# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/complex-number/modulus-argument-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/modulus-argument-complex-numbers/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: "Modulus and Argument of Complex Numbers",
  description: "Calculate modulus |z| = √(x²+y²) and argument θ using quadrant rules. Master distance and angle measurements for polar form conversions.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## What are Modulus and Argument?

A complex number <InlineMath math="z = x + iy" /> can be represented as a point <InlineMath math="(x, y)" /> on the complex plane (similar to the Cartesian plane). Besides being a point, we can also view it as a **vector** starting from the origin <InlineMath math="(0, 0)" /> to the point <InlineMath math="(x, y)" />.

This vector has a **length** and a **direction**. This length and direction are what we call the **Modulus** and **Argument**.

## Modulus of a Complex Number

The **Modulus** of a complex number <InlineMath math="z = x + iy" />, written as <InlineMath math="|z|" />, is the **distance** from the origin <InlineMath math="(0,0)" /> to the point <InlineMath math="(x, y)" /> on the complex plane. This is the same as the **length of the vector** representing <InlineMath math="z" />.

<LineEquation
  title="Modulus Visualization"
  description={
    <>
      The modulus <InlineMath math="|z|" /> is the length of the vector from the
      origin to the point <InlineMath math="z" />. We can see it as the
      hypotenuse of a right-angled triangle.
    </>
  }
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
  data={[
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 4, z: 0 },
      ],
      color: getColor("SKY"),
      labels: [
        { text: "z = 3+4i", at: 1, offset: [0.5, 0.5, 0] },
        {
          text: "|z|",
          at: 1,
          offset: [-2, -1.5, 0],
          color: getColor("AMBER"),
        },
      ],
      cone: { position: "end" },
    },
    // Helper lines for x and y
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 3, y: 0, z: 0 },
      ],
      color: getColor("ROSE"),
      labels: [{ text: "x = 3", at: 1, offset: [-1.5, -0.5, 0] }],
    },
    {
      points: [
        { x: 3, y: 0, z: 0 },
        { x: 3, y: 4, z: 0 },
      ],
      color: getColor("EMERALD"),
      labels: [{ text: "y = 4", at: 1, offset: [1.5, -1.5, 0] }],
    },
  ]}
/>

To calculate the modulus, we can use the Pythagorean Theorem on the right-angled triangle formed by the real part (<InlineMath math="x" />), the imaginary part (<InlineMath math="y" />), and the modulus (<InlineMath math="|z|" />) as the hypotenuse.

**Definition of Modulus:**

The modulus of the complex number <InlineMath math="z = x + iy" /> is:

<BlockMath math="|z| = \sqrt{x^2 + y^2}" />

The modulus is always **non-negative** (never negative) because it represents a distance.

### Calculating the Modulus

1.  **Find the modulus of <InlineMath math="z_1 = 3 + 4i" />**, with <InlineMath math="x=3, y=4" />

    <BlockMath math="|z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5" />

2.  **Find the modulus of <InlineMath math="z_2 = -1 - 2i" />**, with <InlineMath math="x=-1, y=-2" />

    <BlockMath math="|z_2| = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}" />

3.  **Find the modulus of <InlineMath math="z_3 = 5" />**, with <InlineMath math="x=5, y=0" />

    <BlockMath math="|z_3| = \sqrt{5^2 + 0^2} = \sqrt{25} = 5" />

    (The modulus of a real number is its absolute value).

4.  **Find the modulus of <InlineMath math="z_4 = -2i" />**, with <InlineMath math="x=0, y=-2" />

    <BlockMath math="|z_4| = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2" />

## Argument of a Complex Number

The **Argument** of a non-zero complex number <InlineMath math="z = x + iy" />, written as <InlineMath math="\arg(z)" /> or <InlineMath math="\theta" />, is the **angle** formed by the vector <InlineMath math="z" /> with the **positive real axis** on the complex plane. This angle is usually measured in radians or degrees.

From basic trigonometry on the same right-angled triangle as in the modulus visualization, we know the relationships:

<div className="flex flex-col gap-4">
  <BlockMath math="x = |z| \cos \theta" />
  <BlockMath math="y = |z| \sin \theta" />
  <BlockMath math="\tan \theta = \frac{y}{x} \quad (\text{if } x \neq 0)" />
</div>

To find <InlineMath math="\theta" />, we can use the arctangent function (or <InlineMath math="\tan^{-1}" />):

<BlockMath math="\theta = \arctan\left(\frac{y}{x}\right)" />

Calculators usually give the <InlineMath math="\arctan" /> value in the range <InlineMath math="(-90^\circ, 90^\circ)" /> or <InlineMath math="(-\pi/2, \pi/2)" />. We need to **consider the quadrant** where the point <InlineMath math="(x, y)" /> lies to determine the correct argument.

