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URL: https://nakafa.com/en/subjects/mathematics/complex-number/complex-number-form
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Learn Cartesian, polar, and exponential forms of complex numbers. Convert between representations with Euler's formula and examples.
---
## Cartesian Form (Rectangular)
A complex number has the form $$z = x + iy$$, where $$x$$ is the real part and $$y$$ is the imaginary part. This form $$z = x + iy$$ is called the **Cartesian form** or rectangular form.
Visible text: A complex number has the form , where is the real part and is the imaginary part. This form is called the **Cartesian form** or rectangular form.
```math
z = x + iy
```
- $$x = \text{Re}(z)$$ (Real Part)
- $$y = \text{Im}(z)$$ (Imaginary Part)
Visible text: - (Real Part)
- (Imaginary Part)
We can also view the complex number $$z = x + iy$$ as an ordered pair $$(x, y)$$ on a coordinate plane. This special plane is called the **complex plane** or Argand diagram.
Visible text: We can also view the complex number as an ordered pair on a coordinate plane. This special plane is called the **complex plane** or Argand diagram.
- The horizontal axis ($$x$$-axis) represents the **real** part.
- The vertical axis ($$y$$-axis) represents the **imaginary** part.
Visible text: - The horizontal axis (-axis) represents the **real** part.
- The vertical axis (-axis) represents the **imaginary** part.
### Visualization on the Complex Plane
Let's try plotting some complex numbers on the complex plane. Each number $$z = x + iy$$ is plotted as the point $$(x, y)$$ and is usually represented as a vector (arrow) from the origin $$(0, 0)$$ to that point.
Visible text: Let's try plotting some complex numbers on the complex plane. Each number is plotted as the point and is usually represented as a vector (arrow) from the origin to that point.
Component: LineEquation
Props:
- title: Complex Numbers on the Complex Plane
- description: Visualization of several complex numbers as points and vectors on the complex plane.
- cameraPosition: [0, 0, 10]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 2, z: 0 },
],
color: getColor("ROSE"),
labels: [{ text: "z₁ = 3 + 2i", at: 1, offset: [0.5, 0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: -1, y: -1, z: 0 },
],
color: getColor("EMERALD"),
labels: [{ text: "z₂ = -1 - i", at: 1, offset: [-1, -0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2, y: 0, z: 0 },
],
color: getColor("VIOLET"),
labels: [{ text: "z₃ = 2", at: 1, offset: [0.5, 0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: 1, z: 0 },
],
color: getColor("AMBER"),
labels: [{ text: "z₄ = i", at: 1, offset: [-1, 0.5, 0] }],
},
]
## Polar Form
Besides Cartesian, there's another way to represent complex numbers: the **polar form**. This form uses:
1. **Modulus ($$r$$):** The distance from the origin $$(0, 0)$$ to the point $$(x, y)$$ on the complex plane. Its value is always non-negative.
2. **Argument ($$\theta$$):** The angle formed by the line from the origin to the point $$(x, y)$$ with the positive real axis. This angle is usually measured in radians or degrees.
Visible text: 1. **Modulus ():** The distance from the origin to the point on the complex plane. Its value is always non-negative.
2. **Argument ():** The angle formed by the line from the origin to the point with the positive real axis. This angle is usually measured in radians or degrees.
The relationship between Cartesian form ($$x, y$$) and Polar form ($$r, \theta$$) can be seen from basic trigonometry:
Visible text: The relationship between Cartesian form () and Polar form () can be seen from basic trigonometry:
Component: MathContainer
Children:
```math
x = r \cos \theta
```
```math
y = r \sin \theta
```
From this, we can find $$r$$ and $$\theta$$ if $$x$$ and $$y$$ are known:
Visible text: From this, we can find and if and are known:
Component: MathContainer
Children:
```math
r = |z| = \sqrt{x^2 + y^2}
```
```math
\tan \theta = \frac{y}{x}
```
When finding $$\theta$$ from $$\tan \theta$$, pay attention to the quadrant where the point $$(x, y)$$ lies to determine the correct angle.
Visible text: When finding from , pay attention to the quadrant where the point lies to determine the correct angle.
By substituting $$x$$ and $$y$$ into the Cartesian form, we get the **polar form**:
Visible text: By substituting and into the Cartesian form, we get the **polar form**:
Component: MathContainer
Children:
```math
z = x + iy = (r \cos \theta) + i(r \sin \theta)
```
```math
z = r (\cos \theta + i \sin \theta)
```
Sometimes, the form $$(\cos \theta + i \sin \theta)$$ is abbreviated as $$\text{cis } \theta$$.
