Graph of a complex number and its conjugate
A complex number z = a + ib can be represented as a point on the Euclidean plane coordinates as (Re(z), Im(z)). The Euclidean plane that represents complex numbers as points where X and Y axes represent the real and imaginary parts of a complex number is called an argand plane or a complex plane. From the below graph, we can observe that the conjugate of a complex number is the reflection of a complex number about the real axis (X-axis).
- The polar form of a complex number z = a + ib is z = reix = r(cosθ + isinθ), where r is the modulus and θ is the argument of a complex number. The polar form of a complex number helps to represent and identify it on an argand plane. The polar form of the complex conjugate is z̅ = re−ix = r(cosθ − isinθ).
- The modulus of a complex number is defined as the distance of the complex number z = a + ib from the origin, in an argand plane. It is denoted by |z| and its value is r = √(a2 + b2).
- The angle made in the anticlockwise direction by the line that joins the origin and the point that represents a complex number with the positive X-axis is called the argument of the complex number.
Argz (θ) = tan−1(b/a)