Number System: Introduction of Numbers

(1) Natural numbers: The numbers starting from 1, 2, 3, and so on are counted as natural numbers. They are 1, 2, 3, 4,...

Exceptions: Zero, negative and decimal numbers are not counted in this list.

(2) Whole numbers: Zero and all other numbers are known as natural numbers. They are 0, 1, 2, 3, 4,...

(3) Integers: These are the numbers that include all the whole numbers and their negatives. These are –4, –3, –2, –1, 0, 1, 2, 3, 4,....

(4) Rational Numbers: All numbers which are terminating, repeating, and can be written in the form p/q, where p and q are integers and q should not be equal to 0 are termed as rational numbers.

Example: 0.12121212...

(5) Irrational Numbers: All the numbers which are non-terminating, non-repeating, and cannot be written in the form p/q, where p and q are integers and q should not be equal to 0 are termed as irrational numbers.

Example: pie, e

(6) Real numbers: All the numbers existing on the number line are real numbers. The group is made up of all rational and irrational numbers.

(7) Imaginary numbers: Imaginary numbers are the numbers formed by the product of real numbers and the imaginary unit 'i'.

This imaginary unit is defined as follows:

i2 = –1, multiplication of this 'i' is calculated according to the value given above. Example: 8i

(8) Complex number: The numbers formed by the combination of real numbers and imaginary numbers are called complex numbers. Every complex number is written in the following form:

A + iB, where A is the real part of the number, and B is the imaginary part.

(9) Prime numbers: All the numbers with only two divisors, 1 and the number itself, are called prime numbers. Hence, a prime number can be written as the product of the number itself and 1.

For example, 2, 3, 5, 7, etc.

(10) Composite numbers: All numbers which are not prime numbers are called composite numbers. This number has factors other than one and itself.

For example, 4, 10, 99, 105, 1782, etc.

(11) Even and odd numbers: All the numbers divided by 2 are even numbers. Whereas, the ones not divisible by 2 are odd numbers.

For example, 4, 6, 64, 100, 10004, etc., are all even numbers.

3, 7, 11, 91, 99, 1003 are all odd numbers.

(12) Relative prime numbers/ co-prime numbers: Numbers that do not have any common factor other than 1 are called co-prime numbers.

Example: 5 and 17 are co-primes.

(13) Perfect numbers: All the numbers are called perfect numbers if the sum of all the factors of that number, excluding the number itself and including 1, are equal to the number itself then the number is termed as a perfect number.

Example: 6 is a perfect number as the factors of 6 are 2 and 3.

As per the rule of perfect numbers, sum = 2 + 3 + 1 = 6. Hence, 6 is a perfect number.

Click Here to Watch Free Session on Last Lap to CAT 2022

Some important properties of numbers:

The number 1 is neither prime nor composite.
The only even prime number is 2.
All the prime numbers greater than 3 can be written in the form of (6k + 1) or (6k – 1), where k is an integer.
The square of every natural number can be written in the form 3n or (3n + 1) and 4n or (4n + 1).
The tens digit of every perfect square is even unless the square is ending in 6, in which case, the tens digit is odd.
The product of n consecutive natural numbers is always divisible by n! , where n! = 1 × 2 × 3 × 4 ×….× n (known as factorial n).