# Conversation between Complex Number Forms

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The two forms rectangular and polar are related by the following equations, as illustrated in [Fig. 1].
Fig. 1: Conversion between forms.

### Rectangular to Polar

The rectangular form of complex numbers can be written as
$$C = X +jY$$
While Polar form can be written as
$$C = Z \angle \theta$$
The conversion from rectangular to polar can be done easily by finding Z from X and Y points of rectangular form. The Pythagoras theorem is given as:
$$Z^2 = X^2+Y^2$$
$$\bbox[10px,border:1px solid grey]{ Z = \sqrt{X^2+Y^2}} \tag{1}$$
and
$$\bbox[10px,border:1px solid grey]{\theta = \tan ^{-1} {Y \over X}} \tag{2}$$

#### Polar to Rectangular

As we know from vector analysis, that given quantity of hypotenuse and angle between x-axis can be used to find the X-component and Y-component.
$$\bbox[10px,border:1px solid grey]{X = Z \cos \theta } \tag{3}$$
$$\bbox[10px,border:1px solid grey]{Y = Z \sin \theta } \tag{4}$$
Example 1: Convert the following from
a. $C = 3 + j4$ to Polar form
b. $C = 10 \angle 45^{\circ}$ to rectangular form.
Fig. 2: Example 1(a)
Fig. 3: Example 1(b)
Solution:
a.
$$Z = \sqrt{3^2+4^2} = \sqrt{25} = 5$$
$$\theta = \tan ^{-1} {4 \over 3} = 53.13^\circ$$
and
$$C = 5 \angle 53.13^\circ$$
b.
$$X = 10 \cos 45^{\circ} = 10(0.707) = 7.07$$
$$Y = 10 \sin 45^{\circ} = 10(0.707) = 7.07$$
and
$$C = 7.07 + j 7.07$$
If the complex number should appear in the second, third, or fourth quadrant, simply convert it in that quadrant, and carefully determine the proper angle to be associated with the magnitude of the vector.
Example 2: Convert the following from rectangular to polar form: [Fig. 4].
$$C = -6 + j 3$$
Fig. 4: Example 2
Solution:
$$Z = \sqrt{(-6)^2+3^2} = \sqrt{45} = 6.71$$
$$\beta = \tan ^{-1} {3 \over 6} = 26.57^\circ$$
$$\theta = 180^\circ- 26.57^\circ = 153.43^\circ$$
and
$$C= 6.71 \angle 153.43^\circ$$

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