Real Numbers Chapter Notes for CBSE Class 10 Science

Key Points to Remember

• Real Numbers include all rational and irrational numbers.
• Fundamental Theorem of Arithmetic (FTA) states every composite number can be uniquely expressed as a product of primes, apart from the order of factors.
• Prime Factorization Method is used to find HCF (product of smallest powers of common prime factors) and LCM (product of highest powers of all prime factors).
• For any two positive integers 'a' and 'b', HCF(a, b) × LCM(a, b) = a × b.
• A number 'p' (prime) divides a² implies 'p' divides 'a' (where 'a' is a positive integer) is key for proving irrationality.
• Proof by Contradiction is the standard method to prove irrationality (e.g., √2, √3, √5).
• A rational number x = p/q (q ≠ 0, p and q coprime) has a terminating decimal expansion if the prime factorization of 'q' is of the form 2^m5ⁿ, where m and n are non-negative integers.
• If the prime factorization of 'q' is not of the form 2^m5ⁿ, the rational number x = p/q has a non-terminating repeating decimal expansion.

Key Definitions

Real Numbers

The set of all rational and irrational numbers. They can be represented 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. Their decimal expansions are either terminating or non-terminating repeating.

Irrational Numbers

Numbers that cannot be expressed in the form 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 (e.g., 2, 3, 5, 7).

Composite Number

A natural number greater than 1 that is not prime; it has at least one divisor other than 1 and itself (e.g., 4, 6, 8, 9).

HCF (Highest Common Factor)

The largest positive integer that divides two or more integers without leaving a remainder. Also known as GCD (Greatest Common Divisor).

LCM (Least Common Multiple)

The smallest positive integer that is a multiple of two or more integers.

Terminating Decimal

A decimal expansion that ends after a finite number of digits (e.g., 0.5, 0.25).

Non-terminating Repeating Decimal

A decimal expansion that continues infinitely but with a repeating block of digits (e.g., 0.333..., 0.142857142857...).

Understanding the Fundamental Theorem of Arithmetic

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 primes, apart from the order in which the prime factors occur. This means that no matter how you factorize a composite number, you will always end up with the same set of prime factors. For example, the number 12 can be written as 2 × 2 × 3 or 3 × 2 × 2, but the prime factors are always two 2s and one 3.

This theorem has practical applications, especially in finding the HCF (Highest Common Factor) and LCM (Least Common Multiple) of two or more numbers. To find the HCF, we list the prime factors common to all the given numbers and take the lowest power of each common prime factor. For instance, if you have 12 (2² × 3) and 18 (2 × 3²), the common prime factors are 2 and 3. The lowest power of 2 is 2¹ and of 3 is 3¹. So, HCF(12, 18) = 2¹ × 3¹ = 6.

To find the LCM, we take all the prime factors involved in any of the numbers (whether common or not) and raise them to their highest power. For 12 (2² × 3) and 18 (2 × 3²), the prime factors involved are 2 and 3. The highest power of 2 is 2² and of 3 is 3². So, LCM(12, 18) = 2² × 3² = 4 × 9 = 36. An important relationship to remember is that for any two positive integers 'a' and 'b', the product of their HCF and LCM is equal to the product of the numbers themselves: HCF(a, b) × LCM(a, b) = a × b. This relationship is a powerful tool for solving problems related to HCF and LCM.

Steps to Prove Irrationality (by Contradiction)

1. — Assume that the number (e.g., √2) is rational. This means it can be written in the form a/b, where a and b are coprime integers (they have no common factors other than 1) and b ≠ 0.
2. — Square both sides of the equation. For √2 = a/b, this becomes 2 = a²/b², leading to a² = 2b². This implies that a² is an even number (or a multiple of 2).
3. — If a² is even, then 'a' must also be even. (If 'a' were odd, a² would be odd). So, we can write a = 2c for some integer c.
4. — Substitute a = 2c back into the equation from Step 2: (2c)² = 2b² => 4c² = 2b² => 2c² = b². This implies that b² is also an even number (or a multiple of 2).
5. — If b² is even, then 'b' must also be even. Now we have that both 'a' and 'b' are even. This means they both have a common factor of 2. This contradicts our initial assumption in Step 1 that 'a' and 'b' are coprime (have no common factors other than 1).
6. — Since our assumption led to a contradiction, our initial assumption must be false. Therefore, √2 (or the number we started with) is irrational.

Worked Examples

• {"title":"HCF and LCM using Prime Factorization","description":"Find the HCF and LCM of 96 and 404.","solution":"1. Prime factorization:\n 96 = 2⁵ × 3¹\n 404 = 2² × 101¹\n2. HCF: Smallest power of common prime factors.\n Common prime factor is 2. Smallest power is 2². So, HCF(96, 404) = 2² = 4.\n3. LCM: Highest power of all prime factors.\n LCM(96, 404) = 2⁵ × 3¹ × 101¹ = 32 × 3 × 101 = 9696."}
• {"title":"Decimal Expansion Type","description":"Without actual division, state whether 13/3125 has a terminating or non-terminating repeating decimal expansion.","solution":"1. Factorize the denominator: 3125 = 5⁵.\n2. The denominator is of the form 2⁰ × 5⁵ (i.e., 2^m5ⁿ where m=0, n=5).\n3. Since the prime factorization of the denominator contains only powers of 5 (and implicitly 2⁰), the rational number 13/3125 has a terminating decimal expansion."}

Exam Tip for Real Numbers

Pay close attention to questions involving proofs of irrationality. These are common and require a clear, logical step-by-step presentation using the proof by contradiction method. Ensure you explicitly state your assumption, show the derivation leading to the contradiction, and clearly conclude why the original assumption must be false. For HCF/LCM problems, remember the relationship HCF × LCM = product of numbers as it can often simplify calculations or help verify your answer. Always reduce fractions to their simplest form (coprime numerator and denominator) before checking decimal expansion properties based on the denominator's prime factors.

Practice Questions with Solutions

• Q: State the Fundamental Theorem of Arithmetic. A: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
• Q: If HCF(a, b) = x and LCM(a, b) = y, what is the relation between a, b, x, and y? A: The relation is: a × b = HCF(a, b) × LCM(a, b), so a × b = x × y.
• Q: What is the condition for a rational number p/q (q ≠ 0, p and q coprime) to have a terminating decimal expansion? A: The prime factorization of the denominator 'q' must be of the form 2^m5ⁿ, where m and n are non-negative integers.
• Q: What method is generally used to prove that √5 is irrational? A: The method of Proof by Contradiction is generally used.

Frequently Asked Questions