Real Numbers

Real Numbers Notes : Formulas, Theorems & Examples (Class 10 Maths)

Ch 1 : REAL NUMBERS

Welcome to the world of Real Numbers! In this chapter, we’ll explore the fascinating system of numbers we use every day. From counting to complex calculations, Real Numbers are the foundation of mathematics. We’ll understand different types of real numbers and some important ideas about them.

SUMMARY OF TOPICS WE’LL COVER:

  • What are Real Numbers?
  • Rational Numbers and their Properties
  • Irrational Numbers and their Properties
  • The Fundamental Theorem of Arithmetic (and its applications)
  • Decimal Expansions of Rational Numbers (Terminating and Non-Terminating Repeating)
  • HCF and LCM of Numbers (including Rational Numbers)

1. RATIONAL NUMBERS:

  • DEFINITION: A rational number is any number that can be expressed in the form p/q, where:
    • ‘p’ and ‘q’ are INTEGERS (whole numbers and their negatives, including zero).
    • ‘q’ is NOT EQUAL TO ZERO (q ≠ 0).

Think about it: Why can’t ‘q’ be zero? Division by zero is undefined in mathematics!

  • EXAMPLES:
    • Fractions: 1/2, -2/3, 7/9, 15/4
    • Integers: 0, 1, -5, 10 (because any integer ‘n’ can be written as n/1)
    • Terminating Decimals: 0.5 (which is 1/2), 1.75 (which is 7/4), -0.2 (which is -1/5)
    • Repeating Decimals: 0.333… (which is 1/3), 1.666… (which is 5/3), 0.142857142857… (which is 1/7)

Key Point: Rational numbers, when expressed as decimals, are either TERMINATING (end after a certain number of digits) or NON-TERMINATING REPEATING (digits repeat in a pattern).

2. IRRATIONAL NUMBERS:

  • DEFINITION: Irrational numbers are numbers that CANNOT be written in the form p/q, where p and q are integers and q ≠ 0.
  • EXAMPLES:
    • Square roots of non-perfect squares: √2, √3, √5, √7, √15 (and many more!)
    • Pi (π): Approximately 3.14159…, but its decimal representation goes on forever without repeating.
    • Non-terminating, Non-repeating decimals: 0.10110111011110… (This pattern ensures it never repeats).

Why are numbers like √2 irrational? It’s a bit more complex to prove, but essentially, you can’t find two integers that, when you divide them, will exactly equal √2.

  • DECIMAL EXPANSION OF IRRATIONAL NUMBERS: Irrational numbers have decimal expansions that are NON-TERMINATING AND NON-REPEATING. They go on forever without any repeating pattern.

3. REAL NUMBERS:

  • DEFINITION: The collection of ALL RATIONAL NUMBERS AND ALL IRRATIONAL NUMBERS together makes up the set of Real Numbers.

Think of it like this: Real Numbers are the “big family” that includes both Rational and Irrational numbers.

  • REAL NUMBER LINE: All real numbers can be represented on a number line.

4. THE FUNDAMENTAL THEOREM OF ARITHMETIC:

  • STATEMENT: Every composite number (a whole number greater than 1 that is not prime) can be expressed (factorized) as a product of prime numbers. This factorization is UNIQUE, except for the order in which the prime factors occur.

Why is this “Fundamental”? It’s a cornerstone of number theory! It tells us that prime numbers are the building blocks of all composite numbers.

  • PRIME NUMBERS: Numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, 13…).
  • COMPOSITE NUMBERS: Numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9, 10, 12…).
  • EXAMPLES:
    • 12 = 2 × 2 × 3 = 22 × 3 (Prime factors are 2 and 3)
    • 30 = 2 × 3 × 5 (Prime factors are 2, 3, and 5)
    • 100 = 2 × 2 × 5 × 5 = 22 × 52 (Prime factors are 2 and 5)
  • APPLICATIONS: The Fundamental Theorem of Arithmetic is used to find:
    • HCF (Highest Common Factor)
    • LCM (Lowest Common Multiple)

5. REVISITING IRRATIONAL NUMBERS (THEOREMS):

  • THEOREM 1.3: Let ‘p’ be a prime number. If ‘p’ divides a2, then ‘p’ also divides ‘a’ (where ‘a’ is a positive integer).
    • Example: If 3 divides 62 (which is 36), then 3 also divides 6.
  • THEOREM 1.4: √2 is irrational. (Similarly, √3, √5, √7, etc., are also irrational).
    • This is a very important result! It confirms that numbers like √2 are indeed irrational and cannot be expressed as simple fractions.

