Argand Plane
The complex number $x+iy$ which corresponds to the ordered pair (x,y) can be represented geometrically as the unique point P(x,y) in the XY-plane and vice-versa.
Some complex numbers such as $2+i4$, $-2+i3$, $0+i1$, $2+i0$, $-5-i2$, and $1-i2$ which corresponds to the ordered pairs (2,4), (-2,3), (0,1), (2,0), (-5,-2) and (1,-2) respectively, have been represented geometrically by the points A, B, C, D, E and F respectively.
The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane.
If the point P(x,y) represent the complex number $z=x+iy$, then in Argand plane, the modulus of the complex number $|z|=\sqrt{x^{2}+y^{2}}$ is the distance between the point P(x,y) and the origin O(0,0).
Note that every real number $x+i0$ is represented by the point (x,0) lying on x-axis is called the real axis, and every purely imaginary number $0+iy$ is represented by the point (0,y) lying on y-axis is called the imaginary axis.
The representation of a complex number $z=x+iy$ and its conjugate $z=x-iy$ in the Argand plane are respectively, the points P(x,y) and Q(x,-y).
Geometrically, the point Q is the mirror image of the point P on the real axis.
Polar representation of a complex number
Let the point P(x,y) represent the non-zero complex number $z=x+iy$ in the Argand plane. Let the directed line segment OP be the length $r$ and $\theta$ be the radian measure of the angle which OP makes with the positive direction of x-axis.
Then $r=\sqrt{x^{2}+y^{2}}=|z|$ and is called modulus of $z$ and $\theta$ is called the amplitude or argument of $z$ and is written as amp($z$) or arg($z$).
We know that, $x=r\cos \theta$ and $y=r\sin \theta$
∴ $z=x+iy$ $=r\cos \theta +ir\sin \theta$ $=r(\cos \theta +i\sin \theta )$
Thus, $z=r(\cos \theta +i\sin \theta )$ is called the polar form of the complex number $z$.
For any non-zero complex number $z=x+iy$, there corresponds only one value of $\theta$ in $-\pi < \theta \leq \pi$. The unique value of $\theta$ such that $-\pi < \theta \leq \pi$ is called principal value of amplitude or argument.
Thus every complex number $z=x+iy$ can be uniquely expressed as $z=r(\cos \theta +i\sin \theta )$ where $r>0$ and $-\pi < \theta \leq \pi$ and conversely, for every $r>0$ and $\theta$ such that $-\pi < \theta \leq \pi$, we get a unique complex number $z=r(\cos \theta +i\sin \theta )$$=x+iy$.
The complex number $x+iy$ which corresponds to the ordered pair (x,y) can be represented geometrically as the unique point P(x,y) in the XY-plane and vice-versa.
Some complex numbers such as $2+i4$, $-2+i3$, $0+i1$, $2+i0$, $-5-i2$, and $1-i2$ which corresponds to the ordered pairs (2,4), (-2,3), (0,1), (2,0), (-5,-2) and (1,-2) respectively, have been represented geometrically by the points A, B, C, D, E and F respectively.
The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane.
If the point P(x,y) represent the complex number $z=x+iy$, then in Argand plane, the modulus of the complex number $|z|=\sqrt{x^{2}+y^{2}}$ is the distance between the point P(x,y) and the origin O(0,0).
The representation of a complex number $z=x+iy$ and its conjugate $z=x-iy$ in the Argand plane are respectively, the points P(x,y) and Q(x,-y).
Geometrically, the point Q is the mirror image of the point P on the real axis.
Let the point P(x,y) represent the non-zero complex number $z=x+iy$ in the Argand plane. Let the directed line segment OP be the length $r$ and $\theta$ be the radian measure of the angle which OP makes with the positive direction of x-axis.
We know that, $x=r\cos \theta$ and $y=r\sin \theta$
∴ $z=x+iy$ $=r\cos \theta +ir\sin \theta$ $=r(\cos \theta +i\sin \theta )$
Thus, $z=r(\cos \theta +i\sin \theta )$ is called the polar form of the complex number $z$.
For any non-zero complex number $z=x+iy$, there corresponds only one value of $\theta$ in $-\pi < \theta \leq \pi$. The unique value of $\theta$ such that $-\pi < \theta \leq \pi$ is called principal value of amplitude or argument.
Thus every complex number $z=x+iy$ can be uniquely expressed as $z=r(\cos \theta +i\sin \theta )$ where $r>0$ and $-\pi < \theta \leq \pi$ and conversely, for every $r>0$ and $\theta$ such that $-\pi < \theta \leq \pi$, we get a unique complex number $z=r(\cos \theta +i\sin \theta )$$=x+iy$.
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