Graphical Representation of complex numbers

 

Graphical Representation of complex numbers

Representation of a Complex Number

A complex number of the form z = a + ib can be represented on the argand plane by considering its coordinates as (Re(z), Im(z)) = (a, ib). An Argand plane or a complex plane is a Euclidean plane concerning complex numbers where the real part of a complex number “a” is represented on the X-axis and the imaginary part “ib” is represented on the Y-axis. The modulus of the complex number (r) is the distance of the complex number represented as a point in the argand plane (a, ib), i.e., the linear distance between the origin (0, 0) and the point (a, ib). 

r = √(a2 + b2)

The argument of the complex number is the angle in the anticlockwise direction made by the line joining the geometric representation of the complex number and the origin, with the positive x-axis.

Argz (θ) = tan−1(b/a)

Polar Form of a Complex Number

A complex number can also be represented and identified on the argand plane by using its polar form. To represent the complex number on the argand plane, the polar form makes use of the modulus and argument of the complex number. A complex number z = a + ib is expressed as z = r(cosθ + isinθ) in its polar form, where r is the modulus and θ is the argument of a complex number. Here, r is equal to √(a2 + b2), whereas θ is equal to tan-1(b/a).

Conjugate of a Complex Number

The conjugate of a complex number is another complex number that has the same real part as the original complex number, and the magnitude of the imaginary part is also the same but with the opposite sign. Two complex numbers are said to be each other’s conjugates if their sum and product are real numbers. 

• The conjugate of the complex number z = a + ib is z̅ = a − ib.

• Sum of a complex number and its conjugate = z + z̅ = (a + ib) + (a − ib) = 2a
• Product of a complex number and its conjugate = z × z̅ = (a + ib)×(a − ib)= a2+b2