Complex Number Form

Cartesian or Rectangular Form

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

Each complex number $z = x + iy$ can be placed at the point $(x, y)$ in the complex plane. The position vector from the origin $(0, 0)$ to that point represents the same number geometrically.

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.

For $z=0$, the modulus is $0$, but no direction is defined. Therefore, zero has no argument and must be handled separately when using polar or exponential form.

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 $\text{cis } \theta$.

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. $z = -i$

Worked solutions:

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

   • Identify $x = 1$ and $y = \sqrt{3}$.

   • Calculate the modulus $r$:

     $$\begin{aligned} r &= \sqrt{1^2 + (\sqrt{3})^2} \\ &= \sqrt{1 + 3} \\ &= \sqrt{4} \\ &= 2 \end{aligned}$$

   • 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, so its Principal Argument is $\theta = -90^\circ$.

   • Polar form using the Principal Argument:

     $$z = 1(\cos (-90^\circ) + i \sin (-90^\circ))$$

     The coterminal angle $270^\circ$ gives the equivalent form

     $$z = 1(\cos 270^\circ + i \sin 270^\circ)$$

Exponential Form

The exponential form follows from Euler's formula:

Here, $e \approx 2.71828\ldots$ 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:

For nonzero factors, this form turns multiplication into multiplying moduli and adding arguments. Division uses the corresponding quotient of moduli and difference of arguments.

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:

     $$\begin{aligned} 315^\circ &= 315 \times \frac{\pi}{180} \\ &= \frac{7 \times 45 \times \pi}{4 \times 45} \\ &= \frac{7\pi}{4} \end{aligned}$$

     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. $z = \cos 15^\circ + i \sin 15^\circ$

Worked solutions:

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:

     $$z = e^{i \pi/12}$$

Choosing a Suitable Form

Cartesian form is usually the clearest choice for addition and subtraction. Polar or exponential form makes multiplication, division, powers, and roots easier because moduli and arguments can be combined directly.

To return from polar form to Cartesian form, evaluate the sine and cosine. For example:

Changing form is a problem-solving choice. The complex number itself stays the same.

Equality of Two Complex Numbers

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

Equality Example

The example below compares two numbers with different real parts but equally large imaginary parts, so the sign of the imaginary part decides the answer.

• $z_1 = 3 - 2i$ 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 $z_2 = i + 1$

Worked solutions:

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 ($1 \neq -1$).

   The unequal imaginary parts show that $z_1$ 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

Each item asks you to accept or reject a statement about the three standard forms, so name the form before you decide.

1. True or False. Every real number is a complex number.
2. True or False. The three standard forms used here are 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 $x_2 = 1 - i$.

Worked Solutions

1. True. A real number $a$ can be written as $a + 0i$.

2. False. The three standard forms used here are Cartesian, polar, and exponential. A logarithm is an operation applied to a complex number. It is not one of these three forms.

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 $r$:

     $$\begin{aligned} r &= \sqrt{x^2+y^2} \\ &= \sqrt{2^2+2^2} \\ &= \sqrt{4+4} \\ &= \sqrt{8} \\ &= 2\sqrt{2} \end{aligned}$$

   • 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:

     $$\begin{aligned} z &= r(\cos \theta + i \sin \theta) \\ &= 2\sqrt{2}(\cos 45^\circ + i \sin 45^\circ) \end{aligned}$$

   • 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 $a=1, b=-2, c=6$:

   $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}$$

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

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

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

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

   $$x = 1 \pm i\sqrt{5}$$

   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:

     $$\begin{aligned} x_1 + x_2 &= (1+i) + (1-i) \\ &= 1 + i + 1 - i \\ &= 2 \end{aligned}$$

   • Calculate the product of the roots:

     $$\begin{aligned} x_1 x_2 &= (1+i)(1-i) \\ &= 1^2 - i^2 \\ &= 1 - (-1) \\ &= 1 + 1 \\ &= 2 \end{aligned}$$

   • Construct the quadratic equation:

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

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