Geometric Interpretation and Modulus

Geometric Interpretation

$2+3i,3- 2i,- 3+i,- 2- 3i$

Solution:

The complex plane is also called "The Argand Diagram" after the French- Swiss mathematician Jean Robert Argand (1768- 1822) who was the first to represent a complex number geometrically as point in the plane.

We observe that the complex numbers appear in all the four quadrants due to the negative and positive signs of their real and imaginary parts.

If $x = 0$ , then $x + y i = y i$ is an imaginary number located on vertical axis. If $y = 0$ , then $x + y i = x$ is a real number located on horizontal axis.

Absolute Value or Modulus of a Complex Number

Let $z = (a,b) = a + i b$ be a complex number.

The absolute value or modulus of $z$ , denoted by $|z|$ is defined as:

which is a real number. From the diagram, it is clear that the modulus of a complex number is the distance OP of complex number from the origin.

If, $z = a + i b$ , then $\mathrm{\mid }z\mathrm{\mid }=\sqrt{(\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}{)}^2+(\mathrm{i}\mathrm{m}\mathrm{g}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}{)}^2}=\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}$$|z| = \sqrt{(\mathrm{real~part})^{2} + (\mathrm{img~part})^{2}} = \mathrm{Real~Number}$

​=Real Number∣z∣=(real part)2+(img part)2

​=Real Number

Example: Find the absolute value of the complex numbers:

a. 5i

b. 2

c. 4-7i

d. -3+2i

Solution: