Complex Numbers & Vectors: A Practical Introduction with Real-World Applications

1 Introduction to Complex Numbers

Complex numbers are mathematical entities composed of a real part and an imaginary part. Represented as a + bi, where a is the real component, b is the imaginary component, and i is the square root of -1; i = \sqrt{-1}, i^2 =-1, they extend our understanding beyond real numbers.

2 The Basics:

Basic operations with complex numbers:
- Sum: (a + bi) + (c + di) = (a + c) + (b + d)i
- Difference: (a + bi) - (c + di) = (a - c) + (b - d)i
- Product: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
- Quotient: \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}
Magnitude (or modulus or length or distance or … ) of a complex number z = a + bi is given by:
- |z| = \sqrt{a^2 + b^2}
The conjugate of a complex number z = a + bi is:
- \overline{z} = a - bi
Euler’s formula relates complex numbers, exponentials, and trigonometric functions:
- |z|e^{i\theta} = |z|(\cos(\theta) + i\sin(\theta))
- e^{i\theta} = \cos(\theta) + i\sin(\theta)

3 Warm Up:

Given two complex numbers: z_1 = 3 + 4i z_2 = 1 - 2i

Find their sum, difference, product, and quotient.
Find the magnitude (or modulus) of both z_1 and z_2.
Find the conjugate of both z_1 and z_2.
Using the Euler’s formula, express z_1 , z_2 and z_1\times z_2 in the form r \times e^{i\theta}, where r is the magnitude and \theta is the argument (angle) of the complex number. What do you notice?

4 Real World Applications:

These next four examples underscore the versatility of complex numbers, bridging diverse domains from banking to aviation to electrical circuits. They emphasize the interplay between tangible (real) and transitional (imaginary) components, illustrating the profound connections woven through various real-world scenarios.

5 Electrical Engineering Problem: Complex Numbers in AC Circuits.

In electrical engineering, complex numbers offer a framework to analyze alternating current (AC) circuits. They capture both magnitude (real part) and phase (imaginary part) of sinusoidal waveforms. By encompassing these two aspects, complex numbers can provide a comprehensive understanding of AC phenomena.

Electrical Engineering Problem: In an electrical circuit, the voltage V and current I are given by complex numbers. An engineer measures the voltage as V = 5 + 3i volts and the current as I = 2 + 4i amperes. The power P in the circuit is given by P = V \times I^*, where I^* is the conjugate of I. What is the power in the circuit?

Problem Statement: In an AC circuit, voltage V and current I are represented by complex numbers. Given V = 5 + 3i volts and I = 2 + 4i amperes, calculate the complex power P = V \times I^*.

Solution: In electrical circuits, especially those with alternating current (AC), power has both real and imaginary components. The complex power S, given in “volt-amperes” (VA), is a combination of:

Real Power (P): This is the power that actually does work, like lighting a bulb or turning a motor. It’s what you’re billed for on your electricity bill, and it’s measured in watts (W). This component is the real part of the complex power.
Reactive Power (Q): This doesn’t do any real work but is necessary for the operation of devices like motors and transformers. It’s related to the storage and return of energy to the source. It’s measured in “volt-amperes reactive” (VAR). This component is the imaginary part of the complex power.

The complex power representation S = P + jQ (where j is the imaginary unit, often used in electrical engineering instead of i) helps engineers account for both the work-doing capability and the energy storage aspects of electrical systems.

Given a complex power of 22 - 14i watts (or VA to be more precise):

The real power P = 22 W means there’s a power loss or consumption of 2 watts. The negative sign is unusual for simple circuit problems and might indicate power being supplied back into the system, but without full context, it’s hard to say definitively.
The reactive power Q = -14 VAR indicates that 14volt-amperes of power is being cyclically stored and returned to the system. This is typical for circuits with inductive or capacitive elements.

reactive power (often denoted by Q) can be positive or negative, depending on whether the load is capacitive or inductive:
- Inductive loads (like motors) have a positive reactive power. This means they absorb reactive power from the circuit.
- Capacitive loads (like capacitors) have a negative reactive power. This indicates they supply reactive power back to the circuit.

To visualize this, imagine a waterwheel in a stream. The real power is the water that turns the wheel and does work (grinding grain, for example). The reactive power is like the swirling eddies and whirlpools in the stream: they don’t turn the wheel, but they are still part of the water’s overall movement.

In practical terms, while consumers are usually billed only for real power, both real and reactive powers are crucial for utilities and engineers to manage and understand, ensuring the efficient and safe operation of the power system.

