Question

Find the product of additive inverse and multiplicative inverse of \[ - \dfrac{1}{3}\].

Answer 1

Hint: To find the product of additive inverse and multiplicative inverse of \[ - \dfrac{1}{3}\], we need to first find the additive inverse and multiplicative inverse. After finding, we need to multiply both to obtain the answer. To find the additive inverse of \[a\], we need to find a number \[b\] such that \[a + b = 0\]. And, to find the multiplicative inverse of \[x\], we need to find a number \[y\] such that \[x \times y = 1\].

Complete step-by-step solution:
We need to find the product of the additive inverse and multiplicative inverse of \[ - \dfrac{1}{3}\].
Let us first find the Additive Inverse.
Let \[x\] be the additive inverse of \[ - \dfrac{1}{3}\]. So, it should satisfy \[ - \dfrac{1}{3} + x = 0\] as \[b\] is said to be the inverse of \[a\] if and only if \[a + b = 0\].
Hence, \[ - \dfrac{1}{3} + x = 0\] which is equal to
\[ \Rightarrow x - \dfrac{1}{3} = 0\] as addition is commutative.
Now, reshuffling the terms, we get
\[ \Rightarrow x = \dfrac{1}{3}\]
Hence, the additive inverse of \[ - \dfrac{1}{3}\] is \[\dfrac{1}{3} - - - - - - (1)\]
Now, finding the Multiplicative inverse of \[ - \dfrac{1}{3}\].
Let \[y\] be the multiplicative inverse of \[ - \dfrac{1}{3}\]. So, it should satisfy \[ - \dfrac{1}{3} \times y = 1\] as \[m\] is said to be the multiplicative inverse of \[n\] if and only if \[m \times n = 1\].
Hence, \[ - \dfrac{1}{3} \times y = 1\]
Multiplying the numerators, we get
\[ \Rightarrow - \dfrac{y}{3} = 1\]
Reshuffling the terms, we have
\[ \Rightarrow - y = 1 \times 3\]
\[ \Rightarrow - y = 3\]
Now, multiplying with negative sign, we have
\[ \Rightarrow - \left( { - y} \right) = - \left( 3 \right)\]
Using \[ - \left( { - f\left( x \right)} \right) = f\left( x \right)\], we have
\[ \Rightarrow y = - 3\]
Hence, multiplicative inverse of \[ - \dfrac{1}{3}\] is \[ - 3 - - - - - - (2)\]
Now, we need to find the product of the additive inverse and multiplicative inverse of \[ - \dfrac{1}{3}\].
Multiplying additive inverse and multiplicative inverse i.e. multiplying (1) and (2), we get
\[ \Rightarrow \dfrac{1}{3} \times \left( { - 3} \right)\]
Multiplying the numerators, we get
\[ \Rightarrow \dfrac{{\left( { - 3} \right)}}{3}\]
Now cancelling out the terms, we get
\[ \Rightarrow \left( { - 1} \right)\]
Hence, the product of additive inverse and multiplicative inverse of \[ - \dfrac{1}{3}\] is \[\left( { - 1} \right)\].

Note: The conditions to be fulfilled for additive inverse and multiplicative inverse are both ways i.e. they are if and only if statements. Here \[0\] is called the additive identity and \[1\] is called the multiplicative identity. So, for multiplicative inverse we need to keep the product equal to multiplicative identity and for additive inverse, we need to keep the sum equal to additive identity.