By Eben van Tonder, 25 March 25
Introduction: A Thread That Runs Through the Universe
This is the story of a line. A line that began with a thought in the mind of a mathematician and now quietly supports entire industries, ecosystems, bodies, and societies. It is the simplest function in mathematics, yet it offers a doorway into a deeper truth: that change can be tracked, that outcomes can be shaped, and that the future can, in some cases, be predicted.
The linear function, f(x) = mx + b, sits at the heart of our modern world. When Leonhard Euler first began expressing relationships using functional notation, he gifted us a new way of seeing reality. He took the humble equation y = mx + b and reframed it as a process: a mechanism, an engine. Something dynamic. Something alive.
This lesson is not about complicating maths. It is about rediscovering one of its simplest tools. And in doing so, it is about realising just how much power a single variable can hold. Together with Christa and Armin, we will explore this concept through the lens of science, philosophy, society, biology, meat science, gaming, and even love. Because once you start looking through this frame, you begin to see it everywhere.
The Birth of a Function: Euler’s Insight
Before Euler, equations like y = mx + b were used to describe balances—two sides held equal by a rule. Euler transformed this. By writing f(x) = mx + b, he did something revolutionary: he introduced the idea of a function as an abstract machine that takes an input (x), applies a rule (m and b), and delivers an output (f(x)).
This shift allowed humans to predict outcomes from changing inputs. It was no longer a static truth! It became a dynamic process. You could plug in any value and watch the consequences unfold.
This is more than mathematical elegance. It is the structural language of all modelling, from physics to social sciences, from economics to personal growth. It is the grammar of planning.
In the Human World: Sociology and Psychology
Let’s begin with human society. Sociologists build models to understand the world. One of the most basic questions they ask is: “What happens to one thing when another changes?”
Consider a simple model: income = m(education level) + b. Although real life is more complex, this equation captures a clear intuition: that more education often leads to higher income. The gradient, m, reflects how much change in income we might expect from each unit of education. The b term reflects inherited advantages or disadvantages—what someone starts with.
Sociologists also use linear functions to understand crime rates in relation to poverty, or life expectancy in relation to access to healthcare. In each case, the model isn’t claiming to tell the full story—it is offering a first approximation. And this approximation allows governments, policy-makers, and social scientists to plan. To test policies. To explore outcomes without risking lives.
In psychology, response models use linear functions to map stimuli to behavioural response. Reaction times, learning outcomes, even the formation of habits can often be modelled—at least initially—as linear systems.
Biology and Medicine: Linear Thinking in Living Systems
Biological systems are often nonlinear and chaotic. But even here, linear models play a crucial foundational role.
In pharmacology, early-stage modelling often uses f(x) = mx + b to describe how a drug dose affects the concentration in blood over time. Doctors and researchers begin with the assumption that doubling a dose will double its effect—until they see where this breaks down.
In endocrinology, linear models describe how insulin output relates to glucose levels, especially in early diagnostic frameworks. In oncology, risk increases for certain cancers are plotted as linear functions against age or exposure to toxins.
Gene expression, particularly in simplified bacterial systems, can often be modelled linearly with environmental factors like temperature or nutrient availability.
The simplicity of these models makes them powerful tools for quick estimation, especially in emergency settings or rural clinics without full diagnostic suites. It’s better to have a reliable linear approximation than to wait for a complex model that requires days or weeks.
Quantum Physics and the Language of Operators
Quantum mechanics may seem like the realm of chaos, randomness, and abstract uncertainty, but it is built almost entirely on linear mathematics. The wave function, the central object of quantum physics, is manipulated using linear operators. The famous Schrödinger equation is linear.
This means that if you know how a system behaves at time x = 1, you can predict its behaviour at x = 2 using a function. The linearity of quantum systems allows us to make astonishingly precise predictions, even though the outcomes themselves are probabilistic.
In this way, f(x) = mx + b becomes a symbolic stand-in for all functional relationships: those that preserve structure and allow reliable extrapolation.
Engineering and Technological Systems
No field makes more constant use of linear modelling than engineering. Whether in mechanical, civil, electrical, or systems engineering, linear approximations are everywhere.
