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.
0 Comments