JEE (Main) – Notes of Number System

• In the early days, there were no instruments while counting, people used to count the numbers with the help of stones, sticks, etc.
• These methods were not sufficient and had many limitations.
• We talk about the number of people coming to school, the number of modules taken per student, etc.
• Since our childhood, we have studied numbers (1,2,3,…), and arithmetic operations such as addition, subtraction, multiplication, and division of numbers.
• Numbers play a great role in various activities.

Number System:

The number system is a set of numbers in which one number is related to another number using one or more algebraic operations. There are different number systems which are as follows:

• Natural Numbers
• Whole Number
• Integer

1. Natural Numbers

• The natural (or counting) numbers are {1,2,3,4,5,…… N}.
• These numbers are also called Positive Numbers.
• In this case, counting starts from 1 and contains infinitely many natural numbers.
• Set of Natural numbers is represented by N = {1,2,3,4,5,…… }.
• Even Natural Number: Exactly divisible by 2 and infinitely many even numbers.
• Set of Even Natural numbers are {2,4,6,8,…… }.
• Odd Natural Number: Not exactly divisible by 2 and infinitely many odd numbers.
• Set of Odd Natural numbers are {1,3,5,7,…… }.

2. Whole Numbers

• The numbers which start from zero (0) are called whole numbers.
• In other words, whole numbers are the natural numbers together with 0.
• Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on).
• Set of whole numbers is represented by W = {1,2,3,4,5,…… }.

3. Integers

• Integers are the collection of whole numbers and negative numbers.
• The numbers. …… -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, + 7 …. etc.
• Integers do not include fraction numbers.
• Integers are represented by the symbol Z= {…… -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5,….}.
• The sum, product, and difference of any two integers is also an integer. But this is not true for the division.
• Positive Integer: Set of positive numbers is represented by Z⁺ = {1,2,3,4,5,…… }.
• Negative Integer: Set of negative numbers is represented by Z^– = {-1,-2,-3,-4,-5,…… }.
• Integers can be zero {0}.
• Non-Negative Integers = { 0, 1, 2, 3, 4, … } = W
• Non-Positive Integers = { 0, -1, -2, -3, -4, … }
• Even Numbers – n / 2 = 0 where n is counting number; [0,2,4,…]
• Odd Numbers – n / 2 ≠ 0 where n is counting number; [1,3,5,…]

4. Prime Number

• It is a whole number that is greater than 1.
• It is divisible only by itself or 1.
• Set of prime numbers is {2,3,5,7,11,13,17…… }.

5. Composite Number

• It is the opposite of Prime numbers.
• It can be divided exactly by a whole number other than itself.
• The number except 1 and which are not prime.
• A composite number is a number with more than 2 factors.
• Example: Factor 8 are = 1, 2, 4, 8

6. Co-Primes Numbers 

• Two natural numbers are co-primes if their H.C.F. is 1.
• Two consecutive natural numbers are always co-prime.
• Co-prime numbers are also known as relatively prime or mutually prime numbers.
• Example 1: For 21 and 22 :
  1. The factors of 21 are 1, 3, 7, and 21.
  2. The factors of 22 are 1, 2, 11, and 22.
  3. Here 21 and 22 have only one common factor that is 1. Hence, their HCF is 1 and is co-prime.

Recommended Reading

Trigonometric identities and equations

Reference

wikipedia