Number System

We use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. 0 is called insignificant digit.

Natural numbers

Counting numbers 1, 2, 3, 4, 5. .. are known as natural numbers.
The set of all natural numbers can be represented by N= {1, 2, 3, 4, 5…}

Whole numbers

If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 … are called whole numbers.
The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5…}
Every natural number is a whole number but 0 is a whole number which is not a natural number.

Integers

All counting numbers and their negatives including zero are known as integers.
The set of integers can be represented by Z or I = {…-4, -3, -2, -1, 0, 1, 2, 3, 4 …}

Positive Integers

The set I+ = {1, 2, 3, 4…} is the set of all positive integers.
Positive integers and natural numbers are synonyms.

Negative Integers

The set I- = {-1, -2, -3…} is the set of all negative integers.
0 is neither positive nor negative.

Non-negative Integers

The set {0, 1, 2, 3…} is the set all non-negative integers.

Rational Numbers

The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers, e.g. 4/7, 3/2, -5/8, 0/1, -2/3, etc.
The set of all rational numbers is denoted by Q. i.e. Q ={x: x =p/q; p, q belong to I, q≠0}.
Since every natural number ‘a’ can be written as a/1, every natural number is a rational number.

Since 0 can be written as 0/1 and every non-zero integer ‘a’ can be written as a/1, every integer is a rational number.Every rational number has a peculiar characteristic that when expressed in decimal form is expressible rather in terminating decimals or in non-terminating repeating decimals.

For example, 1/5 =0.2, 1/3 = 0.333…22/7 = 3.1428704287, 8/44 = 0.181818…., etc.

Irrational Numbers

Those numbers which when expressed in decimal from are neither terminating nor repeating decimals are known as irrational number, e.g. √2, √3, √5, π, etc.
Note that the exact value of π is not 22/7. 22/7 is rational while π irrational number.
22/7 is approximate value of π. Similarly, 3.14 is not an exact value of it.

Real Numbers

The rational and irrational numbers combined together are called real numbers,
e.g.13/21, 2/5, -3/7; √3, 4 + √2, etc. are real numbers.
The set of all real numbers is denoted by R.
Note that the sum, difference or product of a rational and irrational number is irrational, e.g. 3+ √2, 4-√3, 2/3-√5, 4√3, -7√5 are all irrational.

Even Numbers

All those numbers

8, 10, etc., are even numbers.

Odd Numbers

All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1,
3, 5, 7 etc., are odd numbers.

Prime Numbers

A natural number other than 1 is a prime number if it is divisible by 1 and itself only.

For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers.

Composite Numbers

Natural numbers greater than 1which are not prime, are known as composite numbers.
For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers.