# rational numbers examples | irrational numbers definition

In this article **rational numbers examples **we give the information about **type of number system in mathematics** like natural numbers, odd and even, **odd numbers definition** and **even number definition**.

**rational numbers examples**:

Who many cows, sheep’s, animals, horse’s…..etc….

All things answer in

**number system.**The numbers one, two, three, etc., That are used for counting, as well as their respective symbols, 1, 2, 3, 4, 5, ……

The type of number system :

**Natural numbers: **The group of numbers 1, 2, 3, 4, …. is called natural numbers.The Natural numbers denoted by N.

**N = { 1, 2, 3, 4, . . . . }**

These numbers increase in value as we go from left to right.

The smallest natural number is 1.

1 + 2 + 3 + 4 + ——- + n = n ( n + 1 ) / 2.

#### The natural numbers examples :

**Ex 1)** 1 + 2 + 3 + 4 + ——- + 10 = ?

the Answer : n ( n + 1 ) / 2 = 10 ( 11 ) / 2 = 55.

**Ex 2)**1 + 2 + 3 + 4 + ——- + 50 = ?

the Answer : 50 ( 51 ) / 2 = 1275.

**Ex 3)**11 + 12 + 13 + ——- + 30 = ?

The Answer : ( ( First number + Last number ) / 2 ) * Total natural numbers.

= ( ( 11 + 30 ) / 2 ) * 20

= 410.

**Whole numbers :**

The group of numbers 0, 1, 2, 3, 4, …….is called whole numbers.

The whole numbers are denoted by W.

**W = { 0, 1, 2,3, 4, ——– }**

These numbers increase in value as we go from left to right.

The smallest whole number is 0.

**Integers numbers :**

The group of numbers ——- , -3, -2, -1, 0, 1, 2, 3, —— is called integers numbers.

It is denoted by Z.

**Z = { ——, -3, -2, -1, 0, 1, 2, 3, ——– }**

**Comparing integers :**

- Of any two numbers on the number line, the one on the left is the smaller number.
- any negative integer is smaller than 0 and any positive integer is greater than 0.
- we cannot tell the biggest or the smallest integer.
- You cant give positive or negative sign to zero.

**rational numbers examples :**

If a is any integer and b is any non-zero integer, then a / b is called a rational number.

The Rational number is denoted by Q.

**For example :**

3/5, -6/7, 9, -5, 0, 4/3, —— these are rational numbers.

**Note :**

the integers are contained in the group of rational numbers.

**irrational numbers definition:**

the number whose decimal from is non-terminating but is not recurring are called irrational numbers.

It is denoted by I.

Note :

The square root of numbers that are not perfect squares are irrational numbers.

For example :

√2 = 1.414213……

√3 = 1.732050…….

etc……

**Real numbers :**

The collection of numbers we get by putting together rational and irrational numbers is called the collection of real numbers.

The Real numbers denoted by R.

**Note :**

- each number in this group is called a real number.
- all rational number is a real number.
- every irrational number is also a real number.

**the even number definition :**

If a number is divisible by 2 is known as even number.

**Or**

If a number has any of the digits 0, 2, 4, 6 or 8 in its units place, then the number is called even number.

**For example :**

10, 124, 86, 456 etc.

The Even number is denoted by E.

**E = { 2, 4, 6, 8, ——– }.**

**Ex 1)**To determine sum of first 5 even numbers.

0 + 2 + 4 + 6 + 8 + 10 + 12 = ?

the ans. = N ( N + 1 ) = 6 ( 7 ) = 42.

**Ex 2)**Determine sum of first 50 numbers.

0 + 2 + 4 + 6 + ———- + 100 = ?

the ans. = N ( N + 1 ) = 50 ( 51 ) = 2550.

**Ex 3)**12 + 14 + 16 + ——– + 42 = ?

Ans. = ( ( First number + Last number ) / 2) * Total number of even number.

= ( ( 12 + 42 ) / 2 ) * 16.

= (54 / 2) * 16

= 54 * 8

= (54 / 2) * 16

= 54 * 8

= 432.

**The odd numbers definition:**

If a number is not divisible by 2 is known as odd numbers.

**Or**

If a number has any of the digit’s 1, 3, 5, 7, or 9 in its units place, then that number is known as odd number.

For example :

345, 57, 9, 133, etc.

Odd numbers denoted by O.

**O = { 1, 3, 5, 7, ——— }.**

**Ex 1)**To determine sum of first 5 odd numbers.

1 + 3 + 5 + 7 + 9 + 11 = ?

the Ans.= N square = 6 × 6 = 36.

**Ex 2)**Determine sum of first 20 odd numbers.

1 + 3 + 5 + ——– + 39 = ?

the Ans. = N square = 20 × 20 = 400.

**Ex 3)**11 + 13 + 15 + —— + 49 = ?

the Ans. = ( ( First number + second number ) / 2 ) * Total number of odd numbers.

= ( ( 11 + 49 ) / 2 ) * 20.

= ( 60 / 2 ) * 20

= 30 * 20

= ( 60 / 2 ) * 20

= 30 * 20

= 600.

#### prime and composite numbers :

The numbers whose only divisors are 1 and the number itself are called prime numbers.

For example :

3, 7, 11, 19 etc.

The Prime numbers denoted by P.

P = { 2, 3, 5, 7, 11, 13, ——– }.

**Note:**

- The number 1 is not a prime number also not a composite number.
- The numbers 12 and 24 have more than two divisors. Numbers that have more than two divisors are called composite numbers.
- small prime number is 2.
- 2 is only even prime number.
- 25 prime numbers between 1 to 100.
- 15 prime numbers between 1 to 50.
- the sum of first 25 prime numbers is 1060.
- sum of first 15 prime numbers is 328.

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