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Learn principal argument rules for complex number multiplication and division in polar form with range adjustments and solved practice problems.

---

## 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:

Component: MathContainer
Children:

```math
z_1 = r_1(\cos \theta_1 + i\sin \theta_1)
```

```math
z_2 = r_2(\cos \theta_2 + i\sin \theta_2)
```

Where $$\theta_1$$ is one argument of $$z_1$$ and $$\theta_2$$ is one argument of $$z_2$$.

Visible text: Where is one argument of and is one argument of .

### Argument of Product

The argument of the product of two complex numbers ($$z_1 \times z_2$$) is the **sum** of the arguments of the individual complex numbers.

Visible text: The argument of the product of two complex numbers () is the **sum** of the arguments of the individual complex numbers.

Mathematically, the relationship between the sets of arguments is:

```math
\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)
```

This means if $$\theta_1$$ is an argument of $$z_1$$ and $$\theta_2$$ is an argument of $$z_2$$,

Visible text: This means if is an argument of and is an argument of ,

then $$\theta_1 + \theta_2$$ is one of the arguments of $$z_1 z_2$$.

Visible text: then is one of the arguments of .

**To find the Principal Argument $$\text{Arg}(z_1 z_2)$$:**

Visible text: **To find the Principal Argument :**

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

Visible text: 1. Calculate .
2. If the result is already within the range (or ), that is the Principal Argument.
3. If the result is outside the range, add or subtract multiples of (or ) to bring it into the range.

### Argument of Quotient

The argument of the quotient of two complex numbers ($$\frac{z_1}{z_2}$$, with $$z_2 \neq 0$$) is the **difference** between the argument of the numerator complex number ($$z_1$$) and the argument of the denominator complex number ($$z_2$$).

Visible text: The argument of the quotient of two complex numbers (, with ) is the **difference** between the argument of the numerator complex number () and the argument of the denominator complex number ().

The relationship between the sets of arguments:

```math
\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)
```

This means if $$\theta_1$$ is an argument of $$z_1$$ and $$\theta_2$$ is an argument of $$z_2$$,

Visible text: This means if is an argument of and is an argument of ,

then $$\theta_1 - \theta_2$$ is one of the arguments of $$\frac{z_1}{z_2}$$.

Visible text: then is one of the arguments of .

**To find the Principal Argument $$\text{Arg}\left(\frac{z_1}{z_2}\right)$$:**

Visible text: **To find the Principal Argument :**

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

Visible text: 1. Calculate .
2. If the result is already within the range (or ), that is the Principal Argument.
3. If the result is outside the range, add or subtract multiples of (or ) to bring it into the range.

## Using Argument Properties

Given two complex numbers:

Component: MathContainer
Children:

```math
z_1 = 2(\cos 45^\circ + i\sin 45^\circ)
```

```math
z_2 = 3(\cos 95^\circ + i\sin 95^\circ)
```

Find the Principal Argument of $$z_1 \times z_2$$ and $$\frac{z_1}{z_2}$$.

Visible text: Find the Principal Argument of and .

**Solution:**

We know the Principal Arguments are:

Component: MathContainer
Children:

```math
\text{Arg}(z_1) = 45^\circ
```

```math
\text{Arg}(z_2) = 95^\circ
```

1.  **Argument of Product ($$z_1 \times z_2$$):**

    Sum of Principal Arguments:

    
    
    ```math
    \text{Arg}(z_1) + \text{Arg}(z_2) = 45^\circ + 95^\circ = 140^\circ
    ```

    Since $$140^\circ$$ is already within the range $$[0^\circ, 360^\circ)$$, then:

    
    
    ```math
    \text{Arg}(z_1 \times z_2) = 140^\circ
    ```

    The set of all arguments is $$\{140^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}$$

2.  **Argument of Quotient ($$\frac{z_1}{z_2}$$):**

    Difference of Principal Arguments:

    
    
    ```math
    \text{Arg}(z_1) - \text{Arg}(z_2) = 45^\circ - 95^\circ = -50^\circ
    ```

    Since $$-50^\circ$$ is outside the range $$[0^\circ, 360^\circ)$$, we need to add $$360^\circ$$:

    
    
    ```math
    -50^\circ + 360^\circ = 310^\circ
    ```

    Therefore:

    
    
    ```math
    \text{Arg}\left(\frac{z_1}{z_2}\right) = 310^\circ
    ```

    The set of all arguments is $$\{-50^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}$$, which is the same as $$\{310^\circ + k \cdot 360^\circ : k \in \mathbb{Z}\}$$.

Visible text: 1. **Argument of Product ():**

 Sum of Principal Arguments:

 
 

 Since is already within the range , then:

 
 

 The set of all arguments is 

2. **Argument of Quotient ():**

 Difference of Principal Arguments:

 
 

 Since is outside the range , we need to add :

 
 

 Therefore:

 
 

 The set of all arguments is , which is the same as .

## Exercise

Given $$z_a = 4(\cos 120^\circ + i\sin 120^\circ)$$ and $$z_b = 2(\cos 50^\circ + i\sin 50^\circ)$$. Find:

Visible text: Given and . Find:

1.  $$\text{Arg}(z_a \times z_b)$$
2.  $$\text{Arg}\left(\frac{z_a}{z_b}\right)$$

Visible text: 1. 
2.

### Answer Key

Given $$\text{Arg}(z_a) = 120^\circ$$ and $$\text{Arg}(z_b) = 50^\circ$$.

Visible text: Given and .

1.  **Argument of Product:**

    
    
    ```math
    \text{Arg}(z_a) + \text{Arg}(z_b) = 120^\circ + 50^\circ = 170^\circ
    ```

    Since $$170^\circ \in [0^\circ, 360^\circ)$$, then $$\text{Arg}(z_a z_b) = 170^\circ$$.

2.  **Argument of Quotient:**

    
    
    ```math
    \text{Arg}(z_a) - \text{Arg}(z_b) = 120^\circ - 50^\circ = 70^\circ
    ```

    Since $$70^\circ \in [0^\circ, 360^\circ)$$, then $$\text{Arg}\left(\frac{z_a}{z_b}\right) = 70^\circ$$.

Visible text: 1. **Argument of Product:**

 
 

 Since , then .

2. **Argument of Quotient:**

 
 

 Since , then .