Complex Number Form

Cartesian Form (Rectangular)

A complex number has the form $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part. This form $z = x + iy$ is called the Cartesian form or rectangular form.

• $x = \text{Re}(z)$ (Real Part)
• $y = \text{Im}(z)$ (Imaginary Part)

We can also view the complex number $z = x + iy$ as an ordered pair $(x, y)$ on a coordinate plane. This special plane is called the complex plane or Argand diagram.

• The horizontal axis (x-axis) represents the real part.
• The vertical axis (y-axis) represents the imaginary part.

Visualization on the Complex Plane

Let's try plotting some complex numbers on the complex plane. Each number $z = x + iy$ is plotted as the point $(x, y)$ and is usually represented as a vector (arrow) from the origin (0,0) to that point.

Polar Form

Besides Cartesian, there's another way to represent complex numbers: the polar form. This form uses:

1. Modulus ($r$): The distance from the origin (0,0) to the point $(x, y)$ on the complex plane. Its value is always non-negative.
2. Argument ($\theta$): The angle formed by the line from the origin to the point $(x, y)$ with the positive real axis. This angle is usually measured in radians or degrees.

The relationship between Cartesian form ($x, y$) and Polar form ($r, \theta$) can be seen from basic trigonometry:

From this, we can find $r$ and $\theta$ if $x$ and $y$ are known:

When finding $\theta$ from $\tan \theta$, pay attention to the quadrant where the point $(x, y)$ lies to determine the correct angle.

By substituting $x$ and $y$ into the Cartesian form, we get the polar form:

Sometimes, the form $(\cos \theta + i \sin \theta)$ is abbreviated as $\text{cis } \theta$.

Example: Conversion to Polar Form

Suppose we have $z = 1 + i$.

• Real part $x = 1$.
• Imaginary part $y = 1$.

Find $r$:

Find $\theta$:

Since $x$ and $y$ are positive, the point $(1, 1)$ is in quadrant $I$. The angle whose $\tan$ is 1 in quadrant $I$ is $45^\circ$ or $\pi/4$ radians.

So, the polar form is:

Polar Form Exercise

Express the following complex numbers in polar form:

1. $z = 1 + i\sqrt{3}$
2. $z = -i$

Answer Key:

1. For $z = 1 + i\sqrt{3}$:

   • Identify $x = 1$ and $y = \sqrt{3}$.
   • Calculate the modulus $r$:
     $$r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$$
   • Calculate the argument $\theta$:
     $$\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}$$
     Since $x$ and $y$ are positive, the point $(1, \sqrt{3})$ is in quadrant $I$ , so $\theta = 60^\circ$.
   • Polar Form:
     $$z = 2(\cos 60^\circ + i \sin 60^\circ)$$

2. For $z = -i$:

   • Identify $x = 0$ and $y = -1$.
   • Calculate the modulus $r$:
     $$r = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1$$
   • Determine the argument $\theta$: The point $(0, -1)$ lies on the negative imaginary axis. The angle is $\theta = 270^\circ$ or it can also be written as $\theta = -90^\circ$.
   • Polar Form (choose one angle):
     $$z = 1(\cos 270^\circ + i \sin 270^\circ)$$
     or
     $$z = 1(\cos (-90^\circ) + i \sin (-90^\circ))$$

Exponential Form

There's one more important form: the exponential form. This form comes from the magical Euler's Formula:

Here, $e \approx 2.71828...$ is Euler's number (the base of the natural logarithm).

If we substitute Euler's Formula into the polar form $z = r(\cos \theta + i \sin \theta)$, we get the exponential form:

This form is very useful for multiplying and dividing complex numbers.

Example: Conversion to Exponential Form

Take the previous examples:

1. For $z = 1 + i$, we already have the polar form $\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)$.

   • Modulus $r = \sqrt{2}$.
   • Argument $\theta = 45^\circ = \pi/4$ radians.
   • Exponential Form:
     $$z = r e^{i\theta} = \sqrt{2} e^{i \pi/4}$$

2. For $z = \sqrt{2}(\cos 315^\circ + i \sin 315^\circ)$:

   • Modulus $r = \sqrt{2}$.
   • Argument $\theta = 315^\circ$. Convert to radians:
     $$315^\circ = 315 \times \frac{\pi}{180} = \frac{7 \times 45 \times \pi}{4 \times 45} = \frac{7\pi}{4}$$
     Or use the negative angle $315^\circ - 360^\circ = -45^\circ = -\pi/4$ radians.
   • Exponential Form (choose one angle):
     $$z = \sqrt{2} e^{i 7\pi/4}$$
     or
     $$z = \sqrt{2} e^{-i \pi/4}$$

Exponential Form Exercise

Express the following complex numbers in exponential form (use radian angles):

1. $z = 2(\cos 60^\circ + i \sin 60^\circ)$
2. $z = \cos 15^\circ + i \sin 15^\circ$

Answer Key:

1. For $z = 2(\cos 60^\circ + i \sin 60^\circ)$:
   • Modulus $r = 2$.
   • Argument $\theta = 60^\circ$. Convert to radians:
     $$\theta = 60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$$
   • Exponential Form:
     $$z = 2 e^{i \pi/3}$$
2. For $z = \cos 15^\circ + i \sin 15^\circ$:
   • Modulus $r = 1$ (because there is no coefficient in front of $\cos$ and $\sin$).
   • Argument $\theta = 15^\circ$. Convert to radians:
     $$\theta = 15^\circ = 15 \times \frac{\pi}{180} = \frac{\pi}{12}$$
   • Exponential Form:
     $$z = e^{i \pi/12}$$

Equality of Two Complex Numbers

Two complex numbers $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$ are said to be equal if and only if their real parts are equal AND their imaginary parts are also equal.