So to summarize; First, we find the conjugate of I, denoted as I^* = 2 - 4i.

Next, we compute the complex power: P = (5 + 3i)(2 - 4i) , P = 22 - 14i watts.

Physically, this denotes: - Real Power P = 22 W: The effective power performing work. - Reactive Power Q = -14 VAR: Represents cyclically stored and returned power, crucial for devices like capacitors (negitive imaginary parts) and inductors (positive imaginary parts).

This concept is parallel to a bank account with actual balance and pending transactions. While the real balance reflects the money you can use, pending transactions are amounts in transition, not immediately accessible but still relevant.

6 Flight Problem: Complex Numbers as Position.

Flight Problem: Two airplanes start from the same airport. The first airplane flies 4 km North and then 3 km East, represented by the complex number z_1 = 3 + 4i. The second airplane flies 2 km South and then 1 km East, represented by the complex number z_2 = 1 - 2i. Find the distance between the two airplanes after their respective flights.

Problem Statement: Two planes start from an airport. One flies 4 km North then 3 km East (represented by z_1 = 3 + 4i. The other flies 2 km South and 1 km East (given by z_2 = 1 - 2i. Determine their distance apart post-flight.

Solution: First, we find their relative position in the complex plane: z = z_1 - z_2, z = (3 + 4i) - (1 - 2i),z = 2 + 6i

The distance between them is the magnitude of z: d = |z| = \sqrt{2^2 + 6^2} = 2\sqrt{10} km.

This scenario mirrors the electrical engineering concept where spatial positions (akin to voltages) can be added or subtracted, producing resultant vectors (or combined voltages).

7 Airplane Rotation Problem: Navigation with Complex Numbers

Rotation Problem: Two airplanes are flying relative to an observation tower at an airport. The first airplane is located at a position represented by the complex number z = 4 + 2i kilometers, where the real part represents the eastward distance and the imaginary part represents the northward distance from the tower. Due to incoming weather, the air traffic controller instructs the airplane to change its course by rotating its position pi/4 radians counterclockwise about the observation tower without changing its distance from the tower. Where will the airplane be after adjusting its course?

Problem Statement: An airplane is positioned relative to an observation tower at an airport, represented by the complex number z = 4 + 2i kilometers. The air traffic controller instructs the airplane to rotate its position pi/4 radians counterclockwise about the observation tower. Where will the airplane be after adjusting?

Solution: When you multiply two complex numbers, you multiply their lengths (distances from the origin) and add their angles (the angles they make with the positive real axis).

For example, consider a complex number z and the point representing e^{i\theta} on the unit circle. When you multiply z by e^{i\theta}, you’re essentially rotating z by an angle \theta around the origin. This is because the length of e^{i\theta} is 1 (it’s on the unit circle), so the magnitude of z remains unchanged. However, the angle of z increases by \theta due to the addition of angles during multiplication. Thus, multiplying a complex number by e^{i\theta} rotates the number counterclockwise by an angle \theta on the complex plane without changing its distance from the origin. Hence if z' = z \times e^{i\theta} then |z'| = |z|. This is why rotating a complex number counterclockwise by an angle \theta is equivalent to multiplying it by e^{i\theta}.

In the context of the Electrical Engineering Problem. When we multiply two complex numbers, we combining their magnitudes (lengths) and adding their angles (phases). In the context of the complex plane, this multiplication can be visualized as a rotation and scaling operation.

So for rotations in the complex plane, the formula for a complex number z rotated by an angle \theta is:

z' = z \times e^{i\theta}

Here, e^{i\theta} is a complex number on the unit circle with an angle \theta to the positive real axis. The magnitude of e^{i\theta} is 1, so when you multiply z by it, the magnitude of z remains unchanged, but its angle increases by \theta.

Using Euler’s formula, e^{i\theta} can be expanded as:

e^{i\theta} = \cos(\theta) + i\sin(\theta)

So first, we represent the rotation of \pi/4 radians using Euler’s formula:

e^{i\pi/4} = \cos(\pi/4) + i\sin(\pi/4)

e^{i\pi/4}=\frac{\sqrt{2}}{2}\ + i\frac{\sqrt{2}}{2}

Then, to find the new position after rotation, multiply the original complex number z by e^{i\pi/4}:

z' = z \times e^{i\pi/4}

z' = (4 + 2i) \times (\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2})

Expanding this multiplication: z' = 4 \times \frac{\sqrt{2}}{2} + 4 \times i\frac{\sqrt{2}}{2} + 2i \times \frac{\sqrt{2}}{2} - 2 \times \frac{\sqrt{2}}{2}

Hence z' = 3\sqrt{2} + 6\sqrt{2}i after rotating the point z=4 + 2i by \pi/4 radians about the origin.