6. REVISITING RATIONAL NUMBERS AND THEIR DECIMAL EXPANSIONS (THEOREMS):

  • THEOREM 1.5: Let ‘x’ be a rational number whose decimal expansion terminates. Then ‘x’ can be expressed in the form p/q, where p and q are co-prime (have no common factors other than 1), and the prime factorization of ‘q’ is of the form 2n 5m (where n and m are non-negative integers).
    • In simpler terms: If a rational number has a terminating decimal, its denominator (in simplest form) will only have prime factors of 2 and/or 5.
    • Example: 0.25 = 25/100 = 1/4. Denominator 4 = 22 (form 2n 5m where n=2, m=0).
  • THEOREM 1.6: Let x = p/q be a rational number such that the prime factorization of ‘q’ is of the form 2n 5m (where n, m are non-negative integers). Then ‘x’ has a decimal expansion which terminates.
    • Converse of Theorem 1.5: If the denominator (in simplest form) of a rational number only has prime factors of 2 and/or 5, then its decimal expansion will terminate.
    • Example: 7/8. Denominator 8 = 23. Decimal expansion is 0.875 (terminates).
  • THEOREM 1.7: Let x = p/q be a rational number where p and q are co-prime. If the prime factorization of ‘q’ is NOT of the form 2n 5m, then ‘x’ has a decimal expansion which is non-terminating repeating (recurring).
    • If the denominator has prime factors other than 2 and 5, the decimal will repeat.
    • Example: 1/3. Denominator 3 (not of the form 2n 5m). Decimal expansion is 0.333… (repeating).

7. HCF AND LCM OF NUMBERS:

  • FOR ANY TWO POSITIVE INTEGERS ‘A’ AND ‘B’:
  1. HCF (a, b): Product of the smallest power of each common prime factor in the numbers.
  2. LCM (a, b): Product of the greatest power of each common and uncommon prime factor in the numbers.
  3. Relationship: HCF (a, b) × LCM (a, b) = Product of the numbers (a × b)
    • Example: Let a = 12 (22 × 3) and b = 18 (2 × 32)
      • HCF (12, 18) = 21 × 31 = 6 (Common prime factors with smallest powers)
      • LCM (12, 18) = 22 × 32 = 36 (All prime factors with greatest powers)
      • HCF × LCM = 6 × 36 = 216. Product of numbers = 12 × 18 = 216. Relationship verified!
  • LCM AND HCF OF RATIONAL NUMBERS (FRACTIONS):
    • LCM of Rational numbers = (LCM of numerators) / (HCF of denominators)
    • HCF of Rational numbers = (HCF of numerators) / (LCM of denominators)
    • Example: Find LCM and HCF of 1/2 and 3/4
      • LCM = LCM(1, 3) / HCF(2, 4) = 3 / 2 = 1.5
      • HCF = HCF(1, 3) / LCM(2, 4) = 1 / 4 = 0.25

SUMMARY OF KEY TAKEAWAYS:

  • Real Numbers encompass both Rational and Irrational numbers.
  • Rational numbers can be written as p/q and have terminating or repeating decimals.
  • Irrational numbers cannot be written as p/q and have non-terminating, non-repeating decimals.
  • The Fundamental Theorem of Arithmetic is crucial for understanding prime factorization.
  • Theorems 1.5, 1.6, and 1.7 link the decimal expansion of a rational number to the prime factors of its denominator.
  • HCF and LCM are important tools for working with numbers, and there are specific formulas for rational numbers.