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Convert the given complex number in polar form: $1-i$

seo-qna
Last updated date: 13th Jun 2024
Total views: 402k
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Answer
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Hint: In order to solve this question, we must have a basic knowledge of imaginary number, complex numbers, the Euler form of a complex number and how we can convert a given complex number in Euler form.

Complete step-by-step answer:
Definition of imaginary numbers: The numbers which give a negative number upon squaring it, is termed as imaginary number. In other words, the square root of a negative number gives an imaginary number as a result.
Definition of complex number: Complex numbers are the combination of real and imaginary numbers. It is generally denoted as: $a+ib$ , where the term before the addition symbol is called the real part of the number and the term after the addition symbol is called the imaginary part.
Euler form: Any complex number can be written in the Euler form. The Euler form of a complex number is also known as the polar form. It can be written as:
$a+ib=r{{e}^{i\theta }}$ ,
where $a$ is the real part of the complex number, $ib$ is the imaginary part of the complex number, $r$ is the norm of the complex number and $\theta $ is called the argument of the complex number.
$r$ is given by the following formula:
$r=\sqrt{{{(a)}^{2}}+{{(b)}^{2}}}$ .
$\theta $ is given by the following formula:
$\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$ .
Therefore, using these properties of a complex number, we can solve the given question.
Hence, the norm of the given complex number is:
$r=\sqrt{{{1}^{2}}+{{1}^{2}}}=\sqrt{2}$ .
And the argument of the complex number is:
$\theta ={{\tan }^{-1}}\left( \dfrac{-1}{1} \right)=-\dfrac{\pi }{4}$
Therefore, $1-i=\sqrt{2}{{e}^{-\dfrac{\pi }{4}i}}$.

Note: While calculating the argument of the complex, we must be very careful regarding the location of the complex number, i.e. the quadrant in which the complex number lies. Many times, we tend to make the mistake of finding the argument, and in actual the location might be in some other quadrant. Therefore, we must take care of the sign of the imaginary and the real parts of the complex number.