So to summarize Using Euler’s formula for rotation:

z' = (4 + 2i)(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}),

z' = 3\sqrt{2}+ 6\sqrt{2}i kilometers.

This problem demonstrates how rotations in the complex plane can be applied to real-world scenarios like adjusting the course of an airplane. Rotations in the complex plane are also analogous to phase shifts in electrical signals: the angle changes, but the magnitude remains the same.

8 Banking Problem: Complex Numbers in Financial Transactions.

Banking Problem: John has a savings account with a balance represented by the complex number z_1 = 6 + 8i dollars, where the real part represents the actual balance and the imaginary part represents the pending transactions. On the other hand, Lisa has a savings account with a balance represented by the complex number z_2 = 3 - 5i dollars. If both of them combine their savings, what will be their total balance including the pending transactions?

Problem Statement: John’s savings account balance is represented as z_1 = 6 + 8i dollars, where the real part is the actual balance, and the imaginary part denotes pending transactions. If Lisa’s savings are z_2 = 3 - 5i dollars, what’s their combined balance?

Solution: Combined balance is: z = z_1 + z_2 ,z = 9 + 3i dollars.

The real balance, like the real power in electrical circuits, represents readily usable resources. In contrast, pending transactions, similar to reactive power, are amounts in transition—not immediately accessible but still pertinent.

9 Complex Numbers Conclusion

The proceeding examples, express the interplay between the real and imaginary parts of complex numbers in real world scenarios. Complex numbers can be used to represent tangible and transitional states. Next we will take a look at two dimensional vectors.

10 Introduction to Vectors

Vectors are mathematical constructs that describe both magnitude and direction. Within a two-dimensional space they are defined as \vec{v} = \begin{bmatrix}a \\ b\end{bmatrix} where (a) is the component along the x-axis and (b) is the component along the y-axis, these vectors can represent positions, velocities, forces, and other quantities within a plane. Unlike scalars, which only have magnitude, vectors encompass both magnitude and direction. This dual characteristic makes them invaluable in numerous domains, including physics, engineering, computer graphics, and many others.

11 The Basics:

These operations provide a understanding of how vectors are manipulated in a two-dimensional space.

Vector Addition, Subtraction , Scalar Multiplication & Dot Product:

Given two vectors: \vec{a} = \begin{bmatrix}a_1 \\ a_2\end{bmatrix} and \vec{b} =\begin{bmatrix}b_1 \\b_2\end{bmatrix}

Sum: \vec{a} + \vec{b} = \begin{bmatrix}a_1 + b_1 \\a_2 + b_2\end{bmatrix}
Difference: \vec{a} - \vec{b} = \begin{bmatrix}a_1 - b_1 \\a_2 - b_2\end{bmatrix}
Dot Product: \vec{a} \cdot \vec{b} = a_1 \cdot b_1 + a_2 \cdot b_2.
Dot Product: \vec{a} \times \vec{b} = \det\begin{bmatrix}a_1 \space \space b_1 \\a_2 \space \space b_2\end{bmatrix}= a_1 \cdot b_2 - a_2 \cdot b_1.

Magnitude of a Vector:

For vector \vec{a} = \langle a_1, a_2 \rangle, its magnitude is: |\vec{a}| = \sqrt{a_1^2 + a_2^2}.

Multiplying a Matrix by a Vector

Given a 2x2 matrix: M = \begin{bmatrix}m_{11} & m_{12} \\m_{21} & m_{22} \\\end{bmatrix}and a 2D vector: \vec{v} = \left[ \begin{array}{c}a \\b \\\end{array} \right] The multiplication of matrix M by vector \vec{v} is: M \vec{v} = \left[ \begin{array}{c} m_{11} \times a + m_{12} \times b \\ m_{21} \times a + m_{22} \times b \\ \end{array} \right]
Geometric Interpretation: Rotation: If M is a rotation matrix, then multiplying it with a vector will rotate the vector in the plane. Scaling: If M is a scaling matrix, the vector will be scaled (stretched or compressed). Reflection: Reflection matrices will reflect the vector across a specific axis. Shearing: Shearing matrices will skew the vector in a particular direction. Matrix-vector multiplication provides a powerful way to transform vectors, making it a key operation in many scientific and engineering applications.

