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.