A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Rational numbers are terminating decimals but irrational numbers are non-terminating.​

The properties of rational numbers are:

• Closure Property

• Commutative Property

• Associative Property

• Distributive Property

• Identity Property

• Inverse Property

Closure Property

​When any two rational numbers are added, subtracted, or multiplied the result of all three cases will also be a rational number.

Why division is not under closure property?

The 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.​

• ​Commutative Property: x + y = y + x

• Associative Property: x + (y+z) = (x+y) + z

• Distributive Property: x.(y+z) = xz + xz

• Identity Property: 0 is an additive identity and 1 is a multiplicative identity for rational numbers.

  Examples:

  • 1/2 + 0 = 1/2    [Additive Identity]

  • 1/2 x 1 = 1/2   [Multiplicative Identity]​

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• Inverse Property: For a rational number x/y, the additive inverse is -x/y and y/x is the multiplicative inverse.

  Examples:

  The additive inverse of 1/3 is -1/3. Hence, 1/3 + (-1/3) = 0

  The multiplicative inverse of 1/3 is 3. Hence, 1/3 x 3 = 1

​Problem 1: Let a be a rational number other than 0, b an irrational number. Prove that ab is irrational.

​Suppose that a and b are natural numbers and sqrt(a)+sqrt(b) is rational number. Show that both sqrt(a) and sqrt(b) are integers.

​Suppose that a and b are natural numbers and sqrt(a)+sqrt(b)+sqrt(c) is rational number. Show that all sqrt(a), sqrt(b) and sqrt(c) are integers.