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Find unique principal argument Arg(z) in range [0°,360°). Convert infinite angle possibilities to one standard value for complex number equality.

---

## Understanding the Principal Argument

The argument $$\theta$$ of a complex number $$z = x + iy$$ is the angle formed by the vector $$z$$ with the positive real axis.

Visible text: The argument of a complex number is the angle formed by the vector with the positive real axis.

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

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

Visible text: If is an argument of , then (where is an integer: ) is also an argument of , because adding multiples of or radians results in the same angle on the complex plane.

**Example:**

The angles $$45^\circ$$, $$405^\circ$$ ($$45^\circ + 360^\circ$$), and $$-315^\circ$$ ($$45^\circ - 360^\circ$$) all indicate the same direction.

Visible text: The angles , (), and () 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 $$z = r(\cos \theta + i\sin \theta)$$ is the unique value of the argument $$\theta$$ that satisfies a specific range.

Visible text: The Principal Argument of a complex number is the unique value of the argument that satisfies a specific range.

**Principal Argument** (denoted $$\text{Arg}(z)$$) is defined as the argument $$\theta$$ that satisfies:

Visible text: **Principal Argument** (denoted ) is defined as the argument that satisfies:

```math
0 \leq \theta < 2\pi \quad \text{or} \quad 0^\circ \leq \theta < 360^\circ
```

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

Visible text: Note: Other definitions sometimes use the range or . 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 $$[0, 2\pi)$$ or $$[0^\circ, 360^\circ)$$.

Visible text: Determining the Principal Argument is the same as finding the regular argument, but we need to ensure the final result is within the range or .

### Finding the Principal Argument

1.  **Find the Principal Argument of $$z = 1 + i$$**

    The point $$(1, 1)$$ is in Quadrant $$I$$.

    <MathContainer>
      
    
    ```math
    \tan \theta = \frac{y}{x} = \frac{1}{1} = 1
    ```

      
    
    ```math
    \theta = \arctan(1) = 45^\circ
    ```

    </MathContainer>

    Since $$45^\circ$$ is already within the range $$[0^\circ, 360^\circ)$$, the Principal Argument is:

    
    
    ```math
    \text{Arg}(z) = 45^\circ \text{ or} \frac{\pi}{4} \text{ radians}
    ```

2.  **Find the Principal Argument of $$z = \sqrt{3} + i$$**

    The point $$(\sqrt{3}, 1)$$ is in Quadrant $$I$$.

    <MathContainer>
      
    
    ```math
    \tan \theta = \frac{y}{x} = \frac{1}{\sqrt{3}}
    ```

      
    
    ```math
    \theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ
    ```

    </MathContainer>

    Since $$30^\circ$$ is already within the range $$[0^\circ, 360^\circ)$$, the Principal Argument is:

    
    
    ```math
    \text{Arg}(z) = 30^\circ \text{ or} \frac{\pi}{6} \text{ radians}
    ```

Visible text: 1. **Find the Principal Argument of **

 The point is in Quadrant .

 <MathContainer>
 
 

 
 

 </MathContainer>

 Since is already within the range , the Principal Argument is:

 
 

2. **Find the Principal Argument of **

 The point is in Quadrant .

 <MathContainer>
 
 

 
 

 </MathContainer>

 Since is already within the range , the Principal Argument is:

Component: ContentBlock
Children:
Component: LineEquation
Props:
- title: Principal Argument Visualization
- description: Showing vectors for $$z_1=1+i$$ and
$$z_2=\sqrt{3}+i$$, along with their Principal Arguments
($$45^\circ$$ and $$30^\circ$$
).
  Visible text: Showing vectors for and
, along with their Principal Arguments
( and 
).
- 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"),
},
]

## Equality of Two Complex Numbers in Polar Form

Two complex numbers $$z_1 = r_1(\cos \theta_1 + i\sin \theta_1)$$ and $$z_2 = r_2(\cos \theta_2 + i\sin \theta_2)$$ are said to be **equal** if and only if:

Visible text: Two complex numbers and are said to be **equal** if and only if:

1.  Their moduli are equal:

    $$r_1 = r_2$$ (or $$|z_1| = |z_2|$$)

2.  Their arguments are the same or differ by a multiple of $$2\pi$$ (or $$360^\circ$$):

    $$\theta_1 = \theta_2 + 2k\pi$$ or $$\theta_1 - \theta_2 = 2k\pi$$ for
    some integer $$k$$.

