Question

The numerical expression \[\dfrac{3}{8} + \dfrac{{( - 5)}}{7} = \dfrac{{ - 19}}{{56}}\] shows:
A) Rational numbers are closed under addition.
B) Rational numbers are not closed under addition.
C) Rational numbers are closed under multiplication.
D) Addition of Rational numbers is not commutative.

Answer 1

Hint: We will first notice the binary operation used in the given expression and eliminate the obvious options. Now, we will see if the binary operation on two rational numbers results in a rational number or not and then mark accordingly.

Complete step-by-step answer:
Let us first see what is being closed.
Let us say we have a set X and * be the binary operation on X.
Now if $a*b \in X\forall a,b \in X$, then X is said to be closed under *.
Now, let us see what do we mean by commutative.
If $a*b = b*a\forall a,b \in X$ , then X is said to be commutative for *.
Now, let us come back to our question.
We have: \[\dfrac{3}{8} + \dfrac{{( - 5)}}{7} = \dfrac{{ - 19}}{{56}}\] .
We see that addition is going on here. So, that means this has nothing to do with the multiplication.
Hence, (C) is discarded.
Since, in commutative property, we need to show that $a*b = b*a$.
But if we look at the given expression, we do not have two terms a and b on the RHS.
Hence, (D) is discarded as well.
Now, we just have to look whether it is closed under addition or not. What is it showing?
If we notice clearly, on the LHS of the expression, we have two rational numbers.
[Rational numbers are such real numbers which can we written in the form of \[\dfrac{p}{q}\], where
$p,q \in Q,q \ne 0$]
Now on the RHS, we also have a rational number which means addition of two rational numbers are resulting in a rational number which according to the definition of being closed matches perfectly.
Hence, this shows that Rational numbers are closed under addition.

Hence, the correct option is (A).

Note: The students must note that the correct option came out to be (A) that does not mean that the other options cannot be valid for the rational numbers. The point is just that the expression given to does not imply any other property.
In fact, if we generally take Rational numbers, they are closed under multiplication as well.