Personal Learning Path #1

Hello everyone!

This is my first blog post, a very exciting time indeed. My name is Foster Harlfinger, and I am a 5th year student in Fordham's 5-year education program. Last year, I received my Bachelor's Degree in Math, and I will graduate with a Masters Degree in Teaching Adolescent Mathematics in Spring 2022. I am currently acting as a student teacher at West Bronx Academy for the Future in a 9th grade geometry class. I have not had any experience teaching in front of a classroom, but I worked as a one-on-one math tutor with an organization in the Bronx and as a homework grader for Fordham's math department for several years. My first official teaching year should begin in Fall 2022.

Background

I am working from the New York State Teacher Certification Examinations (NYSTCE) standards, Field 004: Mathematics. This week, we will be discussing a topic from Competency 0001 - Number and Quantity. Within this Competency, I have chosen the following performance indicator: "performs arithmetic operations with complex numbers."

Here are the NY State Common Core Learning Standards for Mathematics which align with my selected topic:

My goal is to develop an intuitive explanation for students (and teachers!) as to why complex numbers are necessary. I will not be including problem sets, since those are very easy to work through and teach once you have a basic understanding of complex numbers. My main goal here is to answer the question: "Why do imaginary/complex numbers exist?"

It will be assumed that students already understand and can define real numbers and the imaginary unit, i. For your reference, the imaginary unit, i, is simply given by the square root of -1. Real numbers include whole numbers (positive and negative), rational numbers (numbers that can be represented with a fraction), and irrational numbers (square roots, pi, etc).

The Idea

The core idea is that complex numbers are necessary in order to solve certain types of equations. We have two tools in our toolbelt: counting numbers (1, 2, 3, etc) and algebra (addition, multiplication, exponents, etc). Using only these two tools, we can naturally (math pun intended) arrive at complex numbers.

I have attached a chart which represents this idea below. I have modified mine from Eddie Woo's video on complex numbers, but charts like these can be found in many places. Here is how the chart works: In the first column, we are given an equation. In the second column, we are given the solution to that equation. In the third column, we are told from which set of numbers our solutions originate. As an example, the first row shows us that our solution is x = 9 which is a natural number. Simple.

We will take a ground-up approach to understanding numbers in order to explain this concept verbally as follows:

1. We are able to solve our first equation simply using the counting (natural) numbers.
2. When presented with our second equation, we are presented with a problem. The natural numbers begin at 0 or 1 (depending on who you ask). Therefore, we need to adapt our understanding of numbers to include negative numbers. In other words, we need the integers!
3. When presented with our third equation, we are presented with another problem. Our solution is not a whole number, yet the integers only include positive and negative whole numbers. Therefore, we need to adapt our understanding of numbers to include fractions. In other words, we need the rational numbers!
4. When presented with our fourth equation, we are presented with yet another problem (if you can believe it!). Our solution is neither a whole number nor a fraction, yet the rational numbers only include numbers which can be represented by a quotient. Therefore, we need to adapt our understanding of numbers to include square roots. In other words, we need the real numbers!
5. When presented with our fifth equation, we are presented with our final problem. Our solution is not a real number! Hmm... How can this be? Well, our pattern has worked well enough for us so far. There's no reason to give up now. Why don't we try adapting our understanding of numbers to include the imaginary numbers. In other words, we need complex numbers!

This explanation almost feels too simple, but students are often taught how to add/subtract complex numbers before they are ever given an intuitive understanding of what complex numbers are and why they are necessary. Put simply: Complex numbers are the logical next step when solving increasingly difficult equations, and solving these equations is necessary for many practical applications.

For teachers: You should work through the chart (as seen above) with your class as a whole. You can present the equations in the first column, but allow student volunteers to share the solutions for the second and the number set for the third column. Once you get to the final row, take back the reins and verbally explain the intuition behind the final steps.

Once this intuitive explanation has been given, the teacher should give a few examples to show how complex numbers look, etc. This part of the lesson is fairly straightforward, and there are many resources available online. You should also address the unfortunate naming of these numbers as "imaginary," since that often gives students the wrong impression.

My blog post answers the question, "Why are complex numbers necessary?" However, students may still be thinking, "Okay, I understand why they are necessary, but what exactly is an imaginary or complex number? What does it look like?" This linked article should help you to answer that question for your students.

Reflection

The goal of this post was to offer an intuitive explanation for why complex numbers are necessary. Students are often taught this material without being given any sort of intuition behind this complex (pun intended) topic. I believe that the short activity I have described above will help teachers (including myself) offer students this intuitive explanation they are missing. One piece of advice for teachers would be to complement this activity with a brief explanation of where these complex equations appear. For example, a quick internet search notes that complex equations can be found in fluid dynamics, quantum mechanics, signal processing, and vibration analysis, to name a few examples. To reiterate, the purpose of this activity should be to help students understand that the math they are learning is not merely some scheme devised by testing companies to torture students. This math is used practically throughout all aspects of life, and without it, we wouldn't have electricity, space travel, computers, film animation, or architecture/engineering, to name a few obvious examples.