PROPERTIES OF RATIONAL NUMBER

PROPERTIES OF RATIONAL NUMBER

Closure Property

1.     Addition : When two rational numbers are added, their sum is always a rational number.

For example,

                                    5/6 + 4/5 = 49/30,

Which is also a rational number.

Therefore, rational numbers are closed under addition. It means for any two rational numbers a and b, a + b is also a rational number.

2.     Subtraction: When two rational numbers are subtracted, the result is always a rational number.

 For example,

                                                      3/4 – 2/3 = 1/12,

Which is also a rational number.

Therefore, rational numbers are closed under subtraction. It means for any two rational numbers, a and b, a-b is also a rational number.

3.      Multiplication: When two rational numbers are multiplied, their product is always a rational number. For example

4/11 × 3/7 = 12/77

Which is also a rational number.

Therefore, rational number are closed under multiplication. It means for any two rational number a and b a × b is also a rational number.

4.      Division: As for any rational number a, a ÷ 0 is not defined, therefore not all rational numbers are closed under division. We can say that except zero, all rational numbers are closed under division. Take a look at some examples of division of rational numbers.

5/7 ÷ 3/8 = 5/7 × 8/3 =40/21,

Which is a rational number.

-4/5 ÷ -6/7 = -4/5 × 7/-6 = 14/15,

which is a rational number.

Note: For any rational number a/b, b/a is called its reciprocal.

Commutative

1.      Addition: Addition is commutative for a rational numbers. In general, for any two rational numbers a and b,

a + b = b + a

The following examples prove the commutativity of addition for rational numbers.

                                    3/7 + 5/7 = 8/7

and       5/7 + 3/7 = 8/7

-4/9 + -7/9 = -11/9

                         and       -7/9 + -4/9 = -11/9

2.      Multiplication: Multiplication is also commutative for rational numbers. In general, for any two rational numbers a and b,

a × b = b × a

The following examples prove the commutativity of multiplication for rational numbers.

                        2/7 × 5/9 = 10/63               and                       5/9 × 2/7 = 10/63

                        -3/5 × -8/11 =24/55           and                      -8/11 × -3/5 = 24/55

Subtraction: Subtraction is not commutative for rational numbers. In general, for any tow rational numbers a and b,

           

                                                a-b ≠ b-a

Look at the following example showing that subtraction of rational numbers is not commutative.

5/6 – 2/3 = 1/6 but 2/3 -5/6 = -1/6

4.Division: Division is not commutative for rational numbers. In general, for any rational numbers a and b,

a ÷ b ≠ b ÷a

Look at the following example showing that division of rational numbers is not commutative.

8/11 ÷ 4/5 = 10/11 but 4/5 ÷ 8/11 = 11/10