Real Numbers
Real numbers include all the rational and irrational numbers. They are used to represent distances, measurements, and quantities in real-world problems. In this chapter, we will study important properties and theorems related to real numbers.
1. Euclid’s Division Lemma
Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
a = bq + r, where 0 ≤ r < b
This lemma is the foundation for finding the HCF (Highest Common Factor) of two numbers using the Euclidean Algorithm.
2. The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every composite number can be expressed (or factorized) as a product of primes, and this factorization is unique, except for the order of the factors. For example:
60 = 2 × 2 × 3 × 5
This theorem is useful in understanding divisibility, finding LCM, and HCF of numbers.
3. Irrational Numbers
Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. They have non-terminating and non-repeating decimal expansions. Examples of irrational numbers include:
- √2, √3, √5, etc.
- π (pi) and e (Euler’s number)
4. Decimal Expansion of Real Numbers
The decimal expansion of real numbers can be classified into two types:
- Terminating: When the decimal expansion ends after a finite number of digits (e.g., 0.25, 0.5).
- Non-terminating: When the decimal expansion continues indefinitely.
Non-terminating decimals can be further classified as:
- Repeating (Rational): The decimal repeats in a pattern (e.g., 0.666… = 2/3).
- Non-repeating (Irrational): The decimal has no repeating pattern (e.g., π).
5. Rationalizing the Denominator
Rationalizing the denominator involves converting an expression to remove any irrational number from the denominator. For example:
To rationalize 1/√2, multiply both the numerator and denominator by √2: (1/√2) × (√2/√2) = √2/2
