# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/complex-number/complex-number-form
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/complex-number-form/en.mdx

Output docs content for large language models.

---

import { LineEquation } from "@repo/design-system/components/contents/line-equation";
import { getColor } from "@repo/design-system/lib/color";

export const metadata = {
  title: "Complex Number Form",
  description: "Master Cartesian, polar, and exponential forms of complex numbers. Learn conversions between representations with Euler's formula and examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## Cartesian Form (Rectangular)

A complex number has the form <InlineMath math="z = x + iy" />, where <InlineMath math="x" /> is the real part and <InlineMath math="y" /> is the imaginary part. This form <InlineMath math="z = x + iy" /> is called the **Cartesian form** or rectangular form.

<BlockMath math="z = x + iy" />

- <InlineMath math="x = \text{Re}(z)" /> (Real Part)
- <InlineMath math="y = \text{Im}(z)" /> (Imaginary Part)

We can also view the complex number <InlineMath math="z = x + iy" /> as an ordered pair <InlineMath math="(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 <InlineMath math="z = x + iy" /> is plotted as the point <InlineMath math="(x, y)" /> and is usually represented as a vector (arrow) from the origin (0,0) to that point.

<LineEquation
  title="Complex Numbers on the Complex Plane"
  description="Visualization of several complex numbers as points and vectors on the complex plane."
  cameraPosition={[0, 0, 10]}
  showZAxis={false}
  data={[
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 2, z: 0 },
      ],
      color: getColor("ROSE"),
      labels: [{ text: "z₁ = 3 + 2i", at: 1, offset: [0.5, 0.5, 0] }],
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: -1, y: -1, z: 0 },
      ],
      color: getColor("EMERALD"),
      labels: [{ text: "z₂ = -1 - i", at: 1, offset: [-1, -0.5, 0] }],
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 2, y: 0, z: 0 },
      ],
      color: getColor("VIOLET"),
      labels: [{ text: "z₃ = 2", at: 1, offset: [0.5, 0.5, 0] }],
    },
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 0, y: 1, z: 0 },
      ],
      color: getColor("AMBER"),
      labels: [{ text: "z₄ = i", at: 1, offset: [-1, 0.5, 0] }],
    },
  ]}
/>

## Polar Form

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

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

The relationship between Cartesian form (<InlineMath math="x, y" />) and Polar form (<InlineMath math="r, \theta" />) can be seen from basic trigonometry:

<div className="flex flex-col gap-4">
  <BlockMath math="x = r \cos \theta" />
  <BlockMath math="y = r \sin \theta" />
</div>

From this, we can find <InlineMath math="r" /> and <InlineMath math="\theta" /> if <InlineMath math="x" /> and <InlineMath math="y" /> are known:

<div className="flex flex-col gap-4">
  <BlockMath math="r = |z| = \sqrt{x^2 + y^2}" />
  <BlockMath math="\tan \theta = \frac{y}{x}" />
</div>

When finding <InlineMath math="\theta" /> from <InlineMath math="\tan \theta" />, pay attention to the quadrant where the point <InlineMath math="(x, y)" /> lies to determine the correct angle.

By substituting <InlineMath math="x" /> and <InlineMath math="y" /> into the Cartesian form, we get the **polar form**:

<div className="flex flex-col gap-4">
  <BlockMath math="z = x + iy = (r \cos \theta) + i(r \sin \theta)" />
  <BlockMath math="z = r (\cos \theta + i \sin \theta)" />
</div>

Sometimes, the form <InlineMath math="(\cos \theta + i \sin \theta)" /> is abbreviated as <InlineMath math="\text{cis } \theta" />.

<BlockMath math="z = r \text{ cis } \theta" />

### Example: Conversion to Polar Form

Suppose we have <InlineMath math="z = 1 + i" />.

- Real part <InlineMath math="x = 1" />.
- Imaginary part <InlineMath math="y = 1" />.

