Question

Write the rational numbers that are equal to their reciprocals.

Answer 1

Hint: Take examples of rational numbers and find their reciprocals. Check if the rational number and reciprocal are equal. Thus find the numbers that are equal to their reciprocals.

Complete step-by-step answer:

A rational number is a number that can be expressed as the quotient or fraction \[{}^{p}/{}_{q}\]of two integers, where p and q are integers and \[q\ne 0.\] If \[q=1\], then every integer is a rational number.
So what we want is the reciprocal of rational numbers. Now for every non-zero rational number \[{}^{a}/{}_{b}\], there exists a rational number \[{}^{b}/{}_{a}\] such that,
\[\dfrac{a}{b}\times \dfrac{b}{a}=1=\dfrac{b}{a}\times \dfrac{a}{b}.\]
The rational number \[{}^{b}/{}_{a}\] is called multiplicative inverse or reciprocal of \[{}^{a}/{}_{b}\] and is denoted by \[{{\left( {}^{a}/{}_{b} \right)}^{-1}}.\]
Now let us consider a few examples of rational numbers and let us take their reciprocal. Check if the rational number and their reciprocal are some.
The reciprocal of 13 is \[{}^{1}/{}_{13}\Rightarrow \] not equal.
The reciprocal of \[{}^{1}/{}_{2}\] is \[2\Rightarrow \]not equal.
The reciprocal of \[{}^{3}/{}_{4}\] is \[{}^{4}/{}_{3}\Rightarrow \] not equal.
The reciprocal of \[{}^{16}/{}_{5}\] is \[{}^{5}/{}_{16}\Rightarrow \] not equal.
The reciprocal of -8 is \[{}^{-1}/{}_{8}\Rightarrow \] not equal.
The reciprocal of 1 is \[1\Rightarrow \] equal.
The reciprocal of -1 is \[-1\Rightarrow \] equal.
Thus from the above examples, we can make out that the reciprocal of 1 is 1 and the reciprocal of (-1) is (-1). 1 and -1 are the only rational numbers which are their own reciprocal. No other rational number is equal to its own reciprocal.
So the rational numbers that are equal to their reciprocals are 1 and -1.

Note: There is no rational number which when multiplied with zero, gives 1.
Therefore, rational number zero has no reciprocal or multiplicative inverse.
The product of rational numbers with its reciprocal is always equal to 1.
E.g.:- The reciprocal of \[{}^{15}/{}_{23}\] is \[{}^{23}/{}_{15}\]. Their product is \[{}^{15}/{}_{23}\times {}^{23}/{}_{15}=1.\]