Visible text: Sometimes, the form is abbreviated as .
```math
z = r \text{ cis} \theta
```
### Example: Conversion to Polar Form
Suppose we have $$z = 1 + i$$.
Visible text: Suppose we have .
- Real part $$x = 1$$.
- Imaginary part $$y = 1$$.
Visible text: - Real part .
- Imaginary part .
Find $$r$$:
Visible text: Find :
```math
r = \sqrt{1^2 + 1^2} = \sqrt{2}
```
Find $$\theta$$:
Visible text: Find :
```math
\tan \theta = \frac{1}{1} = 1
```
Since $$x$$ and $$y$$ are positive, the point $$(1, 1)$$ is in quadrant $$I$$. The angle whose $$\tan$$ is $$1$$ in quadrant $$I$$ is $$45^\circ$$ or $$\pi/4$$ radians.
Visible text: Since and are positive, the point is in quadrant . The angle whose is in quadrant is or radians.
So, the polar form is:
```math
z = \sqrt{2} (\cos 45^\circ + i \sin 45^\circ)
```
### Polar Form Exercise
Express the following complex numbers in polar form:
1. $$z = 1 + i\sqrt{3}$$
2. $$z = -i$$
Visible text: 1.
2.
**Answer Key:**
1. For $$z = 1 + i\sqrt{3}$$:
- Identify $$x = 1$$ and $$y = \sqrt{3}$$.
- Calculate the modulus $$r$$:
```math
r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
```
- Calculate the argument $$\theta$$:
```math
\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}
```
Since $$x$$ and $$y$$ are positive, the
point $$(1, \sqrt{3})$$ is in quadrant $$I$$
, so $$\theta = 60^\circ$$.
- Polar Form:
```math
z = 2(\cos 60^\circ + i \sin 60^\circ)
```
2. For $$z = -i$$:
- Identify $$x = 0$$ and $$y = -1$$.
- Calculate the modulus $$r$$:
```math
r = \sqrt{0^2 + (-1)^2} = \sqrt{1} = 1
```
- Determine the argument $$\theta$$:
The point $$(0, -1)$$ lies on the negative imaginary axis. The angle is $$\theta = 270^\circ$$ or it can also be written as $$\theta = -90^\circ$$.
- Polar Form (choose one angle):
```math
z = 1(\cos 270^\circ + i \sin 270^\circ)
```
or
```math
z = 1(\cos (-90^\circ) + i \sin (-90^\circ))
```
Visible text: 1. For :
- Identify and .
- Calculate the modulus :
- Calculate the argument :
Since and are positive, the
point is in quadrant
, so .
- Polar Form:
2. For :
- Identify and .
- Calculate the modulus :
- Determine the argument :
The point lies on the negative imaginary axis. The angle is or it can also be written as .
- Polar Form (choose one angle):
or
## Exponential Form
There's one more important form: the **exponential form**. This form comes from the magical **Euler's Formula**:
```math
e^{i\theta} = \cos \theta + i \sin \theta
```
Here, $$e \approx 2.71828...$$ is Euler's number (the base of the natural logarithm).
Visible text: Here, is Euler's number (the base of the natural logarithm).
If we substitute Euler's Formula into the polar form $$z = r(\cos \theta + i \sin \theta)$$, we get the **exponential form**:
Visible text: If we substitute Euler's Formula into the polar form , we get the **exponential form**:
```math
z = r e^{i\theta}
```
This form is very useful for multiplying and dividing complex numbers.
### Example: Conversion to Exponential Form
Take the previous examples:
1. For $$z = 1 + i$$, we already have the polar form $$\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)$$.
- Modulus $$r = \sqrt{2}$$.
- Argument $$\theta = 45^\circ = \pi/4$$ radians.
- Exponential Form:
```math
z = r e^{i\theta} = \sqrt{2} e^{i \pi/4}
```
2. For $$z = \sqrt{2}(\cos 315^\circ + i \sin 315^\circ)$$:
- Modulus $$r = \sqrt{2}$$.
- Argument $$\theta = 315^\circ$$. Convert to radians:
```math
315^\circ = 315 \times \frac{\pi}{180} = \frac{7 \times 45 \times \pi}{4 \times 45} = \frac{7\pi}{4}
```
Or use the negative angle $$315^\circ - 360^\circ = -45^\circ = -\pi/4$$ radians.