Find <InlineMath math="r" />:

<BlockMath math="r = \sqrt{1^2 + 1^2} = \sqrt{2}" />

Find <InlineMath math="\theta" />:

<BlockMath math="\tan \theta = \frac{1}{1} = 1" />

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

So, the polar form is:

<BlockMath math="z = \sqrt{2} (\cos 45^\circ + i \sin 45^\circ)" />

### Polar Form Exercise

Express the following complex numbers in polar form:

1.  <InlineMath math="z = 1 + i\sqrt{3}" />
2.  <InlineMath math="z = -i" />

**Answer Key:**

1.  For <InlineMath math="z = 1 + i\sqrt{3}" />:

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

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

<BlockMath math="e^{i\theta} = \cos \theta + i \sin \theta" />

Here, <InlineMath math="e \approx 2.71828..." /> is Euler's number (the base of the natural logarithm).

If we substitute Euler's Formula into the polar form <InlineMath math="z = r(\cos \theta + i \sin \theta)" />, we get the **exponential form**:

<BlockMath math="z = r e^{i\theta}" />

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

### Example: Conversion to Exponential Form

Take the previous examples:

1.  For <InlineMath math="z = 1 + i" />, we already have the polar form <InlineMath math="\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)" />.

    - Modulus <InlineMath math="r = \sqrt{2}" />.
    - Argument <InlineMath math="\theta = 45^\circ = \pi/4" /> radians.
    - Exponential Form:
      <BlockMath math="z = r e^{i\theta} = \sqrt{2} e^{i \pi/4}" />

2.  For <InlineMath math="z = \sqrt{2}(\cos 315^\circ + i \sin 315^\circ)" />:
    - Modulus <InlineMath math="r = \sqrt{2}" />.
    - Argument <InlineMath math="\theta = 315^\circ" />. Convert to radians:
      <BlockMath math="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 <InlineMath math="315^\circ - 360^\circ = -45^\circ = -\pi/4" /> radians.
    - Exponential Form (choose one angle):
      <BlockMath math="z = \sqrt{2} e^{i 7\pi/4}" />
      or
      <BlockMath math="z = \sqrt{2} e^{-i \pi/4}" />

### Exponential Form Exercise

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

1.  <InlineMath math="z = 2(\cos 60^\circ + i \sin 60^\circ)" />
2.  <InlineMath math="z = \cos 15^\circ + i \sin 15^\circ" />

**Answer Key:**

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

## Equality of Two Complex Numbers

Two complex numbers <InlineMath math="z_1 = x_1 + iy_1" /> and <InlineMath math="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.

<BlockMath math="z_1 = z_2 \quad \iff \quad x_1 = x_2 \text{ and } y_1 = y_2" />

### Equality Example

- <InlineMath math="z_1 = 3 - 2i" /> and <InlineMath math="z_2 = 4 + 2i" /> are **different**.

  because <InlineMath math="\text{Re}(z_1) = 3 \neq \text{Re}(z_2) = 4" /> (even though <InlineMath math="|\text{Im}(z_1)| = |-2| = 2 = |\text{Im}(z_2)| = |2|" />, their imaginary signs differ).

- <InlineMath math="z_1 = -1 + i" /> and <InlineMath math="z_2 = i - 1" /> are **equal**.

  because <InlineMath math="\text{Re}(z_1) = -1 = \text{Re}(z_2) = -1" /> and <InlineMath math="\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.  <InlineMath math="z_1 = 4 - (-2i)" /> and <InlineMath math="z_2 = 4 + 2i" />
2.  <InlineMath math="z_1 = i" /> and <InlineMath math="z_2 = 1 - i" />
3.  <InlineMath math="z_1 = -1 + i" /> and <InlineMath math="z_2 = i + 1" />

**Answer Key:**

1.  <InlineMath math="z_1 = 4 - (-2i) = 4 + 2i" />.

    Thus, <InlineMath math="z_1" /> is **equal** to <InlineMath math="z_2 = 4 + 2i" />.

2.  <InlineMath math="z_1 = 0 + 1i" /> and <InlineMath math="z_2 = 1 - 1i" />.

    The real parts are different (<InlineMath math="0 \neq 1" />) and the imaginary parts are different (<InlineMath math="1 \neq -1" />).

    Thus, <InlineMath math="z_1" /> is **different** from <InlineMath math="z_2" />.

3.  <InlineMath math="z_1 = -1 + 1i" /> and <InlineMath math="z_2 = 1 + 1i" />.

