Question

Fill in the blanks: The reciprocal of a negative rational number is ______. \[\]

Answer 1

Hint: We recall the definition of rational number and reciprocal of a rational number. We take two positive integers $ a,b $ and assume the reciprocal of negative rational number $ \dfrac{-a}{b} $ as $ x $ . We solve for $ x $ in the equation $ \dfrac{-a}{b}\times x=1 $ and check whether $ x $ is positive or negative. \[\]

Complete step by step answer:
We know that a rational number is always in the form $ \dfrac{a}{b} $ where $ a $ , an integer is called the numerator and $ b $ a non-zero integer, is called the denominator. A rational number is negative when either numerator is negative or the denominator is negative. \[\]

We know that multiplicative identity of rational number set is 1 as we can multiply any rational number with 1 and get the same number. The multiplicative inverse otherwise known as reciprocal of a rational number say $ \dfrac{a}{b} $ is a rational number with whom by multiplying we get the multiplicative identity 1. Let us assume both $ a,b $ as both positive intgers then $ \dfrac{-a}{b} $ is a negative rational number. Let $ x $ be the multiplicative inverse of $ \dfrac{-a}{b} $ . So we have
\[\begin{align}
  & \dfrac{-a}{b}\times x=1 \\
 & \Rightarrow -ax=b\left( \text{cross-multiplication} \right) \\
 & \Rightarrow x=\dfrac{b}{-a}(\text{by dividing }-a\text{ both side}) \\
\end{align}\]
So the reciprocal of $ \dfrac{-a}{b} $ is $ \dfrac{b}{-a} $ which is a negative rational number because both $ a,b $ are positive integers. So we need to fill ‘negative’ to fill the blank. \[\]

Note:
We note that we cannot find the reciprocal of 0. Similarly, we can prove that the reciprocal of a positive rational number is positive. We can directly conclude the reciprocal of a negative rational number is negative using the property that if we can get the product as a positive number when the two multiplying numbers are both positive and both negative. Since the product of $ \dfrac{-a}{b},x $ is 1, a positive number, the unknown $ x $ has to be negative.