## Argand Plane

We all know that the pair of numbers (x, y) can be represented on the XY-plane, where x is called abscissa and y is called the ordinate. Similarly, we can represent complex numbers also on a plane called Argand plane or complex plane. Similar to the X-axis and Y-axis in two-dimensional geometry, there are two axes in Argand plane.

- The axis which is horizontal is called the real axis
- The axis which is vertical is called the imaginary axis

The complex number x+iy which corresponds to the ordered pair(x, y)is represented geometrically as the unique point (x, y) in the XY-plane.

For example,

The complex number, 2+3i corresponds to the ordered pair (2, 3) geometrically.

Similarly, -3+2i corresponds to the ordered pair (-3, 2).

- Complex numbers in the form 0+ai, where “a” is any real number will lie on the imaginary axis.
- Complex numbers in the form a+0i, where “a” is any real number will lie on the real axis.

It is obvious that the modulus of complex number x+iy, \( \sqrt{x^2+y^2} \) is the distance between the origin (0, 0) and the point (x, y).

- The conjugate of z = x+iy is z = x-iy which is represented as (x, -y) in the Argand plane. Point (x, -y) is the mirror image of the point (x, y) across the real axis in the Argand plane.

Example: Find the distance between the complex number z = 3 – 4i and the origin in the Argand plane.

Distance between the origin and z= 3 – 4i is equal to the modulus of z.

|z|=\( \sqrt{3^2+(-4)^2 }\)= \( \sqrt{9+16} \) = \(\sqrt{25}\)=5 units

Polar representation of complex numbers:

Let “A” represent the non-zero complex number x + iy. OA is the directed line segment of length r and makes an angle Î¸ with the positive direction of X-axis.

Ordered pair (r, Î¸) is called as the polar coordinates of the point A, as the point “A” is uniquely determined by (r, Î¸). The origin is called the pole and the positive X-axis is called the initial line.

Then,

x = r cosÎ¸

y = r sinÎ¸

We can write z = x + iy as z = r cosÎ¸ + ir sinÎ¸ = r(cosÎ¸ + i sinÎ¸), which is called the polar form of complex number.

- Here, r = |z| = \( \sqrt{x^2+y^2}\) is modulus of z and Î¸ is known as the argument or amplitude of z denoted as arg z
- For any non-zero complex number z, there corresponds one value of Î¸, in the interval [0, 2Ï€)
- In any other interval of length 2Ï€, for example, consider the interval -Ï€ < Î¸ â‰¤ Ï€, then the value of Î¸ is called the principal argument of z.

Example: Represent z = \( \sqrt{3} \) + i in the polar form

\( \sqrt{3} \) = r cosÎ¸

1=r sinÎ¸

r = |z| = \( \sqrt{3 + 1} \) = 2

sin Î¸ = \( \frac 12 \)

cos Î¸ = \( \frac {\sqrt{3}}{2}\)

Which gives,

Î¸ = \( \frac {\pi}{6} \)

Therefore, polar form of z is,

z = \( 2~(cos~\left(\frac {\pi}{6} \right)~+~i~sin~\left(\frac{\pi}{6}\right)) \)

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