Equality Example

• $z_1 = 3 - 2i$ and $z_2 = 4 + 2i$ are different.

  because $\text{Re}(z_1) = 3 \neq \text{Re}(z_2) = 4$ (even though $|\text{Im}(z_1)| = |-2| = 2 = |\text{Im}(z_2)| = |2|$, their imaginary signs differ).

• $z_1 = -1 + i$ and $z_2 = i - 1$ are equal.

  because $\text{Re}(z_1) = -1 = \text{Re}(z_2) = -1$ and $\text{Im}(z_1) = 1 = \text{Im}(z_2) = 1$.

Equality Exercise

Determine if the following pairs of complex numbers are equal or different:

1. $z_1 = 4 - (-2i)$ and $z_2 = 4 + 2i$
2. $z_1 = i$ and $z_2 = 1 - i$
3. $z_1 = -1 + i$ and $z_2 = i + 1$

Answer Key:

1. $z_1 = 4 - (-2i) = 4 + 2i$.

   Thus, $z_1$ is equal to $z_2 = 4 + 2i$.

2. $z_1 = 0 + 1i$ and $z_2 = 1 - 1i$.

   The real parts are different ($0 \neq 1$) and the imaginary parts are different ($1 \neq -1$).

   Thus, $z_1$ is different from $z_2$.

3. $z_1 = -1 + 1i$ and $z_2 = 1 + 1i$.

   The real parts are different ($-1 \neq 1$).

   Thus, $z_1$ is different from $z_2$.

Exercises

1. True or False. Every real number is a complex number.
2. True or False. Complex numbers have 3 forms: Cartesian, exponential, and logarithmic.
3. True or False. If the complex number $z = 1 - 3i$ is plotted on the complex plane, it lies in quadrant III.
4. Express the complex number $2+2i$ in polar and exponential forms.
5. Find the numbers $x$ and $y$ such that $z_1 = x + 3i$ and $z_2 = 3 - yi$ satisfy $z_1 = z_2$!
6. Find the solutions to the quadratic equation $x^2 - 2x + 6 = 0$!
7. Find the quadratic equation whose solutions are $x_1 = 1 + i$ and $x_2 = 1 - i$!

Answer Key

1. True. A real number $a$ can be written as $a + 0i$.
2. False. The common forms of complex numbers are Cartesian, Polar, and Exponential. The complex logarithm form exists but is not typically considered one of the three main forms studied at this level.
3. False. $z = 1 - 3i$ has a positive real part ($x=1$) and a negative imaginary part ($y=-3$). The point $(1, -3)$ lies in Quadrant IV.
4. For $z = 2 + 2i$:
   • Calculate the modulus $r$:
     $$r = \sqrt{x^2+y^2} = \sqrt{2^2+2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}$$
   • Calculate the argument $\theta$:
     $$\tan \theta = \frac{y}{x} = \frac{2}{2} = 1$$
     Since $x=2$ and $y=2$ (both positive), the point is in Quadrant I. Thus, $\theta = 45^\circ$ or $\pi/4$ radians.
   • Polar Form:
     $$z = r(\cos \theta + i \sin \theta) = 2\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)$$
   • Exponential Form:
     $$z = r e^{i\theta} = 2\sqrt{2} e^{i \pi/4}$$
5. For $z_1 = x + 3i$ to equal $z_2 = 3 - yi$, the real parts must be equal and the imaginary parts must be equal:
   • Real Part: $x = 3$
   • Imaginary Part: $3 = -y \implies y = -3$ So, $x=3$ and $y=-3$.
6. To solve $x^2 - 2x + 6 = 0$, use the quadratic formula:
   $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
   with $a=1, b=-2, c=6$:
   $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}$$

   $$x = \frac{2 \pm \sqrt{4 - 24}}{2}$$

   $$x = \frac{2 \pm \sqrt{-20}}{2}$$

   $$x = \frac{2 \pm \sqrt{20}\sqrt{-1}}{2}$$

   $$x = \frac{2 \pm 2\sqrt{5}i}{2}$$

   $$x = 1 \pm i\sqrt{5}$$
   The solutions are $x_1 = 1 + i\sqrt{5}$ and $x_2 = 1 - i\sqrt{5}$ .
7. If the roots of a quadratic equation are $x_1 = 1 + i$ and $x_2 = 1 - i$, the equation can be formed from $(x - x_1)(x - x_2) = 0$ or $x^2 - (x_1 + x_2)x + (x_1 x_2) = 0$.
   • Calculate the sum of the roots:
     $$x_1 + x_2 = (1+i) + (1-i) = 1 + i + 1 - i = 2$$
   • Calculate the product of the roots:
     $$x_1 x_2 = (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2$$
   • Construct the quadratic equation:
     $$x^2 - (2)x + (2) = 0$$

     $$x^2 - 2x + 2 = 0$$