Structural engineers model stress versus strain linearly, at least until a material yields. Electrical engineers use Ohm’s law (V = IR) which is a linear function of current and resistance. Thermal engineers model heat flow as a linear function of temperature difference.
Even in software engineering, linear approximations help model performance bottlenecks, energy usage, and algorithmic efficiency.
Control theory, which underlies robotics, vehicle dynamics, aircraft autopilot systems, and chemical processing plants, uses linear equations to keep systems stable.
Linear models are also used to predict failures. If you know how quickly wear and tear accumulates, you can estimate when a part will need replacing.
Nutrition, Meat Science, and Biological Yield
Let’s go deeper into our own domain: meat science. Here, linear models are used constantly—though we may not always describe them as such.
In sausage formulation:
- Yield = m(fat content) + b
- Water retention = m(salt content) + b
- Protein extractability = m(mixing time) + b
When you tweak salt levels, temperature, or fat ratios, linear models can give you a reliable first sense of what will happen to texture, juiciness, shrinkage, and cost.
In nutrition science:
- Caloric intake vs weight gain is often modelled linearly (until adaptive metabolism shifts things).
- Protein requirements in athletes scale linearly with body mass and training intensity—again, until saturation.
- The bioavailability of iron and zinc in different meat cuts can be plotted against total intake to identify cost-effective strategies for supplementation.
Linear functions help food technologists reduce wastage, predict shelf life, balance costs, and meet nutrition targets—all in real time.
Sport and Athletic Progression
Coaches and sports scientists use f(x) = mx + b all the time, whether consciously or not. Linear models underpin how training volume relates to performance gains:
- Sprint time improves by x per week of training.
- VO2 max increases linearly with time spent at 85% max effort.
- Injury risk increases linearly with training intensity that exceeds recovery capacity.
Even psychological models in sports—like motivation vs burnout or rest vs mental clarity—start with linear relationships before moving into curves.
Predicting how fast someone will run in 3 months or how much weight they’ll lift can be done with simple linear planning. For young athletes like Armin, this gives a tangible feeling of control.
The Philosophy of One Variable
At the heart of this is a philosophical stance: that reality, though complex, can sometimes be shaped by identifying one key variable and understanding how it behaves.
f(x) = mx + b is a statement about reality. It says: if you can find the variable that matters, and if the relationship is clear, then you can predict the future.
It is the language of:
- Strategy
- Planning
- Design
- Influence
- Intention
The line doesn’t claim to know everything. But it claims to know enough. Enough to build bridges, feed cities, guide economies, and care for patients.
Family Lesson: Teaching This to Armin
Now imagine this: a quiet evening at home. Christa and you are at the kitchen table. Armin is there too, curious but unsure. You bring out a notebook and a fresh steak.
You say: “Let’s build a formula.”
Start simple: “x is the grams of protein in this steak. m is how much it costs per gram. b is the packaging and transport fee. What happens when x increases?”
You show him a line. Then you graph it. Then you ask: “What if x is your number of study hours? What happens to your grades?”
Then you personalise it: “What if x is your reaction speed in your favourite game, and f(x) is your kill ratio?”
Let him calculate. Let him graph it. Let him feel how control of one variable can shift the outcome.
Then say: “You just built a model. You just predicted the future.”
And finally: “This is how the world is built.”
Conclusion: The Beginning of All Modelling
Before there were simulations, spreadsheets, and machine learning, there was the line.
Before we worry about parabolas, sine waves, logistic growth, or exponential decay—we begin with the linear.
f(x) = mx + b is where children first glimpse order in the universe. It is where adults return when they want clarity. It is the alphabet of modelling, the first line in the song of strategy.
From building bridges to building a future, from meat processing to medicine, from sociology to gaming, the world is shaped by simple changes, one variable at a time.
And when you learn to see that line, you are learning to see possibility.
It’s not just maths. It’s power. It’s freedom. It’s the beginning of wisdom.
In this short series on Beautiful Mathematics:
Euler’s Function: The Birth of a Mathematical Machine
A Deep Dive into f(x) = mx + b