Representation of a complex number in Argand plane
We already know that corresponding to each ordered pair of real numbers (x, y), we get a unique point in the XYplane and vice-versa with reference to a set of mutually perpendicular lines known as the x-axis and the y-axis. 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 + 4i, – 2 + 3i, 0 + 1i, 2 + 0i, – 5 –2i and 1 – 2i which correspond 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 in the Fig 5.1.
The plane having a complex number assigned to each of its point is called the complex plane or theArgand plane.
Obviously, in the Argand plane, the modulus of the complex number x + iy = √(x2 + y2) is the distance between the point P(x, y) to the origin O (0, 0) (Fig 5.2). The points on the x-axis corresponds to the complex numbers of the form a + i 0 and the points on the y-axis corresponds to the complex numbers of the form 0 + i b. The x-axis and y-axis in the Argand plane are called, respectively, the real axis and 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 (x, – y) is the mirror image of the point (x, y) on the real axis (Fig 5.3).