Mathematics
Exploring the Core Properties of Complex Numbers
A World Beyond the Real Numbers
Complex numbers are a pivotal high school math topic. It stands out in many ways compared with other math concepts introduced in a typical high school math curriculum. Due to its highly abstract nature, while some students find it fascinating, others find it daunting.
In this article, I will review the core aspects of the properties of complex numbers, highlighting the possible obstacles that high school students may encounter when they are first introduced to the concept.
An abstract mathematical concept of “numbers”
Almost all young children’s first experience in math class revolves around the concept of counting. Their teachers help them memorize the sequence of natural numbers, from 1 to 10, often through engaging songs. By then, they have acquired the ability to describe and record an abstract concept of “quantity” verbally and on paper.
Complex numbers, on the other hand, are a purely conceptual construct. Students may be surprised to learn that:
- Complex numbers are not primarily used for counting or representing quantities of countable objects.
- Despite being called “numbers”, complex numbers do not even have an “order”, meaning that one cannot compare two arbitrary complex numbers, say z₁ and z₂, using expressions like z₁>z₂ or z₁<z₂.
Complex numbers are considered “numbers” because their algebraic properties, such as addition and multiplication, mirror those of real numbers. Specifically, they satisfy the commutative, associative, and distributive properties. For example:
z₁+ z₂= z₂+ z₁
z₁(z₂+z₃)=z₁z₂+z₁z₃
However, complex numbers transcend real numbers because of their imaginary parts. In a way, complex numbers are like Kinder eggs. If you don’t care about what is hidden inside, you can play with it like a regular toy egg. But, we all know that there’s a surprise hidden within, and we can’t wait to open it. That’s the same with complex numbers, they will surprise you when you explore their full structure.
The transition from learning “numbers” as a measure of quantity to a purely abstract mathematical construct can be tough for some students, and it is completely understandable. Teachers and guardians need to help students realize that in the development of mathematics, when new concepts are introduced their connection to daily life purposes often becomes less obvious.
The imaginary number
To the ancient Greeks, mathematics was very much the study of geometry. As mathematicians studied the surface area and volume of different geometrical shapes, they ended up solving polynomials of different kinds. For about a thousand years, finding the roots of polynomials was closely tied to the study of geometry, until the “imaginary number” was invented, which helped establish algebra as a separate branch of mathematics.
We all know that the imaginary number i is, by definition, a solution of the quadratic equation x²=-1, which has no real root. It is important to understand that in saying “i²=-1”, we implicitly redefine the meaning of -1 from the real number -1 to a representation of the ordered pair (-1,0) within a new kind of “number” system. This system is built on the condition i²=(-1,0) where i=(0,1).
The following representations of complex numbers are all fine:
z=a+bi where a and b are real and i²=-1 or
z=(a, b) where i=(0,1) and i²=(-1,0)
z= a vector in the complex plane with polar co-ordinates (r,𝜑) where r and 𝜑 are, respectively, the modulus and the argument
The addition and multiplication are defined as:
z₁+ z₂ = (a₁+a₂, b₁+b₂)
z₁ z₂= (a₁ b₁— a₂b₂, a₁b₂+ a₂b₁) = ( r₁ r₂, 𝜑₁+ 𝜑₂) — — — — — — — -(*)
The basic algebraic operations of complex numbers are conceptually straightforward but require more mature arithmetic skills than real numbers. The different representations of complex numbers present different arithmetic conveniences and challenges, leading to different levels of satisfaction and frustration for learners depending on their ability. Therefore, when assigning practice problems involving complex numbers to students, careful attention is needed to ensure a progression in the level of difficulty.
Students often wonder how mathematicians conceived the idea of the imaginary number i. Visualization of complex number multiplication may offer a clue.
From equation (*), it’s evident that the multiplication of complex numbers involves two components: the multiplication of their magnitudes and the addition of their arguments. Thus, the relationship i²=-1 can be interpreted as two successive 90-degree counterclockwise rotations from the real number 1.
Before the introduction of complex numbers, imagining real numbers as part of a larger number system defined on a 2-D plane with multiplications corresponding to rotations wouldn’t have been too difficult. In such a system, the equation x²=-1 could naturally have a solution. The real challenge lay in mathematicians’ hesitation to accept a novel number system beyond the real numbers.
The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra(FTA) states that:
Any polynomial of degree n has, with multiplicity, exactly n complex roots. (**)
Solving cubic polynomials played a key role in the emergence of complex numbers. If you’re interested, there are many intriguing accounts online about how ancient mathematicians tackled cubic equations. Given the difficulty of solving cubic equations, which took centuries to address, one can only imagine the challenges posed by polynomials of any degree n.
The Fundamental Theorem of Algebra provides the answer that mathematicians sought for centuries. It is beautiful, powerful, and insightful all at once. However, its proof typically involves mathematical analysis, which is beyond the scope of high school students.
The following concepts should be reinforced when the theorem is introduced to students:
- The root of a polynomial p(z) is the value of z for which p(z)=0.
- The FTA applies to polynomials with either complex or real coefficients.
- For polynomials with real coefficients, if z=a+bi is a root, then its conjugate a-bi is also a root.
An informal graphical visualization can help students grasp why the FTA is true. Simply consider the asymptotic behavior of a polynomial p(z). Start with a random number z₀. If p(z₀)=0, you’ve found a root. If not, visualize how the values of p(z) move on the 2-D complex plane as you move away from z₀ toward |z₀| approaching infinity. Asymptotically, p(z) must be dominated by zⁿ and approaches infinity in all directions. Consequently, at some point, it must cross the origin. In other words, there is at least one root for any p(z). By induction, you would arrive at (**).
The theorem has a great impact on the field of mathematics and beyond. Even though high school students may initially find the theorem rather opaque, hopefully, it would serve as a motivation to pursue more advanced mathematical knowledge rather than as a source of frustration.
Closing Remarks
Teaching the topic of complex numbers is a challenging task, but is also very worthwhile. It could massage students’ ability to approach abstract mathematical concepts and help them build confidence in mathematical deductions. Seeing young minds grow is certainly the greatest reward for those hours spent making your lesson plans or assisting your children with the topic. My salute to you, teachers and guardians!
