Question

Question: One (1) 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 first we know that a rational number are numbers which can be expressed in the form \[\dfrac{p}{q}\] , where \[p\] and \[q\] integers and \[q \ne 0\] ,are called rational numbers. We have to find the element say \[a\] such that \[a * b = b * a = b\] for all values contained in a given domain and also \[a\] is one of elements in the given domain. Then we say that \[a\] is the identity element of the given domain.

Complete step-by-step answer:
A set of rational numbers are denoted by \[\mathbb{Q}\] .

Option A:
Let \[\dfrac{c}{d}\] any non-zero rational number where \[d \ne 0\] . let \[a\] be any number such that
 \[a + \dfrac{c}{d} = \dfrac{c}{d}\]
 \[ \Rightarrow a = 0\] for all \[\dfrac{c}{d} \in \mathbb{Q}\] and \[0 \in \mathbb{Q}\] .
  \[ \Rightarrow \] \[0\] is the identity element for addition of rational numbers. Hence option A is incorrect.

Option B:
Let \[\dfrac{c}{d}\] any non-zero rational number where \[d \ne 0\] . let \[a\] be any number such that
 \[\left| {a - \dfrac{c}{d}} \right| = \left| {\dfrac{c}{d} - a} \right| = \dfrac{c}{d}\] .
 \[ \Rightarrow a = 0\] for all \[\dfrac{c}{d} \in \mathbb{Q}\] and \[0 \in \mathbb{Q}\] .
  \[ \Rightarrow \] \[0\] is the identity element for subtraction of rational numbers under mod function. Hence option B is incorrect.

Option C:
Let \[\dfrac{c}{d}\] any non-zero rational number where \[d \ne 0\] . let \[a\] be any number such that
 \[a \times \dfrac{c}{d} = \dfrac{c}{d}\]
 \[ \Rightarrow a = 1\] for all \[\dfrac{c}{d} \in \mathbb{Q}\]
  \[ \Rightarrow \] \[1\] is the identity element for multiplication of rational numbers. Hence option C is correct.

Option D:
Let \[\dfrac{c}{d}\] any non-zero rational number where \[d \ne 0\] . Assume \[1\] be the identity element for division of rational numbers. Then
 \[\dfrac{1}{{\dfrac{c}{d}}} = \dfrac{c}{d}\] is must true for all \[\dfrac{c}{d} \in \mathbb{Q}\] but \[\dfrac{d}{c} \ne \dfrac{c}{d}\] for all \[\dfrac{c}{d} \in \mathbb{Q}\]
Example \[\dfrac{2}{5}\] be a real number such that \[\dfrac{2}{5} \ne \dfrac{5}{2}\] .
  \[ \Rightarrow \] \[1\] is not the identity element for division of rational numbers. Hence option D is incorrect.
So, the correct answer is “Option C”.

Note: Note that any real number is said to be a rational number if it has finite decimal expansion or infinite repeating decimal expansion. A non-repeating infinite decimal expansion of a given real number is called the irrational number. One \[\left( 1 \right)\] is the multiplicative identity element of a set of real and complex numbers.