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URL: https://nakafa.com/en/subjects/mathematics/complex-number/modulus-argument-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/complex-number/modulus-argument-complex-numbers/en.mdx
Calculate modulus |z| = √(x²+y²) and argument θ using quadrant rules. Learn distance and angle measurements for polar form conversions.
---
## What are Modulus and Argument?
A complex number $$z = x + iy$$ can be represented as a point $$(x, y)$$ 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 $$(0, 0)$$ to the point $$(x, y)$$.
Visible text: A complex number can be represented as a point 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 to the point .
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 $$z = x + iy$$, written as $$|z|$$, is the **distance** from the origin $$(0,0)$$ to the point $$(x, y)$$ on the complex plane. This is the same as the **length of the vector** representing $$z$$.
Visible text: The **Modulus** of a complex number , written as , is the **distance** from the origin to the point on the complex plane. This is the same as the **length of the vector** representing .
Component: LineEquation
Props:
- title: Modulus Visualization
- description: The modulus $$|z|$$ is the length of the vector from the
origin to the point $$z$$. We can see it as the
hypotenuse of a right-angled triangle.
Visible text: The modulus is the length of the vector from the
origin to the point . 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 ($$x$$), the imaginary part ($$y$$), and the modulus ($$|z|$$) as the hypotenuse.
Visible text: To calculate the modulus, we can use the Pythagorean Theorem on the right-angled triangle formed by the real part (), the imaginary part (), and the modulus () as the hypotenuse.
**Definition of Modulus:**
The modulus of the complex number $$z = x + iy$$ is:
Visible text: The modulus of the complex number is:
```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 $$z_1 = 3 + 4i$$**, with $$x=3, y=4$$
```math
|z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
```
2. **Find the modulus of $$z_2 = -1 - 2i$$**, with $$x=-1, y=-2$$
```math
|z_2| = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
```
3. **Find the modulus of $$z_3 = 5$$**, with $$x=5, y=0$$
```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 $$z_4 = -2i$$**, with $$x=0, y=-2$$
```math
|z_4| = \sqrt{0^2 + (-2)^2} = \sqrt{4} = 2
```
Visible text: 1. **Find the modulus of **, with
2. **Find the modulus of **, with
3. **Find the modulus of **, with
(The modulus of a real number is its absolute value).
4. **Find the modulus of **, with
## Argument of a Complex Number
The **Argument** of a non-zero complex number $$z = x + iy$$, written as $$\arg(z)$$ or $$\theta$$, is the **angle** formed by the vector $$z$$ with the **positive real axis** on the complex plane. This angle is usually measured in radians or degrees.
Visible text: The **Argument** of a non-zero complex number , written as or , is the **angle** formed by the vector 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:
Component: MathContainer
Children:
```math
x = |z| \cos \theta
```
```math
y = |z| \sin \theta
```
```math
\tan \theta = \frac{y}{x} \quad (\text{if } x \neq 0)
```
To find $$\theta$$, we can use the arctangent function (or $$\tan^{-1}$$):
Visible text: To find , we can use the arctangent function (or ):
```math
\theta = \arctan\left(\frac{y}{x}\right)
```
Calculators usually give the $$\arctan$$ value in the range $$(-90^\circ, 90^\circ)$$ or $$(-\pi/2, \pi/2)$$. We need to **consider the quadrant** where the point $$(x, y)$$ lies to determine the correct argument.
Visible text: Calculators usually give the value in the range or . We need to **consider the quadrant** where the point lies to determine the correct argument.
- **Quadrant $$I$$** ($$x>0, y>0$$):
```math
\theta = \arctan(y/x)
```
- **Quadrant $$II$$** ($$x<0, y>0$$):
```math
\theta = 180^\circ + \arctan(y/x) \text{ or} \theta = \pi + \arctan(y/x)
```
- **Quadrant $$III$$** ($$x<0, y<0$$):
```math
\theta = 180^\circ + \arctan(y/x) \text{ or} \theta = \pi + \arctan(y/x)
```
- **Quadrant $$IV$$** ($$x>0, y<0$$):
```math
\theta = 360^\circ + \arctan(y/x) \text{ or} \theta = 2\pi + \arctan(y/x)
```
or simply
```math
\theta = \arctan(y/x)
```
if a negative angle is desired
Visible text: - **Quadrant ** ():
- **Quadrant ** ():
- **Quadrant ** ():
- **Quadrant ** ():
or simply
if a negative angle is desired
Often, we are interested in the **Principal Argument** (written $$\text{Arg}(z)$$), which is the argument value in the interval $$(-180^\circ, 180^\circ]$$ or $$(-\pi, \pi]$$.