Visible text: 1. Their moduli are equal:

 (or )

2. Their arguments are the same or differ by a multiple of (or ):

 or for
 some integer .

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

Visible text: If we use the **Principal Argument** (with the range ), the second condition simplifies to: .

### Checking for Equality

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

1.  $$z_1 = \sqrt{2}(\cos 45^\circ + i\sin 45^\circ)$$ and $$z_2 = \sqrt{2}(\cos 95^\circ + i\sin 95^\circ)$$
2.  $$z_1 = \cos 30^\circ + i\sin 30^\circ$$ and $$z_2 = \cos 390^\circ + i\sin 390^\circ$$

Visible text: 1. and 
2. and

**Solution:**

1.  Consider:

    - Modulus: $$|z_1| = \sqrt{2}$$ and $$|z_2| = \sqrt{2}$$. (Equal)
    - Principal Argument: $$\text{Arg}(z_1) = 45^\circ$$ and $$\text{Arg}(z_2) = 95^\circ$$. (Different)

    Since their principal arguments are different ($$45^\circ \neq 95^\circ$$), then $$z_1 \neq z_2$$.

2.  Consider:

    - Modulus: $$|z_1| = 1$$ and $$|z_2| = 1$$. (Equal)
    - Arguments: $$\theta_1 = 30^\circ$$ and $$\theta_2 = 390^\circ$$.
    - Difference of arguments: $$\theta_1 - \theta_2 = 30^\circ - 390^\circ = -360^\circ$$.

    Since the difference of the arguments is a multiple of $$360^\circ$$ ($$-360^\circ = -1 \times 360^\circ$$), then $$z_1 = z_2$$.

    Alternatively, we can see that the Principal Argument of $$z_2$$ is $$390^\circ - 360^\circ = 30^\circ$$, which is the same as the Principal Argument of $$z_1$$.

Visible text: 1. Consider:

 - Modulus: and . (Equal)
 - Principal Argument: and . (Different)

 Since their principal arguments are different (), then .

2. Consider:

 - Modulus: and . (Equal)
 - Arguments: and .
 - Difference of arguments: .

 Since the difference of the arguments is a multiple of (), then .

 Alternatively, we can see that the Principal Argument of is , which is the same as the Principal Argument of .

## Exercise

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

1.  $$1 + \sqrt{3}i$$
2.  $$-i$$

Visible text: 1. 
2.

### Answer Key

1.  **For $$z = 1 + \sqrt{3}i$$:**

    The point $$(1, \sqrt{3})$$ is in Quadrant $$I$$.

    <MathContainer>
      
    
    ```math
    \tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}
    ```

      
    
    ```math
    \theta = \arctan(\sqrt{3}) = 60^\circ
    ```

    </MathContainer>

    Since $$60^\circ \in [0^\circ, 360^\circ)$$, then $$\text{Arg}(z) = 60^\circ$$.

2.  **For $$z = -i$$:**

    Can be written as $$z = 0 - 1i$$. The point $$(0, -1)$$ is on the negative imaginary axis.

    <MathContainer>
      
    
    ```math
    \tan \theta = \frac{y}{x} = \frac{-1}{0} = \infty
    ```

      
    
    ```math
    \theta = \arctan\left(\infty\right) = 90^\circ
    ```

    </MathContainer>

    The argument is $$270^\circ$$ (or $$-90^\circ$$).

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

Visible text: 1. **For :**

 The point is in Quadrant .

 <MathContainer>
 
 

 
 

 </MathContainer>

 Since , then .

2. **For :**

 Can be written as . The point is on the negative imaginary axis.

 <MathContainer>
 
 

 
 

 </MathContainer>

 The argument is (or ).

 Since we are looking for the Principal Argument in the range , then .