- **Quadrant <InlineMath math="I" />** (<InlineMath math="x>0, y>0" />):

  <BlockMath math="\theta = \arctan(y/x)" />

- **Quadrant <InlineMath math="II" />** (<InlineMath math="x<0, y>0" />):

  <BlockMath math="\theta = 180^\circ + \arctan(y/x) \text{ or } \theta = \pi + \arctan(y/x)" />

- **Quadrant <InlineMath math="III" />** (<InlineMath math="x<0, y<0" />):

  <BlockMath math="\theta = 180^\circ + \arctan(y/x) \text{ or } \theta = \pi + \arctan(y/x)" />

- **Quadrant <InlineMath math="IV" />** (<InlineMath math="x>0, y<0" />):

  <BlockMath math="\theta = 360^\circ + \arctan(y/x) \text{ or } \theta = 2\pi + \arctan(y/x)" />

  or simply

  <BlockMath math="\theta = \arctan(y/x)" />

  if a negative angle is desired

Often, we are interested in the **Principal Argument** (written <InlineMath math="\text{Arg}(z)" />), which is the argument value in the interval <InlineMath math="(-180^\circ, 180^\circ]" /> or <InlineMath math="(-\pi, \pi]" />.

### Calculating the Argument

1.  **Find the argument of <InlineMath math="z_1 = 1 + i" />**

    The point <InlineMath math="(1, 1)" /> is in Quadrant <InlineMath math="I" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{1}{1} = 1" />
      <BlockMath math="\theta = \arctan(1) = 45^\circ \text{ or } \frac{\pi}{4} \text{ radians}" />
    </div>

2.  **Find the argument of <InlineMath math="z_2 = -\sqrt{3} + i" />**

    The point <InlineMath math="(-\sqrt{3}, 1)" /> is in Quadrant <InlineMath math="II" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{1}{-\sqrt{3}}" />
      <BlockMath math="\text{Base angle} = \arctan\left(\left|\frac{1}{-\sqrt{3}}\right|\right) = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ" />
      <BlockMath math="\theta = 180^\circ - 30^\circ = 150^\circ \text{ or } \frac{5\pi}{6} \text{ radians}" />
    </div>

    (Because it's in Quadrant <InlineMath math="II" />, we use <InlineMath math="180^\circ - \text{base angle}" />)

3.  **Find the argument of <InlineMath math="z_3 = -1 - i\sqrt{3}" />**

    The point <InlineMath math="(-1, -\sqrt{3})" /> is in Quadrant <InlineMath math="III" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{-\sqrt{3}}{-1} = \sqrt{3}" />
      <BlockMath math="\text{Base angle} = \arctan(\sqrt{3}) = 60^\circ" />
      <BlockMath math="\theta = 180^\circ + 60^\circ = 240^\circ \text{ or } \frac{4\pi}{3} \text{ radians}" />
    </div>

    (Because it's in Quadrant <InlineMath math="III" />, we use <InlineMath math="180^\circ + \text{base angle}" />
    . Principal Argument: <InlineMath math="-120^\circ" /> or <InlineMath math="-2\pi/3" />
    ).

4.  **Find the argument of <InlineMath math="z_4 = 3 - 3i" />**

    The point <InlineMath math="(3, -3)" /> is in Quadrant <InlineMath math="IV" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{-3}{3} = -1" />
      <BlockMath math="\text{Base angle} = \arctan(|-1|) = \arctan(1) = 45^\circ" />
      <BlockMath math="\theta = 360^\circ - 45^\circ = 315^\circ \text{ or } \frac{7\pi}{4} \text{ radians}" />
    </div>

    (Because it's in Quadrant <InlineMath math="IV" />, we use <InlineMath math="360^\circ - \text{base angle}" />
    . Principal Argument: <InlineMath math="-45^\circ" /> or <InlineMath math="-\pi/4" />
    ).

## Exercise

Find the modulus and argument (in degrees) of the following complex numbers:

1.  <InlineMath math="z_a = 2 + 2i" />
2.  <InlineMath math="z_b = -4" />
3.  <InlineMath math="z_c = -i" />

### Answer Key

1.  **For <InlineMath math="z_a = 2 + 2i" />:**

    <InlineMath math="x=2, y=2" /> (Quadrant <InlineMath math="I" />) Modulus:

    <BlockMath math="|z_a| = \sqrt{2^2 + 2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}" />

    Argument:

    <BlockMath math="\tan \theta = \frac{2}{2} = 1 \implies \theta = \arctan(1) = 45^\circ" />

2.  **For <InlineMath math="z_b = -4" />:**

    <InlineMath math="z_b = -4 + 0i" />. <InlineMath math="x=-4, y=0" /> (Negative
    real axis) Modulus:

    <BlockMath math="|z_b| = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4" />

    Argument: The point is on the negative real axis.

    <BlockMath math="\theta = 180^\circ" />

3.  **For <InlineMath math="z_c = -i" />:**

    <InlineMath math="z_c = 0 - 1i" />. <InlineMath math="x=0, y=-1" /> (Negative
    imaginary axis) Modulus:

    <BlockMath math="|z_c| = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1" />

    Argument: The point is on the negative imaginary axis.

    <BlockMath math="\theta = 270^\circ" />

    or <InlineMath math="-90^\circ" /> (Principal Argument).
