# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/complex-number/principal-argument-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/principal-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: "Principal Argument of Complex Numbers",
  description: "Find unique principal argument Arg(z) in range [0°,360°). Convert infinite angle possibilities to one standard value for complex number equality.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## Understanding the Principal Argument

The argument <InlineMath math="\theta" /> of a complex number <InlineMath math="z = x + iy" /> is the angle formed by the vector <InlineMath math="z" /> with the positive real axis.

However, there's an important point: the argument is not a single value!

If <InlineMath math="\theta" /> is an argument of <InlineMath math="z" />, then <InlineMath math="\theta + 2\pi k" /> (where <InlineMath math="k" /> is an integer: <InlineMath math="0, \pm 1, \pm 2, \ldots" />) is also an argument of <InlineMath math="z" />, because adding multiples of <InlineMath math="360^\circ" /> or <InlineMath math="2\pi" /> radians results in the same angle on the complex plane.

**Example:**

The angles <InlineMath math="45^\circ" />, <InlineMath math="405^\circ" /> (<InlineMath math="45^\circ + 360^\circ" />), and <InlineMath math="-315^\circ" /> (<InlineMath math="45^\circ - 360^\circ" />) all indicate the same direction.

Because there are infinitely many arguments for a single complex number, we often need a unique standard value. This value is called the **Principal Argument**.

## Definition of Principal Argument

The Principal Argument of a complex number <InlineMath math="z = r(\cos \theta + i\sin \theta)" /> is the unique value of the argument <InlineMath math="\theta" /> that satisfies a specific range.

**Principal Argument** (denoted <InlineMath math="\text{Arg}(z)" />) is defined as the argument <InlineMath math="\theta" /> that satisfies:

<BlockMath math="0 \leq \theta < 2\pi \quad \text{or} \quad 0^\circ \leq \theta < 360^\circ" />

Note: Other definitions sometimes use the range <InlineMath math="(-\pi, \pi]" /> or <InlineMath math="(-180^\circ, 180^\circ]" />. It's important to always check the definition being used in a specific context.

## Determining the Principal Argument

Determining the Principal Argument is the same as finding the regular argument, but we need to ensure the final result is within the range <InlineMath math="[0, 2\pi)" /> or <InlineMath math="[0^\circ, 360^\circ)" />.

### Finding the Principal Argument

1.  **Find the Principal Argument of <InlineMath math="z = 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" />
    </div>

    Since <InlineMath math="45^\circ" /> is already within the range <InlineMath math="[0^\circ, 360^\circ)" />, the Principal Argument is:

    <BlockMath math="\text{Arg}(z) = 45^\circ \text{ or } \frac{\pi}{4} \text{ radians}" />

2.  **Find the Principal Argument of <InlineMath math="z = \sqrt{3} + i" />**

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

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{1}{\sqrt{3}}" />
      <BlockMath math="\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ" />
    </div>

    Since <InlineMath math="30^\circ" /> is already within the range <InlineMath math="[0^\circ, 360^\circ)" />, the Principal Argument is:

    <BlockMath math="\text{Arg}(z) = 30^\circ \text{ or } \frac{\pi}{6} \text{ radians}" />

<div className="mt-4">
  <LineEquation
    title="Principal Argument Visualization"
    description={
      <>
        Showing vectors for <InlineMath math="z_1=1+i" /> and
        <InlineMath math="z_2=\sqrt{3}+i" />, along with their Principal Arguments
        (<InlineMath math="45^\circ" /> and <InlineMath math="30^\circ" />
        ).
      </>
    }
    cameraPosition={[0, 0, 8]}
    showZAxis={false}
    data={[
      {
        points: [
          { x: 0, y: 0, z: 0 },
          { x: 1, y: 1, z: 0 },
        ],
        color: getColor("SKY"),
        labels: [{ text: "z₁ = 1+i", at: 1, offset: [-1, 0.5, 0] }],
        cone: { position: "end" },
      },
      {
        points: [
          { x: 0, y: 0, z: 0 },
          { x: Math.sqrt(3), y: 1, z: 0 },
        ],
        color: getColor("LIME"),
        labels: [{ text: "z₂ = √3+i", at: 1, offset: [1.5, 0.5, 0] }],
        cone: { position: "end" },
      },
      // Positive real axis line for angle reference
      {
        points: [
          { x: 0, y: 0, z: 0 },
          { x: 2, y: 0, z: 0 },
        ],
        color: getColor("AMBER"),
      },
    ]}
  />
</div>

