Question

Find the principal arguments of the following complex number
\[5 + 5i\].

Answer 1

Hint: Before solving this question, understanding the concept of complex number should be a must and remember the general representation of complex number i.e. $z = a + ib$to compare the given complex number using this information find the principal argument.

Complete step-by-step solution -
According to the given information we know we have complex number i.e. \[5 + 5i\]
As we know that the complex number is represented as $z = a + ib$
By comparing the general equation of complex number by given complex number to find the value of a and b
We get a = 5 and b = 5
The principal argument of a complex number is found using the formula \[\theta = {\tan ^{ - 1}}\left( {\dfrac{b}{a}} \right)\]
Substituting the given values of a and b in the formula of principle argument
\[\theta = {\tan ^{ - 1}}\left( {\dfrac{5}{5}} \right)\]
$ \Rightarrow $\[\theta = {\tan ^{ - 1}}\left( 1 \right)\] (equation 1)
Since we know that $\tan \dfrac{\pi }{4} = 1$
So we can say that \[\dfrac{\pi }{4} = {\tan ^{ - 1}}\left( 1 \right)\]
Now substituting the value of \[{\tan ^{ - 1}}\left( 1 \right)\] in equation 1
\[\theta = \dfrac{\pi }{4}\]
Therefore the principal argument of complex number \[5 + 5i\] is $\dfrac{\pi }{4}$.

Note: The term complex number is a very large concept we know that complex number is expressed in form of $z = a + ib$ here a and b are real number where a represents the horizontal coordinate axis and b represents the vertical axis coordinates whereas the i named as imaginary number, the imaginary number (i) follow the equation ${i^2} = - 1$. Complex numbers are graphically represented in a complex plane which is a two dimensional coordinate plane in the XY plane. Here X axis represents the real value of the complex number and Y axis represents the imaginary value of the complex number.