Visible text: Often, we are interested in the **Principal Argument** (written ), which is the argument value in the interval or .
### Calculating the Argument
1. **Find the argument of $$z_1 = 1 + i$$**
The point $$(1, 1)$$ is in Quadrant $$I$$.
```math
\tan \theta = \frac{y}{x} = \frac{1}{1} = 1
```
```math
\theta = \arctan(1) = 45^\circ \text{ or} \frac{\pi}{4} \text{ radians}
```
2. **Find the argument of $$z_2 = -\sqrt{3} + i$$**
The point $$(-\sqrt{3}, 1)$$ is in Quadrant $$II$$.
```math
\tan \theta = \frac{y}{x} = \frac{1}{-\sqrt{3}}
```
```math
\text{Base angle} = \arctan\left(\left|\frac{1}{-\sqrt{3}}\right|\right) = \arctan\left(\frac{1}{\sqrt{3}}\right) = 30^\circ
```
```math
\theta = 180^\circ - 30^\circ = 150^\circ \text{ or} \frac{5\pi}{6} \text{ radians}
```
(Because it's in Quadrant $$II$$, we use $$180^\circ - \text{base angle}$$)
3. **Find the argument of $$z_3 = -1 - i\sqrt{3}$$**
The point $$(-1, -\sqrt{3})$$ is in Quadrant $$III$$.
```math
\tan \theta = \frac{y}{x} = \frac{-\sqrt{3}}{-1} = \sqrt{3}
```
```math
\text{Base angle} = \arctan(\sqrt{3}) = 60^\circ
```
```math
\theta = 180^\circ + 60^\circ = 240^\circ \text{ or} \frac{4\pi}{3} \text{ radians}
```
(Because it's in Quadrant $$III$$, we use $$180^\circ + \text{base angle}$$
. Principal Argument: $$-120^\circ$$ or $$-2\pi/3$$
).
4. **Find the argument of $$z_4 = 3 - 3i$$**
The point $$(3, -3)$$ is in Quadrant $$IV$$.
```math
\tan \theta = \frac{y}{x} = \frac{-3}{3} = -1
```
```math
\text{Base angle} = \arctan(|-1|) = \arctan(1) = 45^\circ
```
```math
\theta = 360^\circ - 45^\circ = 315^\circ \text{ or} \frac{7\pi}{4} \text{ radians}
```
(Because it's in Quadrant $$IV$$, we use $$360^\circ - \text{base angle}$$
. Principal Argument: $$-45^\circ$$ or $$-\pi/4$$
).
Visible text: 1. **Find the argument of **
The point is in Quadrant .
2. **Find the argument of **
The point is in Quadrant .
(Because it's in Quadrant , we use )
3. **Find the argument of **
The point is in Quadrant .
(Because it's in Quadrant , we use
. Principal Argument: or
).
4. **Find the argument of **
The point is in Quadrant .
(Because it's in Quadrant , we use
. Principal Argument: or
).
## Exercise
Find the modulus and argument (in degrees) of the following complex numbers:
1. $$z_a = 2 + 2i$$
2. $$z_b = -4$$
3. $$z_c = -i$$
Visible text: 1.
2.
3.
### Answer Key
1. **For $$z_a = 2 + 2i$$:**
$$x=2, y=2$$ (Quadrant $$I$$) Modulus:
```math
|z_a| = \sqrt{2^2 + 2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}
```
Argument:
```math
\tan \theta = \frac{2}{2} = 1 \implies \theta = \arctan(1) = 45^\circ
```
2. **For $$z_b = -4$$:**
$$z_b = -4 + 0i$$. $$x=-4, y=0$$ (Negative
real axis) Modulus:
```math
|z_b| = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4
```
Argument: The point is on the negative real axis.
```math
\theta = 180^\circ
```
3. **For $$z_c = -i$$:**
$$z_c = 0 - 1i$$. $$x=0, y=-1$$ (Negative
imaginary axis) Modulus:
```math
|z_c| = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1
```
Argument: The point is on the negative imaginary axis.
```math
\theta = 270^\circ
```
or $$-90^\circ$$ (Principal Argument).
Visible text: 1. **For :**
(Quadrant ) Modulus:
Argument:
2. **For :**
. (Negative
real axis) Modulus:
Argument: The point is on the negative real axis.
3. **For :**
. (Negative
imaginary axis) Modulus:
Argument: The point is on the negative imaginary axis.
or (Principal Argument).