Chapter 1 Real Numbers

 

Chapter 1

                                                                                      Real Numbers 

Natural Numbers

Non-negative counting numbers excluding zero are known as natural numbers.

i.e. 5, 6, 7, 8,  ……….

Whole numbers

All non-negative counting numbers including zero are known as whole numbers.

i.e. 0, 1, 2, 3, 4, 5, …………….

Integers

All negative and non-negative numbers including zero altogether known as integers.

 i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..

The Fundamental Theorem of Arithmetic

We can factorize each composite number as a product of some prime numbers and of course, this prime factorization of a natural number is unique as the order of the prime factors doesn’t matter.

• HCF of given numbers is the highest common factor among all which is also known as GCD i.e. greatest common divisor.

• LCM of given numbers is their least common multiple.

• If we have two positive integers  ‘m’ and ‘n’ then the property of their  HCF and LCM will be:

HCF (m, n) × LCM (m, n) = m × n.

Rational Numbers

The number ‘s’  is known as a rational number if we can write it in the form of m/n where  ‘m' and ‘n’ are integers and n ≠ 0, 2/3, 3/5 etc.

Rational numbers can be written in decimal form also which could be either terminating or non-terminating. E.g. 5/2 = 2.5 (terminating) and(non-terminating)

Irrational Numbers

The number ‘s’ is called irrational if it cannot be written in the form of m/n, where m and n are integers and n≠0 or in the simplest form, the numbers which are not rational are called irrational numbers. Example - √2, √3 etc.

• If p is a prime number and p divides a2 , then p is one of the prime factors of a2 which divides a, where a is a positive integer.

• If p is a positive number and not a perfect square, then √n is definitely an irrational number.

• If p is a prime number, then √p is also an irrational number.