Complex Numbers and Complex Algebra: Polar form of Complex Numbers
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Polar form of complex numbers: Each point $P = (x, \; y)$ of the complex plane, other than the origin, determines a directed segment of geometric vector $\vec{OP}$ (called its position vector), with origin $O$ and endpoint $P$, and conversely. The point $(0, \; 0)$ will be associated with the null vector $\vec{OO}$.

Let MATH be the point corresponding to the non-zero complex number $z = x+iy$ in the plane. let $\vec{OP^{\prime}}$ be the projection of $\vec{OP}$ upon the coordinate axis $OX$ and MATH be the projection of $\vec{OP}$ upon the coordinate axis $OY$. Denoting the signed measures of $\vec{OP^{\prime}}$ and MATH, by MATH and MATH, we have MATH and MATH. (That is, if $P=(-2,\;3)$, then MATH and MATH). The length $r$ of the vector $\vec{OP}$ is given by MATH. The measure $\theta$ in radians of the oriented angle from the positive real axis to the vector $\vec{OP}$ is called the argument or the amplitude of the vector $\vec{OP}$, and we write $\theta = \arg z$.

Figure 3



From trigonometry we have, MATH and MATH. Thus, the number $\theta$ is determined only up to multiples of $2 \pi$ and the set of all such angles is denoted by $\arg z$. However all the values in this set represent the same direction in the complex plane. For example, an argument of the point is MATH and all the values of argument of $i$ are given byMATH where $k$ is any integer.

   
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