1.2 The Complex Plane

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We learn a geometric interpretation of complex numbers.

1 Plotting a Complex Number

Just as the real numbers can be represented visually, or geometrically, by the real number line, complex numbers can be represented by the complex plane. Each complex number will correspond to a point in the plane and vice versa. The plane is analogous to the $xy$ -plane. The difference is that the vertical axis is labeled as $i, 2i, 3i, ...$ instead of $1, 2, 3, ...$ . We plot the complex number $a+bi$ in the complex plane just as we would plot the point $(a,b)$ in the $xy$ -plane.

Example 1 Plot the complex number $1+2i$ in the complex plane.

(Problem 1) Plot the complex numbers in the complex plane: \begin{align*} \quad (i)& \quad 3+i & \\ \quad (ii)& \quad -2-2i \\ \quad (iii)& \quad 4i \\ \quad (iv)& \quad -5 \end{align*}

2 The Real and Imaginary Parts

Geometrically, the real part of a complex number $a+bi$ its horizontal component and the imaginary part is its vertical component. It is important to note that the imaginary part is a real number.

3 The Modulus of a Complex Number

Just as a point in the $xy$ -plane can be thought of as a vector emanating from the origin, a complex number can be interpreted in the same manner. The magnitude of a complex number, considering it as a vector, is called the modulus, denoted $|a+bi|$ . From the Pythagorean Theorem, we can see that $|a+bi| = \sqrt {a^2+b^2}$

The modulus of a real number is just its absolute value.

Example 2 Find the modulus of the complex number $-5+6i$ .
Using the formula $|a+bi| =\sqrt {a^2+b^2}$ , we have $|-5+6i| = \sqrt {(-5)^2 + 6^2} = \sqrt {61}$

(Problem 2a)

Find the modulus of the complex number:
i) $\displaystyle |-3i| = \answer {3}$
ii) $\displaystyle |-12| = \answer {12}$
iii) $\displaystyle |6+8i| = \answer {10}$
iv) $\displaystyle |\overline {2+3i}| = \answer {\sqrt {13}}$

(Problem 2b) Select all of the statements that are true for any complex number, $a+bi$ :

$|\Rep (a+bi)| \leq |a+bi|$ $|\Imp (a+bi)| \leq |a+bi|$ $|a+bi| \leq |\Rep (a+bi)| + |\Imp (a+bi)|$ $\left |\overline {a+bi}\right | = |a+bi|$

(Problem 2c) Prove each of the true statements from problem 2b, above.

4 The Polar Form of a Complex Number

We can describe the position of a complex number in polar form using a magnitude, or radius, and a direction angle rather than using the rectangular form involving horizontal and vertical components. With the standard polar notations $r$ and $\theta$ for the radius and the direction angle, we can represent a complex number in polar form $r\cos \theta + ir\sin \theta = r\left (\cos \theta + i\sin \theta \right ) = r \cis \theta$

The polar radius of a complex number is just its modulus: $r = |a+bi|$

Later, we will discover that $e^{i\theta } = \cis \theta$ . This is known as Euler’s Formula.

Example 3 Find the polar form of the complex number $-\sqrt 3 + i$ .
The polar radius is $r = |-\sqrt 3 + i | =2$ To find the polar angle, $\theta$ , we will use the equation $\tan \theta = -\frac {\ 1}{\sqrt 3}$ We might erroneously conclude that $\theta = \tan ^{-1} \left (-\frac {\ 1}{\sqrt 3}\right ) = -\frac {\pi }{6}$ However, the complex number $-\sqrt 3 + i$ lies in the second quadrant so we add $\pi$ : $\theta = -\frac {\pi }{6} + \pi = \frac {5\pi }{6}$ We can now write the polar form: $-\sqrt 3 +i = 2\cis \left (\frac {5\pi }{6}\right )$

(Problem 3)

Express the complex number in polar form, $r\cis \theta$ where $0 \leq \theta < 2\pi$ :
i) $\displaystyle 2i = \answer {2cis(\pi /2)}$
ii) $\displaystyle -3= \answer {3cis(\pi )}$
iii) $\displaystyle 1+i= \answer {\sqrt 2 cis(\pi /4)}$
iv) $\displaystyle -2-2\sqrt 3i= \answer {4cis(4\pi /3)}$

Here is a video solution to a problem on the polar form of a complex number:

2024-09-27 14:06:17