Number System covers various types of numbers viz. natural numbers, whole numbers, integers, rational and irrational numbers which constitutes the Real number system.

**VARIOUS TYPES OF NUMBERS**

**1. Natural Numbers: **All the positive numbers 1, 2, 3 ,.., etc. that are used in counting are called Natural numbers. Types of Natural Numbers based on divisibility:

**Prime Number:**A natural number larger than unity is a prime number, if it does not have other divisors except for itself and unity.

Examples for Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, etc.

*Note:*** **Unity, i.e. 1 is not prime number.

**Composite Numbers:**Any number other than 1, which is not a prime number, is called a composite number.

Examples for composite numbers are 4, 6, 8, 9, 10, 14, 15, etc.

**2.** **Whole Numbers:** All counting numbers and 0, form the set of whole numbers.

For example: 0, 1, 2, 3…, etc. are whole numbers.

**3.** **Integers: **All counting numbers, Zero and negative of counting numbers, form the set of integers.

Therefore,…-3, -2, -1, 0, 1, 2, 3,.. are all integers.

**4.** **Rational Numbers**

A number which can be expressed in the form p/q, where p and q are integers and q≠0, is called a rational number.

For example 2 can be written as 2/1, therefore 2 is also a rational number.

**5. Irrational Numbers**

Numbers which are not rational but can be represented by points on the number line called irrational numbers.

**6. ****Even and Odd Numbers**

Numbers divisible by 2 are called even numbers whereas numbers that are not divisible by 2 are called odd numbers.

For example 2, 4, 6, etc are even numbers and 3, 5, 7, etc are odd numbers.

The following rules related to Even and Odd numbers are important:

**Read & Understand Practical Application of Number System**

**Factors and Co-Primes**

**Factorization Method for LCM and HCF**

- LCM: Here we can write all the given numbers in their prime factorization format.

For example: 15 = 3 X 5

18 = 3^{2} X 2

24 = 2^{3} X 3

Now take all primes number the given numbers and write their maximum powers. So LCM of 15, 18, 24 =2^{3} X 3^{2 }X 5 = 360

- HCF: 12 = 2
^{2}X 3

18 = 3^{2} X 2

Now, HCF of 12 and 18 = 3 X 2 = 6

**Take Practice Test for Number System – Factors**** **

**TESTS OF DIVISIBILITY OF NUMBERS**

*Divisible by 2:**Divisibility by 3:**Divisibility by 9:*When the sum of its digits is divisible by 9.*Divisibility by 4:*if the sum of its last two digits is divisible by 4.*Divisibility by 8:*If the number formed by hundred’s ten’s and unit’s digit of the given number is divisible by 8.*Divisibility by 10:*When its unit digit is Zero.*Divisibility by 5:*When its unit digit is Zero or five.*Divisibility by 11:*if the difference between the sum of its digits at odd places and the sum of its digits at even places is either o0 or a number divisible by 11.

**Take Practice Test for Remainders Concept in Number System**

**ALGEBRAIC FORMULAE**

- (a + b)
^{2}= a^{2}+ 2ab+ b^{2} - (a - b)
^{2}= a^{2}- 2ab+ b^{2} - (a + b)
^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a - b)
^{3}= a^{3}- b^{3}- 3ab(a - b) - (a + b + c)
^{2}= a^{2}+ b2 + c^{2}+2ab+2bc +2ca - (a + b + c)
^{3}= a^{3}+ b^{3}+ c^{3}+ 3a^{2}b + 3a^{2}c + 3b^{2}c + 3b^{2}a + 3c^{2}a + 3c^{2}b + 6abc - a
^{2}- b^{2}= (a + b)(a – b) - a
^{3}– b^{3}= (a – b) (a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b) (a^{2}- ab + b^{2}) - (a + b)
^{2}+ (a - b)^{2}= 4ab - (a + b)
^{2}- (a - b)^{2}= 2(a^{2}+ b^{2}) - If a + b +c =0, then a
^{3}+ b^{3}+ c^{3}= 3abc

**LAWS OF INDICES**

**BINOMIAL THEOREM**

**Middle Term of Binomial Theorem:**

**Read Practical Problems on Binomial Theorem**

**Take Practice Test for Number System Set 1**

*We hope that these concepts along with their associated practice test would help you in your preparation of Number System chapter. *

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