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# Find the modulus of the complex number $Z = 2 + 3i$.

Last updated date: 18th Mar 2023
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Hint:Let $Z = a + bi$ be a complex number. Then, the modulus of a complex number $Z$ , denoted by $\left| Z \right|$ , is defined to be the non-negative real number $\left| Z \right| = \sqrt {{{\left( {\operatorname{Re} \left( z \right)} \right)}^2} + {{\left( {\operatorname{Im} \left( z \right)} \right)}^2}} = \sqrt {{a^2} + {b^2}}$.

Given, complex number $Z = 2 + 3i$ .
Real part of complex number $Z$ is $\operatorname{Re} \left( Z \right) = a = 2$ .
Imaginary part of complex number $Z$ is $\operatorname{Im} \left( Z \right) = b = 3$ .
Now, we apply the formula of modulus of complex number $Z$ .
$\left| Z \right| = \sqrt {{{\left( {\operatorname{Re} \left( z \right)} \right)}^2} + {{\left( {\operatorname{Im} \left( z \right)} \right)}^2}} = \sqrt {{a^2} + {b^2}}$
$\Rightarrow \left| Z \right| = \sqrt {{{\left( {\operatorname{Re} \left( z \right)} \right)}^2} + {{\left( {\operatorname{Im} \left( z \right)} \right)}^2}} = \sqrt {{a^2} + {b^2}} \\ \Rightarrow \left| Z \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( 3 \right)}^2}} \\ \Rightarrow \left| Z \right| = \sqrt {4 + 9} \\ \Rightarrow \left| Z \right| = \sqrt {13} \\$
So, the modulus of complex number $Z = 2 + 3i$ is $\sqrt {13}$ .