- Exponential Form (choose one angle):
```math
z = \sqrt{2} e^{i 7\pi/4}
```
or
```math
z = \sqrt{2} e^{-i \pi/4}
```
Visible text: 1. For , we already have the polar form .
- Modulus .
- Argument radians.
- Exponential Form:
2. For :
- Modulus .
- Argument . Convert to radians:
Or use the negative angle radians.
- Exponential Form (choose one angle):
or
### Exponential Form Exercise
Express the following complex numbers in exponential form (use radian angles):
1. $$z = 2(\cos 60^\circ + i \sin 60^\circ)$$
2. $$z = \cos 15^\circ + i \sin 15^\circ$$
Visible text: 1.
2.
**Answer Key:**
1. For $$z = 2(\cos 60^\circ + i \sin 60^\circ)$$:
- Modulus $$r = 2$$.
- Argument $$\theta = 60^\circ$$. Convert to radians:
```math
\theta = 60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3}
```
- Exponential Form:
```math
z = 2 e^{i \pi/3}
```
2. For $$z = \cos 15^\circ + i \sin 15^\circ$$:
- Modulus $$r = 1$$ (because there is no coefficient in front of $$\cos$$ and $$\sin$$).
- Argument $$\theta = 15^\circ$$. Convert to radians:
```math
\theta = 15^\circ = 15 \times \frac{\pi}{180} = \frac{\pi}{12}
```
- Exponential Form:
```math
z = e^{i \pi/12}
```
Visible text: 1. For :
- Modulus .
- Argument . Convert to radians:
- Exponential Form:
2. For :
- Modulus (because there is no coefficient in front of and ).
- Argument . Convert to radians:
- Exponential Form:
## Equality of Two Complex Numbers
Two complex numbers $$z_1 = x_1 + iy_1$$ and $$z_2 = x_2 + iy_2$$ are said to be **equal** if and only if their real parts are equal AND their imaginary parts are also equal.
Visible text: Two complex numbers and are said to be **equal** if and only if their real parts are equal AND their imaginary parts are also equal.
```math
z_1 = z_2 \quad \iff \quad x_1 = x_2 \text{ and} y_1 = y_2
```
### Equality Example
- $$z_1 = 3 - 2i$$ and $$z_2 = 4 + 2i$$ are **different**.
because $$\text{Re}(z_1) = 3 \neq \text{Re}(z_2) = 4$$ (even though $$|\text{Im}(z_1)| = |-2| = 2 = |\text{Im}(z_2)| = |2|$$, their imaginary signs differ).
- $$z_1 = -1 + i$$ and $$z_2 = i - 1$$ are **equal**.
because $$\text{Re}(z_1) = -1 = \text{Re}(z_2) = -1$$ and $$\text{Im}(z_1) = 1 = \text{Im}(z_2) = 1$$.
Visible text: - and are **different**.
because (even though , their imaginary signs differ).
- and are **equal**.
because and .
### Equality Exercise
Determine if the following pairs of complex numbers are equal or different:
1. $$z_1 = 4 - (-2i)$$ and $$z_2 = 4 + 2i$$
2. $$z_1 = i$$ and $$z_2 = 1 - i$$
3. $$z_1 = -1 + i$$ and $$z_2 = i + 1$$
Visible text: 1. and
2. and
3. and
**Answer Key:**
1. $$z_1 = 4 - (-2i) = 4 + 2i$$.
Thus, $$z_1$$ is **equal** to $$z_2 = 4 + 2i$$.
2. $$z_1 = 0 + 1i$$ and $$z_2 = 1 - 1i$$.
The real parts are different ($$0 \neq 1$$) and the imaginary parts are different ($$1 \neq -1$$).
Thus, $$z_1$$ is **different** from $$z_2$$.
3. $$z_1 = -1 + 1i$$ and $$z_2 = 1 + 1i$$.
The real parts are different ($$-1 \neq 1$$).
Thus, $$z_1$$ is **different** from $$z_2$$.
Visible text: 1. .
Thus, is **equal** to .
2. and .
The real parts are different () and the imaginary parts are different ().
Thus, is **different** from .
3. and .
The real parts are different ().
Thus, is **different** from .
## Exercises
1. **True or False.** Every real number is a complex number.
2. **True or False.** Complex numbers have $$3$$ forms: Cartesian, exponential, and logarithmic.
3. **True or False.** If the complex number $$z = 1 - 3i$$ is plotted on the complex plane, it lies in quadrant III.