Rotation using a Matrix :

A 2D vector can be rotated counterclockwise by an angle \theta using the rotation matrix: R(\theta) = \left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array} \right]
The rotated vector is obtained by multiplying the original vector by this matrix. Given a 2D vector, it can be rotated counterclockwise by an angle \theta using a rotation matrix.

The rotation matrix for an angle \theta is: R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} To rotate a vector \vec{v} = \begin{bmatrix} x \\ y \end{bmatrix} by the angle \theta, you multiply it by the rotation matrix, hence; \vec{v}' = R(\theta) \vec{v}

When you perform the matrix multiplication: \vec{v}' = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x\cos(\theta) - y\sin(\theta) \\ x\sin(\theta) + y\cos(\theta) \end{bmatrix}

The result \vec{v}' is the rotated vector.

These rules are essential for the vector algebra manipulations required in the problems we’ve discussed.

12 Warm Up:

Given two vectors: \vec{a} = \begin{bmatrix}5 \\ 3\end{bmatrix} and \vec{b} =\begin{bmatrix}2 \\4\end{bmatrix}

Calculate the sum of \vec{v_1} and \vec{v_2}.
Determine the difference between \vec{v_1} and \vec{v_2}.
Compute the dot product of \vec{v_1} and \vec{v_2}.
Compute the cross product of \vec{v_1} and \vec{v_2}.
Compute the magnitude (or length) for: \vec{v_1} and \vec{v_2}
Given: Matrix: M = \left[ \begin{array}{cc}0 & -1 \\1 & 0 \\\end{array}\right] and Vector:\vec{v} = \left[ \begin{array}{c}1 \\2 \\\end{array} \right]
- Multiply matrix M by vector \vec{v} and find the resulting vector.
- Determine the type of transformation applied to vector \vec{v} by matrix M.
- Rotate \vec v counterclockwise by \pi/4 radians , and \pi/2 radians.

13 Real World Applications:

Vectors offer a language to represent and solve problems across myriad fields. With their dual characteristics of magnitude and direction, they encapsulate a vast array of real-world scenarios. The following examples highlight the profound versatility and applicability of vectors, demonstrating their use in domains. These examples show the role that vectors play in solving real world problems, emphasizing the bridge they create between abstract mathematical concepts and tangible real-world phenomena.

14 Electrical Engineering Problem: Power in AC Circuits using Vectors

Problem Statement: Given the voltage vector \vec{V} = \begin{bmatrix}5 \\ 3\end{bmatrix} volts (representing real and reactive components) and the current vector \vec{I} =\begin{bmatrix}2 \\ 4\end{bmatrix}amperes, determine the power vector \vec{P} in the circuit.

Power Vector Formula:
\vec{P} = \begin{bmatrix} \vec{V} \cdot \vec{I} \\ \vec{I} \times \vec{V} \end{bmatrix}

Where: - \vec{V} is the voltage vector. - \vec{I} is the current vector. - \vec{V} \cdot \vec{I} is the dot product of the voltage and current vectors (representing real power). - \vec{I} \times \vec{V} is the cross product of the current and voltage vectors (representing reactive power).

Given: \vec{V} = \begin{bmatrix} 5 \\ 3 \end{bmatrix} \vec{I} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}

Compute the dot product for real power: \vec{V} \cdot \vec{I} = 5(2) + 3(4) = 10 + 12 = 22 \space \text Watts
Compute the cross product for reactive power: \vec{I} \times \vec{V} = 2(3) - 4(5) = 6 - 20 = -14

Thus, the power vector \vec{P} is: \vec{P} = \begin{bmatrix} 22 \\ -14 \end{bmatrix}

So, the power vector \vec{P} in the circuit is: \vec{P} = \begin{bmatrix} 22 \\ -14 \end{bmatrix} \text{VA (Volt-Amperes)}

15 Flight Problem: Vectors as Position

Problem Statement: Two airplanes start from the same airport. The first airplane’s path is represented by the vector \vec{z_1} = \langle 3, 4 \rangle and the second airplane’s path is represented by the vector \vec{z_2} = \langle 1, -2 \rangle . Find the distance between the two airplanes after their respective flights.

Solution: Using vector subtraction to determine the relative position vector: \vec{d} = \vec{a_1} - \vec{a_2} = \langle 3 - 1, 4 + 2 \rangle = \langle 2, 6 \rangle The distance between the two airplanes is the magnitude of the relative position vector: |\vec{d}| = \sqrt{2^ 2 + 6^2}= \sqrt{40} = 2\sqrt{10} Thus, the distance between the two airplanes is 2\sqrt{10} kilometers.

