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
= 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
= 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.

Santosh Nalawade

Work as Assistant Professor and Web Developer.

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