two factors).

Co-prime Numbers

Two integers are co-prime (or relatively prime) if their only common positive divisor is 1 (i.e., their HCF is 1).

Terminating Decimal

A decimal expansion that ends after a finite number of digits. Example: 0.25, 1.75.

Non-terminating Repeating Decimal

A decimal expansion that continues indefinitely with a repeating block of digits. Example: 0.333..., 0.142857142857...

Understanding the Fundamental Theorem of Arithmetic (FTA)

The Fundamental Theorem of Arithmetic is a cornerstone of number theory. It states that every composite number can be uniquely expressed as a product of prime numbers, regardless of the order in which these prime factors appear. For example, the number 12 can be factorised as 2 × 2 × 3, or 2² × 3. No other combination of prime numbers will multiply to give 12. This uniqueness is incredibly powerful.

The theorem implies that prime numbers are the 'building blocks' of all composite numbers. Just as atoms combine to form molecules, prime numbers combine through multiplication to form all other natural numbers greater than 1. This theorem is crucial for various applications, including finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers. To find the HCF, you take the product of the smallest power of each common prime factor involved in the numbers. For LCM, you take the product of the greatest power of each prime factor (common or not) involved in the numbers. The FTA also helps in determining whether a rational number will have a terminating or non-terminating repeating decimal expansion, by examining the prime factors of its denominator. Understanding this theorem deeply helps in solving many number theory problems efficiently.

Steps to Prove Irrationality (Example: √2)

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Illustrative Examples

• Example 1: Find HCF of 135 and 225 using Euclid's Division Algorithm. Step 1: 225 = 135 × 1 + 90 Step 2: 135 = 90 × 1 + 45 Step 3: 90 = 45 × 2 + 0 The remainder is 0. The divisor at this stage is 45. So, HCF(135, 225) = 45.
• Example 2: Check whether 6^n can end with the digit 0 for any natural number 'n'. For a number to end with the digit 0, its prime factorization must contain both 2 and 5. The prime factorization of 6^n is (2 × 3)^n = 2^n × 3^n. * Since the prime factorization of 6^n does not contain 5, by the Fundamental Theorem of Arithmetic (uniqueness of prime factorization), 6^n cannot end with the digit 0 for any natural number 'n'.
• Example 3: Without actual division, determine if 13/3125 has a terminating or non-terminating repeating decimal expansion. The given rational number is 13/3125. The denominator is q = 3125. Prime factorize the denominator: 3125 = 5 × 625 = 5 × 5 × 125 = 5 × 5 × 5 × 25 = 5 × 5 × 5 × 5 × 5 = 5^5. * Since the denominator is of the form 2^m × 5^n (here, 2^0 × 5^5), the decimal expansion of 13/3125 is terminating.

Exam Tip: Avoiding Common Pitfalls

When proving irrationality, always clearly state your initial assumption (e.g., "Let us assume, to the contrary, that √2 is rational..."). Ensure you mention that p and q are co-prime integers. Each step of the derivation to reach the contradiction must be logical and well-explained. For problems involving decimal expansions, remember to simplify the rational number to its simplest form (p/q where p and q are co-prime) before checking the prime factors of the denominator. Clearly state the theorem you are using (Euclid's Division Lemma, Fundamental Theorem of Arithmetic) to justify your steps, as this often carries marks.

Quick Revision Check

• Q: What is the HCF of two prime numbers? A: The HCF of two distinct prime numbers is always 1.
• Q: Can the prime factorization of a composite number be different if we ignore the order of factors? A: No, according to the Fundamental Theorem of Arithmetic, the prime factorization of a composite number is unique, regardless of the order of its prime factors.
• Q: Under what condition does a rational number p/q (q ≠ 0, p and q co-prime) have a terminating decimal expansion? A: It has a terminating decimal expansion if the prime factorization of the denominator 'q' is of the form 2^m × 5^n, where m and n are non-negative integers.
• Q: If HCF(a, b) = 5 and LCM(a, b) = 60, and a = 20, find b. A: We know HCF(a, b) × LCM(a, b) = a × b. So, 5 × 60 = 20 × b. This gives 300 = 20b, so b = 15.

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