Question

Zero $\left( 0 \right)$ is
A. The identity for addition of rational numbers.
B. The identity for subtraction of rational numbers.
C. The identity for multiplication of rational numbers.
D. The identity for division of rational numbers.

Answer 1

Hint: Here we are given an element of a set and we are asked to check whether the given element is an identity element for addition, subtraction, multiplication, or division of rational numbers. First, we need to know the definition of an identity element so that we are able to obtain the required answer for the given question.

Complete step by step solution:
Let us learn about the identity element first. When we combine an element of a set of numbers with another number of the same set under a particular binary operation, we will get the second number unchanged. That element is usually called an identity element.
A binary operation is an operation between two elements of a set to form a third element of the set. The most often used binary operations are addition, subtraction, multiplication, and division.
Here we are given zero from a set. We are asked to find whether zero is an identity for addition, subtraction, multiplication, and division.
A) Let us check whether zero is the identity for the addition of rational numbers.
Let us choose $\dfrac{p}{q}$ from a set of rational numbers.
Now, we add zero with$\dfrac{p}{q}$.
$0 + \dfrac{p}{q} = \dfrac{p}{q} + 0 = \dfrac{p}{q}$
Therefore zero is the identity element for the addition of rational numbers.
B) Let us check whether zero is the identity for the subtraction of rational numbers.
Let us choose $\dfrac{p}{q}$ from a set of rational numbers.
Now, we subtract zero and$\dfrac{p}{q}$.
\[0 - \dfrac{p}{q} = - \dfrac{p}{q}\]
Here we didn’t get the same element. That is the second element that has changed.
Therefore zero is not the identity element for the subtraction of rational numbers.
C) Let us check whether zero is the identity for the multiplication of rational numbers.
Let us choose $\dfrac{p}{q}$ from a set of rational numbers.
Now, we multiply zero and$\dfrac{p}{q}$.
$0 \times \dfrac{p}{q} = \dfrac{p}{q} \times 0 = 0$
Here we didn’t get the same element. That is the second element that has changed.
Therefore zero is not the identity element for the multiplication of rational numbers.
D) Let us check whether zero is the identity for the division of rational numbers.
Let us choose $\dfrac{p}{q}$ from a set of rational numbers.
Now, we divide zero and $\dfrac{p}{q}$.
$0 \div \dfrac{p}{q} = \dfrac{p}{q} \div 0 \ne \dfrac{p}{q}$
Here we didn’t get the same element. That is the second element that has changed.
Therefore zero is not the identity element for the division of rational numbers.
Hence option A is correct.

Note: If we are given an element $1$ from a set and we are asked to find whether element one is the identity for addition, subtraction, multiplication, and division of rational numbers. Element one is the identity for the multiplication of rational numbers. Since we get $1 \times \dfrac{p}{q} = \dfrac{p}{q} \times 1 = \dfrac{p}{q}$ , $1$ is the identity for multiplication of rational numbers.