Rational Numbers
Closure Properties Of Rational Numbers
Consider the two rational numbers as 5/6 and 1/4.
What would we get if we add these two rational numbers, i.e. what is the value of 5/6+1/4?
This means that the sum of two rational numbers 5/6 and 1/4 is a rational number. In other words, we can say that rational
numbers are closed under addition.
Is this true for all
rational numbers?
Yes. We can try for
different rational numbers and see that this property is true for all rational
numbers. Thus, we can say that the sum of two rational numbers is again a
rational number. In other words, we can say that rational numbers are
closed under addition. This property of rational numbers is known as
the closure property for rational numbers and it can be stated as follows.
“If a and b are any two rational
numbers and a + b = c,
then c will always be a rational number”.
|
Are rational numbers closed
under subtraction also?
Let us find out.
Consider two rational numbers -11/12 and 7/8
Thus, rational
numbers are closed under subtraction also.
Closure property of
rational numbers under subtraction can be stated as follows.
“If a and b are any two rational numbers and a − b = c, then c will
always be a rational number”.
|
Now, let us check whether rational numbers are closed under multiplication also. For this, consider two rational numbers 3/4 and -4/11.
Now 
, which is a rational
number.
, which is a rational
number.
Thus, rational
numbers are closed under multiplication also.
Closure property of
rational numbers under multiplication can be defined as follows.
“If a and b are any two rational
numbers, then a × b = c,
then c will always be a rational number”.
|
But rational numbers are not closed under division. If we consider the division of 2/5 by 0, then we will not obtain a rational number.
is
not a rational number because division of a rational number by zero is not
defined.
Thus, we can say that rational
numbers are not closed under division.
We can summarize the above
discussed facts as follows.
|
