# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Properties of Principal Argument of Complex Numbers",
  description: "Learn principal argument rules for complex number multiplication and division in polar form with range adjustments and solved practice problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Complex Number",
};

## Argument Properties in Complex Number Operations

How does the argument behave when complex numbers are multiplied or divided?

These properties are very useful, especially when working with polar or exponential forms.

Suppose we have two complex numbers:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 = r_1(\cos \theta_1 + i\sin \theta_1)" />
  <BlockMath math="z_2 = r_2(\cos \theta_2 + i\sin \theta_2)" />
</div>

Where <InlineMath math="\theta_1" /> is one argument of <InlineMath math="z_1" /> and <InlineMath math="\theta_2" /> is one argument of <InlineMath math="z_2" />.

### Argument of Product

The argument of the product of two complex numbers (<InlineMath math="z_1 \times z_2" />) is the **sum** of the arguments of the individual complex numbers.

Mathematically, the relationship between the sets of arguments is:

<BlockMath math="\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)" />

This means if <InlineMath math="\theta_1" /> is an argument of <InlineMath math="z_1" /> and <InlineMath math="\theta_2" /> is an argument of <InlineMath math="z_2" />,

then <InlineMath math="\theta_1 + \theta_2" /> is one of the arguments of <InlineMath math="z_1 z_2" />.

**To find the Principal Argument <InlineMath math="\text{Arg}(z_1 z_2)" />:**

1.  Calculate <InlineMath math="\text{Arg}(z_1) + \text{Arg}(z_2)" />.
2.  If the result is already within the range <InlineMath math="[0^\circ, 360^\circ)" /> (or <InlineMath math="[0, 2\pi)" />), that is the Principal Argument.
3.  If the result is outside the range, add or subtract multiples of <InlineMath math="360^\circ" /> (or <InlineMath math="2\pi" />) to bring it into the range.

### Argument of Quotient

The argument of the quotient of two complex numbers (<InlineMath math="\frac{z_1}{z_2}" />, with <InlineMath math="z_2 \neq 0" />) is the **difference** between the argument of the numerator complex number (<InlineMath math="z_1" />) and the argument of the denominator complex number (<InlineMath math="z_2" />).

The relationship between the sets of arguments:

<BlockMath math="\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)" />

This means if <InlineMath math="\theta_1" /> is an argument of <InlineMath math="z_1" /> and <InlineMath math="\theta_2" /> is an argument of <InlineMath math="z_2" />,

then <InlineMath math="\theta_1 - \theta_2" /> is one of the arguments of <InlineMath math="\frac{z_1}{z_2}" />.

**To find the Principal Argument <InlineMath math="\text{Arg}\left(\frac{z_1}{z_2}\right)" />:**

1.  Calculate <InlineMath math="\text{Arg}(z_1) - \text{Arg}(z_2)" />.
2.  If the result is already within the range <InlineMath math="[0^\circ, 360^\circ)" /> (or <InlineMath math="[0, 2\pi)" />), that is the Principal Argument.
3.  If the result is outside the range, add or subtract multiples of <InlineMath math="360^\circ" /> (or <InlineMath math="2\pi" />) to bring it into the range.

## Using Argument Properties

Given two complex numbers:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 = 2(\cos 45^\circ + i\sin 45^\circ)" />
  <BlockMath math="z_2 = 3(\cos 95^\circ + i\sin 95^\circ)" />
</div>

Find the Principal Argument of <InlineMath math="z_1 \times z_2" /> and <InlineMath math="\frac{z_1}{z_2}" />.

**Solution:**

We know the Principal Arguments are:

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Arg}(z_1) = 45^\circ" />
  <BlockMath math="\text{Arg}(z_2) = 95^\circ" />
</div>

1.  **Argument of Product (<InlineMath math="z_1 \times z_2" />):**

    Sum of Principal Arguments:

    <BlockMath math="\text{Arg}(z_1) + \text{Arg}(z_2) = 45^\circ + 95^\circ = 140^\circ" />

    Since <InlineMath math="140^\circ" /> is already within the range <InlineMath math="[0^\circ, 360^\circ)" />, then:

    <BlockMath math="\text{Arg}(z_1 \times z_2) = 140^\circ" />

    The set of all arguments is <InlineMath math="\{140^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}" />

2.  **Argument of Quotient (<InlineMath math="\frac{z_1}{z_2}" />):**

    Difference of Principal Arguments:

    <BlockMath math="\text{Arg}(z_1) - \text{Arg}(z_2) = 45^\circ - 95^\circ = -50^\circ" />

    Since <InlineMath math="-50^\circ" /> is outside the range <InlineMath math="[0^\circ, 360^\circ)" />, we need to add <InlineMath math="360^\circ" />:

    <BlockMath math="-50^\circ + 360^\circ = 310^\circ" />

    Therefore:

    <BlockMath math="\text{Arg}\left(\frac{z_1}{z_2}\right) = 310^\circ" />

    The set of all arguments is <InlineMath math="\{-50^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}" />, which is the same as <InlineMath math="\{310^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}" />.

## Exercise

Given <InlineMath math="z_a = 4(\cos 120^\circ + i\sin 120^\circ)" /> and <InlineMath math="z_b = 2(\cos 50^\circ + i\sin 50^\circ)" />. Find:

1.  <InlineMath math="\text{Arg}(z_a \times z_b)" />
2.  <InlineMath math="\text{Arg}\left(\frac{z_a}{z_b}\right)" />

### Answer Key

Given <InlineMath math="\text{Arg}(z_a) = 120^\circ" /> and <InlineMath math="\text{Arg}(z_b) = 50^\circ" />.

1.  **Argument of Product:**

    <BlockMath math="\text{Arg}(z_a) + \text{Arg}(z_b) = 120^\circ + 50^\circ = 170^\circ" />

    Since <InlineMath math="170^\circ \in [0^\circ, 360^\circ)" />, then <InlineMath math="\text{Arg}(z_a z_b) = 170^\circ" />.

2.  **Argument of Quotient:**

    <BlockMath math="\text{Arg}(z_a) - \text{Arg}(z_b) = 120^\circ - 50^\circ = 70^\circ" />

    Since <InlineMath math="70^\circ \in [0^\circ, 360^\circ)" />, then <InlineMath math="\text{Arg}\left(\frac{z_a}{z_b}\right) = 70^\circ" />.