Question

Find multiplicative inverse of \[3 + 2i\].

Answer 1

Hint:
Here, we will use the concept of the multiplicative inverse of a complex number. Multiplicative inverse of a complex number is equal to its inverse. So, we will write its inverse and then we will rationalize it to get the required value of the multiplicative inverse.

Complete step by step solution:
We know that the multiplicative inverse of a complex number \[z\] is \[{z^{ - 1}}\].
Or we can write it as, multiplicative inverse of \[z\]\[ = \dfrac{1}{z}\]
Therefore, multiplicative inverse of \[3 + 2i\] \[ = \dfrac{1}{{3 + 2i}}\]
Now, we have to simplify the above equation by simply rationalizing it. Therefore, we get
\[ \Rightarrow \] Multiplicative inverse of \[3 + 2i\] \[ = \dfrac{1}{{3 + 2i}} \times \dfrac{{3 - 2i}}{{3 - 2i}}\]
\[ \Rightarrow \] Multiplicative inverse of \[3 + 2i\] \[ = \dfrac{{3 - 2i}}{{\left( {3 + 2i} \right)\left( {3 - 2i} \right)}}\]

Now we will use the algebraic identity \[\left( {a + b} \right) \times \left( {a - b} \right) = {a^2} - {b^2}\]. Therefore, we get
\[ \Rightarrow \] Multiplicative inverse of \[3 + 2i\] \[ = \dfrac{{3 - 2i}}{{{{\left( 3 \right)}^2} - {{\left( {2i} \right)}^2}}}\]
Applying the exponent on the terms, we get
\[ \Rightarrow \] Multiplicative inverse of \[3 + 2i\] \[ = \dfrac{{3 - 2i}}{{9 - 4{i^2}}}\]
Now we know that the value of \[{i^2} = - 1\]. So, by putting the value of the \[{i^2}\] in the above equation we will get
\[ \Rightarrow \] Multiplicative inverse of \[3 + 2i\] \[ = \dfrac{{3 - 2i}}{{9 - 4\left( { - 1} \right)}}\]
Adding the terms, we get
\[ \Rightarrow \] Multiplicative inverse of \[3 + 2i\]\[ = \dfrac{{3 - 2i}}{{9 + 4}} = \dfrac{{3 - 2i}}{{13}}\]
We can write it as, multiplicative inverse of \[3 + 2i\] \[ = \dfrac{3}{{13}} - \dfrac{{2i}}{{13}}\]

Hence, the multiplicative inverse of \[3 + 2i\] is \[\dfrac{3}{{13}} - \dfrac{{2i}}{{13}}\].

Note:
Alternate way of finding the multiplicative inverse of a complex number \[z\] is by using the direct formula of the multiplicative inverse of a complex number.
Multiplicative inverse of \[z\]\[ = {z^{ - 1}} = \dfrac{{\bar z}}{{{{\left| z \right|}^2}}}\] where, \[\bar z\] is the complex conjugate of a complex number and \[{\left| z \right|^2}\] is the magnitude of the complex number.
Let us take \[z = 3 + 2i\].
Therefore, \[\bar z = 3 - 2i\]
Magnitude of the complex number, \[{\left| z \right|^2} = {3^2} + {2^2}\]
Apply the exponent on the terms, we get
\[ \Rightarrow {\left| z \right|^2} = 9 + 4\]
Adding the terms, we get
\[ \Rightarrow {\left| z \right|^2} = 13\]
Now, by putting the values in the formula of multiplicative inverse, we get
Multiplicative inverse of \[3 + 2i\] \[ = \dfrac{{3 - 2i}}{{13}} = \dfrac{3}{{13}} - \dfrac{{2i}}{{13}}\]