Representation Of Z Modulus On Argand Plane

Jean-Robert Argand was the first scientist to discover that complex numbers could be used in geometry. He said that complex numbers are similar to the coordinates of a point in the plane. If we take coordinates of any point, say (4,9) and we take a complex number as 4+9i, then the information that will be provided by the point (4,9), the same information will be provided by the complex number 4+9i. Coordinates of a point are separated by comma (,) and the coordinates of a complex number are separated by iota (i). 

Argand insisted on the use of complex numbers in geometry as we use vectors therein. This plane on which coordinates of a complex number are represented is called Argand Plane or Complex Plane in the context of Complex numbers. Every point on the Argand Plane/Complex Plane is a complex number, and every complex number on this plane represents the coordinates of one such complex number on this plane. There are infinite points; hence there are infinitely complex numbers. Corresponding to every point on the complex plane, there would be a corresponding complex number.

Geometrical representation of a complex number

If we take a point P on the Argand Plane/Complex Plane as shown below, we can say that Point P (a,b) can be written as ‘P=a+bi’ in the form of a Complex number. Further the information that we can derive from point P below is that it is at a distance ‘a’ from the y-axis and at a distance ‘b’ from the x-axis. The coordinates of point P are (a,b). The coordinates of point P can be written as a+bi in the form of a complex number. Or, we can also say that the corresponding complex number to point P is a+bi. 

In the Argand plane, the X-axis is called the real axis as complex numbers found on this axis are all real and the y-axis is termed here as Imaginary Axis as complex numbers found on this axis are all imaginary complex numbers. The common point of the real axis and imaginary axis is(0,0) point of origin(O) having properties of both real and imaginary axis. Therefore, we can conclude that Geometrically a+bi will represent the coordinates(a,b) of a given point.

Derivation of modulus of a complex number ’Z’

For a given complex number, Z=a+bi, the modulus of Z is represented by |z|.

The distance of any given complex number Z from its point of Origin(O) is called the modulus of given Complex number|z|.

|z| is a real number.

Let’s take a point P=a+bi on the Argand plane.

Coordinates of point P are (a,b)

Coordinates of Origin (O) will be (0,0)

Using Distance Formula, we can find out the distance between the two points P and O, as below:

= √ (a-0)2 + (b-0)2

= √ a2 + b2

Hence |z|=√ a2 + b2

Example

Find the modulus of -1-√3i

Solution:

a=-1

b=-√3

|z|=√ a2 + b2

|z|=√ (-1)2 +(-√3) 2

= 2

Representation of modulus of complex number |z| on the argand plane

|z| is the distance between the origin (O) and the complex number ’Z’.

Properties of modulus of a complex number

1. z.ˉz = |z|2
2.     |z|= |ˉz|
3. |z1*z2|= |z1|* |z2|

Same property is true for n complex numbers.

4. |z1*z2*z3*…………..zn|= |z1|* |z2|*|z3|*…………………………….. |zn|
5. |z1+z2|= |z1|+ |z2|, this is not true in general
6. |z1/z2|= |z1|/ |z2|, where z2(denominator) should not be equal to zero.
7. |zn|= |z|n

Difference between Cartesian plane and complex plane

A geometrical plane on which a coordinate system has been specified is called a coordinate plane. Coordinates (a,b) for any given point, say P, on a coordinate plane are called Cartesian coordinates. The horizontal axis is called the x-axis and the vertical axis is called the y-axis. X-coordinate is called abscissa and y-coordinate is called ordinate. 

The same coordinate plane is used to draw complex numbers, but when we draw complex numbers on a coordinate plane then that plane is called Argand Plane or Complex Plane or Z-Plane. So we can conclude that it is the plotting of numbers, whether ordered pairs of real numbers or complex numbers, which gives a normal xy plane its specific name as Cartesian Plane or Argand Plane.

Conclusion

The plane which gives the position of a complex number, is called the Argand Plane or Complex Plane in mathematics. Plotting a complex number on the Argand plane is the same as the plotting of any given ordered pair of real numbers on the Cartesian plane (also known commonly as the xy plane). 

This article discussed, in detail, the meaning of the Argand plane (also known as complex plane) and the representation of modulus of a complex number on the Argand plane. Further, we discussed the properties of the modulus of a complex number. We hope this study material would be of great value for anybody seeking basic concept clarity on the topics covered here.