Closure property with reference to Rational Numbers

Closure property states that if for any two numbers a and b, a∗b is also a rational number, then the set of rational numbers is closed under addition.

∗ represents +,−,× or ÷

For eg:-

2

1

and

4

3

2

1

+

4

3

=

2×4

1×4+3×2

=

8

4+6

=

8

10

=

4

5

is a rational number

2

1

−

4

3

=

2×4

1×4−3×2

=

8

4−6

=

8

−2

=

4

−1

is a rational number

2

1

×

4

3

=

2×4

1×3

=

8

3

is a rational number

4

3

2

1

=

2×3

1×4

=

1×3

1×2

=

3

2

is a rational number

Hence, set of rational number is closed under +,−,× and ÷.

DEFINITION

Square Root of Negative Real Numbers

We know that i

2

=−1 and (−i)

2

=i

2

=−1.

Hence, square roots of −1 are i,−i.

But, by symbol

−1

we simply mean i.

Similarly, (−

3

i)

2

=(

3

)

2

i

2

=−3

and (

3

i)

2

=(

3

)

2

i

2

=−3

∴ For any negative real number, −a, the square roots are

a

i and −

a

i

For example, square roots of −5 are

5

i and −

5

i

DEFINITION

Explain closure property and apply it in reference to irrational numbers

Closure property says that a set of numbers is closed under a certain operation if when that operation is performed on numbers from the set, we will get another number from the same set.

where operations are +,−,× or ÷

EXAMPLE

Explain closure property and apply it in reference to irrational numbers

For example:-

2

,−

2

and

3

are irrational numbers

For addition:-

Then,

2

+(−

2

)=

2

−

2

=0 is not an irrational number

For subtraction:-

3

−

3

=0 is not an irrational number

For multiplication:-

Take

3

and

27

as two irrational numbers

3

×

27

=

3×27

=

81

=9 is not an irrational number

For division:-

3

3

=1 is not an irrational number

Hence, the set of irrational numbers is not closed under for any of the operations.

DEFINITION

Definition of closure property

A set of numbers is said to be closed under an operation if and only if the operation on two elements of the set produces another element of the set, however if an element outside the set is obtained, then the set of numbers under that operation is not closed.

EXAMPLE

Irrational numbers are Not closed under multiplication

The product of two irrational numbers may be rational or irrational.

Example: 1)

(

2) is irrational,

2

∗

2

=2

2 is a rational number

2) (2

4

1

)×(2

4

1

) = 2

2

1

=

2

,so, here two irrational numbers multiply to give an irrational number.