**R = Real Numbers:**

All rational and irrational numbers are called real numbers.

**I = Integers:**

All numbers from (…-3, -2, -1, 0, 1, 2, 3…) are called integers.

**Q = Rational Numbers:**

Real numbers of the form , q ≠ 0, p, q ∈ I are rational numbers.

- All integers can be expressed as rational, for example, 5 =
- Decimal expansion of rational numbers terminating or non-terminating recurring.

**Q’ = Irrational Numbers:**

Real numbers which cannot be expressed in the form and whose decimal expansions are non-terminating and non-recurring.

- Roots of primes like √2, √3, √5 etc. are irrational

**N = Natural Numbers:**

Counting numbers are called natural numbers. N = {1, 2, 3, …}

**W = Whole Numbers:**

Zero along with all natural numbers are together called whole numbers. {0, 1, 2, 3,…}

**Even Numbers:**

Natural numbers of the form 2n are called even numbers. (2, 4, 6, …}

**Odd Numbers:**

Natural numbers of the form 2n -1 are called odd numbers. {1, 3, 5, …}

- Why can’t we write the form as 2n+1?

**Remember this!**

- All Natural Numbers are whole numbers.
- All Whole Numbers are Integers.
- All Integers are Rational Numbers.
- All Rational Numbers are Real Numbers.

**Prime Numbers:**

The natural numbers greater than 1 which are divisible by 1 and the number itself are called prime numbers, Prime numbers have two factors i.e., 1 and the number itself For example, 2, 3, 5, 7 & 11 etc.

- 1 is not a prime number as it has only one factor.

**Composite Numbers:**

The natural numbers which are divisible by 1, itself and any other number or numbers are called composite numbers. For example, 4, 6, 8, 9, 10 etc.

Note: 1 is neither prime nor a composite number.

**I. Euclid’s Division lemma**

Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r ≤ b.

Notice this. Each time ‘r’ is less than b. Each ‘q’ and ‘r’ is unique.

**II. Application of lemma**

Euclid’s Division lemma is used to find HCF of two positive integers. Example: Find HCF of 56 and 72 ?

Steps:

- Apply lemma to 56 and 72.
- Take bigger number and locate ‘b’ and ‘r’. 72 = 56 × 1 + 16
- Since 16 ≠ 0, consider 56 as the new dividend and 16 as the new divisor. 56 = 16 × 3 + 8
- Again, 8 ≠ 0, consider 16 as new dividend and 8 as new divisor. 16 = 8 × 2 + 0

**Since remainder is zero, divisor (8) is HCF.**

Although Euclid’s Division lemma is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0.

**III. Constructing a factor tree**

Steps

- Write the number as a product of prime number and a composite number

Example:

Factorize 48 - Repeat the process till all the primes are obtained

∴ Prime factorization of 48 = 2^{4}x 3

**IV. Fundamental theorem of Arithmetic**

Every composite number can be expressed as a product of primes, and this expression is unique, apart from the order in which they appear.

**Applications:**

- To locate HCF and LCM of two or more positive integers.
- To prove irrationality of numbers.
- To determine the nature of the decimal expansion of rational numbers.

**1. Algorithm to locate HCF and LCM of two or more positive integers:**

**Step I:**

Factorize each of the given positive integers and express them as a product of powers of primes in ascending order of magnitude of primes.

**Step II:**

To find HCF, identify common prime factor and find the least powers and multiply them to get HCF.

**Step III:**

To find LCM, find the greatest exponent and then multiply them to get the LCM.

**2. To prove Irrationality of numbers:**

- The sum or difference of a rational and an irrational number is irrational.
- The product or quotient of a non-zero rational number and an irrational number is irrational.

**3. To determine the nature of the decimal expansion of rational numbers:**

- Let x = p/q, p and q are co-primes, be a rational number whose decimal expansion terminates. Then the prime factorization of’q’ is of the form 2
^{m}5^{n}, m and n are non-negative integers. - Let x = p/q be a rational number such that the prime factorization of ‘q’ is not of the form 2
^{m}5^{n}, ‘m’ and ‘n’ being non-negative integers, then x has a non-terminating repeating decimal expansion.

**Alert!**

- 2
^{3}can be written as: 2^{3}= 2^{3}5^{0} - 5
^{2}can be written as: 5^{2}= 2^{0}5^{2}