## Equality of Two Complex Numbers in Polar Form

Two complex numbers <InlineMath math="z_1 = r_1(\cos \theta_1 + i\sin \theta_1)" /> and <InlineMath math="z_2 = r_2(\cos \theta_2 + i\sin \theta_2)" /> are said to be **equal** if and only if:

1.  Their moduli are equal:

    <InlineMath math="r_1 = r_2" /> (or <InlineMath math="|z_1| = |z_2|" />)

2.  Their arguments are the same or differ by a multiple of <InlineMath math="2\pi" /> (or <InlineMath math="360^\circ" />):

    <InlineMath math="\theta_1 = \theta_2 + 2k\pi" /> or <InlineMath math="\theta_1 - \theta_2 = 2k\pi" /> for
    some integer <InlineMath math="k" />.

If we use the **Principal Argument** (with the range <InlineMath math="[0, 2\pi)" />), the second condition simplifies to: <InlineMath math="\text{Arg}(z_1) = \text{Arg}(z_2)" />.

### Checking for Equality

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

1.  <InlineMath math="z_1 = \sqrt{2}(\cos 45^\circ + i\sin 45^\circ)" /> and <InlineMath math="z_2 = \sqrt{2}(\cos 95^\circ + i\sin 95^\circ)" />
2.  <InlineMath math="z_1 = \cos 30^\circ + i\sin 30^\circ" /> and <InlineMath math="z_2 = \cos 390^\circ + i\sin 390^\circ" />

**Solution:**

1.  Consider:

    - Modulus: <InlineMath math="|z_1| = \sqrt{2}" /> and <InlineMath math="|z_2| = \sqrt{2}" />. (Equal)
    - Principal Argument: <InlineMath math="\text{Arg}(z_1) = 45^\circ" /> and <InlineMath math="\text{Arg}(z_2) = 95^\circ" />. (Different)

    Since their principal arguments are different (<InlineMath math="45^\circ \neq 95^\circ" />), then <InlineMath math="z_1 \neq z_2" />.

2.  Consider:

    - Modulus: <InlineMath math="|z_1| = 1" /> and <InlineMath math="|z_2| = 1" />. (Equal)
    - Arguments: <InlineMath math="\theta_1 = 30^\circ" /> and <InlineMath math="\theta_2 = 390^\circ" />.
    - Difference of arguments: <InlineMath math="\theta_1 - \theta_2 = 30^\circ - 390^\circ = -360^\circ" />.

    Since the difference of the arguments is a multiple of <InlineMath math="360^\circ" /> (<InlineMath math="-360^\circ = -1 \times 360^\circ" />), then <InlineMath math="z_1 = z_2" />.

    Alternatively, we can see that the Principal Argument of <InlineMath math="z_2" /> is <InlineMath math="390^\circ - 360^\circ = 30^\circ" />, which is the same as the Principal Argument of <InlineMath math="z_1" />.

## Exercise

Find the Principal Argument (in degrees) for the following complex numbers:

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

### Answer Key

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

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

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}" />
      <BlockMath math="\theta = \arctan(\sqrt{3}) = 60^\circ" />
    </div>

    Since <InlineMath math="60^\circ \in [0^\circ, 360^\circ)" />, then <InlineMath math="\text{Arg}(z) = 60^\circ" />.

2.  **For <InlineMath math="z = -i" />:**

    Can be written as <InlineMath math="z = 0 - 1i" />. The point <InlineMath math="(0, -1)" /> is on the negative imaginary axis.

    <div className="flex flex-col gap-4">
      <BlockMath math="\tan \theta = \frac{y}{x} = \frac{-1}{0} = \infty" />
      <BlockMath math="\theta = \arctan\left(\infty\right) = 90^\circ" />
    </div>

    The argument is <InlineMath math="270^\circ" /> (or <InlineMath math="-90^\circ" />).

    Since we are looking for the Principal Argument in the range <InlineMath math="[0^\circ, 360^\circ)" />, then <InlineMath math="\text{Arg}(z) = 270^\circ" />.