Complex Numbers, in Multiple (Mathematical) Languages

One of the most fascinating aspects of mathematics is the multifaceted nature of a mathematical concept. As we study more and more branches of mathematics, it is interesting to see how a single mathematical object that is commonly associated with a certain branch can appear unexpectedly in other seemingly unrelated branches.

Among such mathematical concepts, complex numbers are worth discussing here. In common high school curricula, complex numbers are usually introduced as an extension of real numbers, with the introduction of the imaginary unit, \(i\), that satisfies \(i^2=-1\), so that equations like \(x^2+1=0\) have solutions. Using the imaginary unit, we obtain a new realm of numbers in the form of \(a+bi\), where real numbers \(a\) and \(b\) respectively represent the real and imaginary part of a complex number.

Indeed, this is a sensible definition of a complex number. However, sometimes this definition may be too elementary to uncover some of its important mathematical structures. Fortunately, by utilising knowledge from other branches of mathematics and studying complex numbers from different perspectives, we can better understand these underlying structures and how they relate to broader areas of mathematics.

Linear Algebra#

In the language of linear algebra, complex numbers can be viewed as a two-dimensional vector space over \(\mathbb{R}\), with a natural orthonormal basis \(\{1,i\}\).

The dual space to the 2D-space of complex numbers has a dual basis of \(\{\mathrm{Re},\mathrm{Im}\}\), which extract the real and imaginary parts of a complex number respectively. Indeed, one can verify that \(\mathrm{Re}(1)=1\), \(\mathrm{Im}(1)=0\), \(\mathrm{Im}(1)=0\) and \(\mathrm{Im}(i)=1\).

In terms of (real) matrices, a complex number \(a+bi\) can also be represented as a \(2 \times 2\) matrix as follows: \[\begin{pmatrix} a & -b \\ b & a \end{pmatrix}=a\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}+b\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.\]

Intuitively, this representation tells us that \(1\) can be ‘treated’ as the identity linear transformation on \(\mathbb{R}^2\) whereas \(i\) can be ‘treated’ as the linear transformation of rotating \(\frac{\pi}{2}\) radians anticlockwise. There is a mathematical jargon for such phenomenon: isomorphisms.

What about complex conjugates? They do not form a dual space to the complex numbers, but can be viewed as transposes of the linear map corresponding to the complex numbers of which they are conjugates, if we are looking at the matrix representation as \(2 \times 2\) real matrices as defined above. This is closely related to the concept of dual spaces.

As a sidenote, complex conjugates also give rise to the notion of the complex conjugate of a vector space.

Abstract Algebra#

From the group-theoretic perspective, complex numbers can be regarded as a group, both additive and multiplicative.

For multiplicative groups, a topic of interest would be the roots of unity, i.e. the roots of the equation \(x^n=1\) where \(n\) is a positive integer, which is a cyclic multiplicative group with order \(n\).

For additive groups, the collection of all complex numbers form an infinite additive group.

Complex Analysis#

As the name suggests, complex analysis is a branch of study devoted to complex numbers. Simply put, it is ‘calculus for complex numbers’. For a straightforward and comprehensive overview of the important topics of complex analysis, I recommend the video series titled Essence of Complex Analysis. Here, rather than complex numbers themselves, functions of complex variables (that take in and output complex numbers) are the main subjects of study. In particular, the roots of multivariate polynomials, e.g. solutions of \(y^2-x^3-x-1=0\), is a major topic of study in this branch of mathematics, and is closely connected to other branches like algebraic geometry and analytic number theory.

There we have it. We can see that complex numbers are not just an artificial concept with a sole aim of solving certain quadratic equations, but are closely related to many other branches of mathematics, as well as physics and engineering, with numerous opportunities of study. Of course, this article only touches the tip of the iceberg, so I hope that this can serve as a motivation for the reader to study more about the power of complex numbers.

Complex Numbers, in Multiple (Mathematical) Languages