Rational Numbers

Rational means anything which is completely logical whereas irrational means anything which is unpredictable and illogical in nature. The word rational has evolved from the word ratio . In general, rational numbers are those numbers that can be expressed in the form of , in which both p and q are integers and q≠0.We can denote these numbers by Q.

Properties of rational numbers

Closure property: For two rational numbers say x and y the results of addition, subtraction and multiplication operations gives a rational number. We can say that rational numbers are closed under addition, subtraction and multiplication. For example:

Rational Numbers

Do you know why division is not under closure property?

Division is not under closure property because division by zero is not defined. We can also say that except ‘0’ all numbers are closed under division. For example:

Division is not under closure property because division by zero is not defined. We can also say that except ‘0’ all numbers are closed under division. For example:

Rational Numbers

Commutative law: For rational numbers, addition and multiplication are commutative.

Commutative law of addition: a+b = b+a

Commutative law of multiplication: a×b = b×a

For example:

Rational Numbers

Rational Numbers

Subtraction is not a commutative property i. e. a-b ≠ b-a. This can be understood clearly with the following example

 Rational Numbers

whereas

 
Rational Numbers

Division is also not commutative i.e. a/b ≠ b/a as,

Rational Numbers 

whereas,

 Rational Numbers

Associative law: Rational numbers follow the associative property for addition and multiplication.

Suppose x, y and z are rational then for addition: x+(y+z)=(x+y)+z

For multiplication: x(yz)=(xy)z.

Some important properties that should be remembered are:

  • 0 is an additive identity and 1 is a multiplicative identity for rational numbers.
  • For a rational number  
    Rational Numbers
     the additive inverse of
     Rational Numbers

To learn more about other topics download Byju’s – The Learning App from Google Play Store and watch interactive videos. Also, take free tests to practice for exams.


Practise This Question

Rational numbers are not closed under which operation?