4. Express the complex number $$2+2i$$ in polar and exponential forms.
5. Find the numbers $$x$$ and $$y$$ such that $$z_1 = x + 3i$$ and $$z_2 = 3 - yi$$ satisfy $$z_1 = z_2$$!
6. Find the solutions to the quadratic equation $$x^2 - 2x + 6 = 0$$!
7. Find the quadratic equation whose solutions are $$x_1 = 1 + i$$ and $$x_2 = 1 - i$$!
Visible text: 1. **True or False.** Every real number is a complex number.
2. **True or False.** Complex numbers have forms: Cartesian, exponential, and logarithmic.
3. **True or False.** If the complex number is plotted on the complex plane, it lies in quadrant III.
4. Express the complex number in polar and exponential forms.
5. Find the numbers and such that and satisfy !
6. Find the solutions to the quadratic equation !
7. Find the quadratic equation whose solutions are and !
### Answer Key
1. **True.** A real number $$a$$ can be written as $$a + 0i$$.
2. **False.** The common forms of complex numbers are Cartesian, Polar, and Exponential. The complex logarithm form exists but is not typically considered one of the three main forms studied at this level.
3. **False.** $$z = 1 - 3i$$ has a positive real part ($$x=1$$) and a negative imaginary part ($$y=-3$$). The point $$(1, -3)$$ lies in **Quadrant IV**.
4. For $$z = 2 + 2i$$:
- Calculate the modulus $$r$$:
```math
r = \sqrt{x^2+y^2} = \sqrt{2^2+2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}
```
- Calculate the argument $$\theta$$:
```math
\tan \theta = \frac{y}{x} = \frac{2}{2} = 1
```
Since $$x=2$$ and $$y=2$$ (both positive),
the point is in Quadrant I. Thus, $$\theta = 45^\circ$$ or $$\pi/4$$ radians.
- Polar Form:
```math
z = r(\cos \theta + i \sin \theta) = 2\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)
```
- Exponential Form:
```math
z = r e^{i\theta} = 2\sqrt{2} e^{i \pi/4}
```
5. For $$z_1 = x + 3i$$ to equal $$z_2 = 3 - yi$$, the real parts must be equal and the imaginary parts must be equal:
- Real Part: $$x = 3$$
- Imaginary Part: $$3 = -y \implies y = -3$$
So, $$x=3$$ and $$y=-3$$.
6. To solve $$x^2 - 2x + 6 = 0$$, use the quadratic formula:
```math
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
```
with $$a=1, b=-2, c=6$$:
```math
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}
```
```math
x = \frac{2 \pm \sqrt{4 - 24}}{2}
```
```math
x = \frac{2 \pm \sqrt{-20}}{2}
```
```math
x = \frac{2 \pm \sqrt{20}\sqrt{-1}}{2}
```
```math
x = \frac{2 \pm 2\sqrt{5}i}{2}
```
```math
x = 1 \pm i\sqrt{5}
```
The solutions are $$x_1 = 1 + i\sqrt{5}$$ and $$x_2 = 1 - i\sqrt{5}$$
.
7. If the roots of a quadratic equation are $$x_1 = 1 + i$$ and $$x_2 = 1 - i$$, the equation can be formed from $$(x - x_1)(x - x_2) = 0$$ or $$x^2 - (x_1 + x_2)x + (x_1 x_2) = 0$$.
- Calculate the sum of the roots:
```math
x_1 + x_2 = (1+i) + (1-i) = 1 + i + 1 - i = 2
```
- Calculate the product of the roots:
```math
x_1 x_2 = (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2
```
- Construct the quadratic equation:
```math
x^2 - (2)x + (2) = 0
```
```math
x^2 - 2x + 2 = 0
```
Visible text: 1. **True.** A real number can be written as .
2. **False.** The common forms of complex numbers are Cartesian, Polar, and Exponential. The complex logarithm form exists but is not typically considered one of the three main forms studied at this level.
3. **False.** has a positive real part () and a negative imaginary part (). The point lies in **Quadrant IV**.
4. For :
- Calculate the modulus :
- Calculate the argument :
Since and (both positive),
the point is in Quadrant I. Thus, or radians.
- Polar Form:
- Exponential Form:
5. For to equal , the real parts must be equal and the imaginary parts must be equal:
- Real Part:
- Imaginary Part:
So, and .
6. To solve , use the quadratic formula:
with :
The solutions are and
.
7. If the roots of a quadratic equation are and , the equation can be formed from or .
- Calculate the sum of the roots:
- Calculate the product of the roots:
- Construct the quadratic equation: