# CHAPTER 1 : Real Numbers

## Revision Checklist

- Introduction
- Euclid's Division Lemma
- Theorem 1.1 :
**Euclid's Division Lemma**

*Given positive integers a and b, there exists unique integers q and r satisfying*

a = bq + r, 0 ≤ r ≤ b. - Definitions:

*Algorithm : An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.*

*Lemma : A lemma is a proven statement used for proving another statement.* **Euclid's Division Algorithm**- Step 1.

Apply Euclid's division lemma, to*c*and*d*. So, we find whole numbers,*q*and*r*such that*c = dq + r, 0 ≤ r < d*. - Step 2.

If*r = 0*,*d*is the HCF of*c*and*d*. If*r ≠ 0*, apply the division lemma to*d*and*r*. - Step 3.

Continue the process till the remainder is zero. The divisor at this stage will be the required HCF. - Exercise 1.1
- The Fundamental Theorem of Arithmetic
- Theorem 1.2 :
**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.*

OR

The prime factorisation of a natural number is unique, except for the order of its factors. - For any two positive integers
*a*and*b*,*HCF (a, b) × LCM (a, b) = a × b*. - Exercise 1.2
- Revisiting Irrational Numbers
- Definitions:

*Irrational Number : 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, √2⁄3, 0.1011011101110…, etc. - Theorem 1.3 :
*Let***p**be a prime number. If**p**divides**a**, then^{2}**p**divides**a**, where**a**is a positive integer.

Check out the explanation and proof in the linked video. - Theorem 1.4 :
**√2**is irrational.

Check out the explanation and proof in the linked video. - The sum or difference of a rational and an irrational number is irrational.
- The product and quotient of a non-zero rational and irrational number is irrational.
- Exercise 1.3
- Revisiting Rational Numbers and Their Decimal Expansions
- 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 coprime, and the prime factorisation of**q**is of the form**2**, where^{n}5^{m}**n**,**m**are non-negative integers. - Theorem 1.6 :
*Let***x = p⁄q**be a rational number, such that the prime factorisation of**q**is of the form**2**, where^{n}5^{m}**n**,**m**are non-negative integers. Then**x**has a decimal expansion which terminates. - Theorem 1.7 :
*Let***x = p⁄q**be a rational number, such that the prime factorisation of**q**is not of the form**2**, where^{n}5^{m}**n**,**m**are non-negative integers. Then**x**has a decimal expansion which is non-terminating repeating (recurring). - The decimal expansion of every rational number is either terminating or non-terminating repeating
- Exercise 1.4