    The real parts are different (<InlineMath math="-1 \neq 1" />).

    Thus, <InlineMath math="z_1" /> is **different** from <InlineMath math="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 <InlineMath math="z = 1 - 3i" /> is plotted on the complex plane, it lies in quadrant III.
4.  Express the complex number <InlineMath math="2+2i" /> in polar and exponential forms.
5.  Find the numbers <InlineMath math="x" /> and <InlineMath math="y" /> such that <InlineMath math="z_1 = x + 3i" /> and <InlineMath math="z_2 = 3 - yi" /> satisfy <InlineMath math="z_1 = z_2" />!
6.  Find the solutions to the quadratic equation <InlineMath math="x^2 - 2x + 6 = 0" />!
7.  Find the quadratic equation whose solutions are <InlineMath math="x_1 = 1 + i" /> and <InlineMath math="x_2 = 1 - i" />!

### Answer Key

1.  **True.** A real number <InlineMath math="a" /> can be written as <InlineMath math="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.** <InlineMath math="z = 1 - 3i" /> has a positive real part (<InlineMath math="x=1" />) and a negative imaginary part (<InlineMath math="y=-3" />). The point <InlineMath math="(1, -3)" /> lies in **Quadrant IV**.
4.  For <InlineMath math="z = 2 + 2i" />:
    - Calculate the modulus <InlineMath math="r" />:
      <BlockMath math="r = \sqrt{x^2+y^2} = \sqrt{2^2+2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}" />
    - Calculate the argument <InlineMath math="\theta" />:
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{2}{2} = 1" />
      Since <InlineMath math="x=2" /> and <InlineMath math="y=2" /> (both positive),
      the point is in Quadrant I. Thus, <InlineMath math="\theta = 45^\circ" /> or <InlineMath math="\pi/4" /> radians.
    - Polar Form:
      <BlockMath math="z = r(\cos \theta + i \sin \theta) = 2\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)" />
    - Exponential Form:
      <BlockMath math="z = r e^{i\theta} = 2\sqrt{2} e^{i \pi/4}" />
5.  For <InlineMath math="z_1 = x + 3i" /> to equal <InlineMath math="z_2 = 3 - yi" />, the real parts must be equal and the imaginary parts must be equal:
    - Real Part: <InlineMath math="x = 3" />
    - Imaginary Part: <InlineMath math="3 = -y \implies y = -3" />
      So, <InlineMath math="x=3" /> and <InlineMath math="y=-3" />.
6.  To solve <InlineMath math="x^2 - 2x + 6 = 0" />, use the quadratic formula:
    <BlockMath math="x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}" />
    with <InlineMath math="a=1, b=-2, c=6" />:
    <div className="flex flex-col gap-4">
      <BlockMath math="x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}" />
      <BlockMath math="x = \frac{2 \pm \sqrt{4 - 24}}{2}" />
      <BlockMath math="x = \frac{2 \pm \sqrt{-20}}{2}" />
      <BlockMath math="x = \frac{2 \pm \sqrt{20}\sqrt{-1}}{2}" />
      <BlockMath math="x = \frac{2 \pm 2\sqrt{5}i}{2}" />
      <BlockMath math="x = 1 \pm i\sqrt{5}" />
    </div>
    The solutions are <InlineMath math="x_1 = 1 + i\sqrt{5}" /> and <InlineMath math="x_2 = 1 - i\sqrt{5}" />
    .
7.  If the roots of a quadratic equation are <InlineMath math="x_1 = 1 + i" /> and <InlineMath math="x_2 = 1 - i" />, the equation can be formed from <InlineMath math="(x - x_1)(x - x_2) = 0" /> or <InlineMath math="x^2 - (x_1 + x_2)x + (x_1 x_2) = 0" />.
    - Calculate the sum of the roots:
      <BlockMath math="x_1 + x_2 = (1+i) + (1-i) = 1 + i + 1 - i = 2" />
    - Calculate the product of the roots:
      <BlockMath math="x_1 x_2 = (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2" />
    - Construct the quadratic equation:
      <div className="flex flex-col gap-4">
        <BlockMath math="x^2 - (2)x + (2) = 0" />
        <BlockMath math="x^2 - 2x + 2 = 0" />
      </div>