16 Airplane Rotation Problem: Navigation with Vectors

Problem Statement: An airplane has a position vector represented by: \vec{p} = \begin{bmatrix} 4 \\ 2 \end{bmatrix} We want to find the new position of the airplane after a rotation of \pi/4: radians counterclockwise.

Solution: For \theta = \pi/4, the rotation matrix becomes: R(\pi/4) = \begin{bmatrix} \cos(\pi/4) & -\sin(\pi/4) \\ \sin(\pi/4) & \cos(\pi/4) \end{bmatrix} = \begin{bmatrix} \sqrt{2}/2 & -\sqrt{2}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 \end{bmatrix}

To rotate the vector \vec{p} by the angle \theta , you multiply it by the rotation matrix: \vec{p'} = R(\pi/4) \vec{p}

Performing the matrix multiplication: \vec{p'} = \begin{bmatrix} \sqrt{2}/2 & -\sqrt{2}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 \end{bmatrix} \begin{bmatrix} 4 \\ 2 \end{bmatrix} = \begin{bmatrix} 2\sqrt{2} \\ 6\sqrt{2} \end{bmatrix}

So after a rotation of \pi/4 radians counterclockwise, the new position of the airplane is given by the vector: \vec{p'} = \begin{bmatrix} 2\sqrt{2} \\ 6\sqrt{2} \end{bmatrix}

This approach uses matrix multiplication to achieve a rotation in 2D space. The matrix rotation approach is fundamental in linear algebra and finds applications in fields like computer graphics and physics.

17 Banking Problem: Vectors in Financial Transactions

Banking Problem: John has a savings account with a balance represented by the vector \vec{z_1} = \begin{bmatrix}6 \\ 8\end{bmatrix} dollars, where the first component represents the actual balance and the second component represents the pending transactions. On the other hand, Lisa has a savings account with a balance represented by the vector \vec{z_1} = \begin{bmatrix}3 \\ -5\end{bmatrix} dollars. If both of them combine their savings, what will be their total balance including the pending transactions?

Problem Statement: John’s savings account balance is represented by the vector\vec{z_1} = \begin{bmatrix}6 \\ 8\end{bmatrix} dollars, and Lisa’s is \vec{z_1} = \begin{bmatrix}3 \\ -5\end{bmatrix} dollars. Find their combined balance.

Solution: Using vector addition: \vec{b} = \vec{z_1} + \vec{z_2} = \begin{bmatrix}6 + 3 \\8 -5\end{bmatrix} Thus, the combined balance, including the pending transactions, is \vec{b} = \begin{bmatrix}9 \\3\end{bmatrix} dollars.

17.1 Vectors Conclusion

The provided examples illustrates the use of vector algebra in diverse real-world situations. They highlight the deep-seated connections that weave through various domains.

17.2 Conclusion

Both complex numbers and vectors provide powerful tools for representation and problem-solving across diverse fields. By understanding their properties and operations, we can gain deeper insights into real-world scenarios.

18 About Me

My name is Joshua Lizardi. For the past 8 years, I have worked for various institutions teaching a wide range of courses in Math, Statistics and Technology. These included Quantitative Reasoning, Calculus ,Applied Technical Mathematics, Remedial Mathematics, Statistics, Computers & Office Automation, Introductory College Algebra, Intermediate College Algebra, Remedial Mathematics, Business Statistics.

I hold a bachelor’s in mathematics (Mercy College), a master’s in applied mathematics (Purdue University), and a master’s in data analytics (Western Governors University). I also hold a few certifications including “SAS Certified Statistical Business Analyst SAS 9”, “SAS Certified Base Programmer SAS 9”, “Oracle Database SQL Certified Associate”.

Subjects like mathematics, statistics, and computer science should not be taught as if they were spectator sports, the best way to learn these subjects is to perform them. Although understanding textbooks and lecture notes is valuable, the learning that comes from one’s own attempts at solving problems is the key to becoming competent in the subject overall. I have always been passionate about mathematics statistics and computer science, and I enjoy encouraging students to see the utility of these subjects.

SPECIALTIES

Applied Mathematics Applied Statistics Data Analytics Data Science Machine Learning Artificial Intelligence

SKILLS

R Python SQL SAS MiniTab Tableau Power BI Microsoft Office

https://www.youracclaim.com/users/joshua-lizardi