Complex Numbers and Complex Algebra: Polar form of Complex Numbers
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Thus, for a given non zero complex number $z$, its modulus $r$ and argument $\theta$ can be computed so that $z$ can be represented as $(r, \; \theta)$. This representation is called the polar representation of $z$, and the values of $r$ and $\theta$ are called polar coordinates of $z$. Writing $z = x+iy$ as MATH is known as the trigonometric form of the complex number $z$.

For the complex number $z = 0$, the modulus is $0$, but the argument is undefined .

If a complex number $z$ is written in the polar form or in the trigonometric form then it is understood that it is a non-zero complex number.

For each non-zero $z$, there is only one value of $\arg z$ say $\Theta$ satisfying MATH. This value will henceforth be denoted by MATH and is called the principal value of $\arg z$. We can establish the relation between $\arg z$ and MATH:

   
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