Throughout my academic journey, I've always been fascinated by the challenge of taking complex problems and expressing them in their simplest possible form. For me, simplifying problems isn't just a preference; it is a way of testing whether I truly understand what I am studying, because if you can explain it simply, then you understand it deeply. And if that instinct resonates with you, then applied mathematics is the perfect home for your curiosity.
"If the instinct of taking complex problems and expressing them in their simplest possible form resonates with you, then applied mathematics is the perfect home for your curiosity.”
From Frustration to Curiosity
Early in my schooling years, I often found myself frustrated because many of the mathematical problems I encountered seemed unnecessarily complicated, as though they had been deliberately dressed up in layers of abstraction.
Very often, I found simpler ways to express the same ideas into something more intuitive and less intimidating, all while preserving the accuracy, assumptions, and theories behind the rationale. This led me to discovering newer, shorter methods of solving some of the most popular mathematical problems in high school mathematics.
Soon I learned that I wasn't the only person with this thought. I remember a conversation with a classmate during my early academic years. He argued that some researchers and educators take relatively simple problems and encode them in highly complex mathematical equations—so complex that they might feel inaccessible to the average person. In his view, education then becomes the process of translating these intimidating expressions back into understandable ideas.
This conversation led me to a question that stayed with me for a long time: Is applied mathematics complex by design, or by necessity? At the time, I didn't have a good answer. But after years of studying the application of mathematics in real-world phenomena—through graduate school and beyond—I believe I finally do.
Why Applied Mathematics Is Complex
The truth is this: applied mathematics is not complex because mathematicians want to make life difficult. It is complex because the world it tries to model is complex. Pure mathematics can choose its assumptions and build elegant structures from them. Applied mathematics does not have that luxury. It is tied to the messy, nonlinear, and often unpredictable nature of reality. Its complexity is a reflection of the world it seeks to describe.
Here's why applied mathematics inevitably becomes intricate:
1. Modeling Real-World Systems
Applied mathematics takes abstract tools and uses them to model real phenomena: climate systems, biological processes, financial markets, and fluid dynamics. These systems are rarely simple. They are inherently complex, involving countless interacting variables, and the mathematics must rise to meet that complexity.
2. Approximation Over Perfection
Exact solutions often do not exist. Instead, applied mathematics relies on approximations, numerical methods, and simulations. These techniques are designed to get close enough to reality, and they introduce layers of complexity because they must balance accuracy, stability, and computational feasibility.
3. A Bridge Between Worlds
Applied mathematics sits at the intersection of theory and application across many disciplines. It demands not only deep mathematical rigor but also domain-specific knowledge in fields such as physics, engineering, biology, economics, and more. That dual fluency naturally increases complexity.
4. Computational Dependence
Modern applied mathematics problems are often too complex to solve analytically by hand. Many are solvable only through computers, algorithms, high-performance computing, and iterative numerical methods—all of which require additional knowledge and experience. For example, consider the recent trend in machine learning, where practitioners increasingly need to understand GPU acceleration because modern deep-learning methods depend on it.
In this sense, applied mathematics complexity is not ornamental—it is functional. It is the price of compressing the real world into something we can analyze.
“Applied mathematics becomes difficult for the same reason reality becomes difficult: every useful model must eventually confront the complexity of the world it describes.”
Why Theory Still Dominates
One might then ask: if applied mathematics is about real-world problems, why do textbooks and research papers feel so theoretical? The answer lies in stability. Applications change constantly, influenced by technology, context, and new discoveries. But the underlying theory remains.
Consider the work of Carl Friedrich Gauss. His development of the normal distribution and the method of least squares provided a theoretical foundation that has endured for centuries. While the applications have evolved—from astronomy to modern machine learning—the core theory remains intact. This is why mathematics is often taught and communicated through abstraction. Theory provides a timeless language for describing problems, even as their applications shift.
Bridging the Gap: Where Mathematical Communication Can Improve
Even though complexity is unavoidable, mathematical communication does not have to be intimidating. Textbooks and research papers can—and should—do more to balance rigor with intuition.
Mathematical communication improves when rigor and intuition work together. This means:
- ➜ Pairing formal derivations with simple, relatable examples
- ➜ Using visual intuition to build mental models
- ➜ Moving back and forth between abstraction and concrete illustrations
- ➜ Showing how complex equations emerge from simple ideas
This dual approach helps learners see not just how the mathematics works, but why it works.
Final Thoughts
It's important to note that these complex equations and theories eventually become real tools and technologies: algorithms, models, devices, and systems—many of which are shaping the modern world.
Unfortunately, much of this translation happens in the research and development labs of institutions, where innovations and discoveries are often protected through patents and trade secrets, which is understandable given the investment required.
Luckily, today there is also an enormous ecosystem of open-source tools, free courses, and accessible resources, which can be utilized; and with enough practice and persistence, the once-intimidating mathematics begins to make sense. Because in the end, every expert was once a beginner—and every complex idea, when understood well enough, reveals a simple core.