Have you ever wondered how we’re able to forecast the weather? Or how your phone can calculate the fastest route to a location? Or even how scientists can predict how a disease will spread? These are all examples of challenges that rely on applied mathematics: the branch of maths that uses mathematical ideas to understand, predict and solve real-world problems.

Different branches of maths

Broadly speaking, maths can be divided into two main strands: pure mathematics and applied mathematics. Although statistics and data science is often grouped with applied mathematics, many mathematicians and statisticians agree that statistics has developed into a substantial discipline in its own right.

Pure mathematics

Pure mathematics develops maths for its own sake, exploring patterns and logical relationships. By introducing precise definitions and rigorously proving theorems, pure mathematicians build the foundations on which many areas of science and engineering depend. These rules essentially tell us why maths works the way that it does.

Applied mathematics

Applied mathematics builds on the tools created by pure mathematicians to model and understand real situations. Applied mathematicians develop equations that describe how a system behaves. Solving these equations allows them to predict how a given situation will change and sometimes even suggest ways to improve it. This can be as complicated as calculating the trajectory of a rocket, or something seemingly simple like modelling heat transfer to estimate how long it takes a cup of coffee to cool down.

What does applied mathematics actually involve?

In essence, applied mathematicians are developing models that describe real-world systems. But what does that actually mean?

By system, we just mean the real-world process, or object, that we want to understand. We might want to know when a cup of coffee is a safe drinking temperature, in which case the system is the coffee in its surroundings; or we might be interested in predicting the weather in which case the system is the Earth’s atmosphere (or some local area of the Earth’s atmosphere).

When we talk about models, we mean the maths that can describe a given system. These models may be equations, but they can also be algorithms or computer simulations, depending on the system they describe. In the case of our cup of coffee, the mathematical model is an equation that relates the coffee temperature to time. For the weather, the model is thousands of differential equations that connect temperature, wind speed, air pressure and many other variables.

The important thing is that once we have a model, we can ask “what if?” questions without needing to perform expensive, dangerous or even impossible experiments.

Let’s suppose that, as an applied mathematician, we want to work out when it’s safe to drink a cup of coffee. There are some key steps that we need to take:

Build a model

Firstly, we want to identify the system and build a mathematical model. In this case, we have a hot object that is losing temperature to its surroundings. A simple model for this is Newton’s law of cooling, a differential equation that relates temperature to time.

Consider our assumptions

The next step is to consider any assumptions we have made. It is almost always the case that the real-world system is far more complicated than a mathematical model we can actually work with, so we need to identify what is important. In our coffee cup example, we might assume that all the coffee is the same temperature, even though the top will cool faster than the bottom.

Solve equations

Solving the equations allows us to predict how the system will behave. We can hope that this is straightforward and we will be able to write down an analytic solution, but often we must turn to computers to numerically solve the system. The dramatic increase in computing power over the past few decades has transformed what applied mathematicians are now able to model and simulate. We are now seeing these transformations in our everyday lives: from highly accurate weather prediction to artificial intelligence (AI) assisting with medical imaging.

Calibrate against data

Once a mathematical model has been developed, we should test it against real-world data. After all, a mathematical model is only useful if it describes the real-world system well enough for its intended purpose. If the model’s predictions agree with the observations, then we can be confident in the model that we have developed. If they do not, then there may be assumptions that we have overlooked or additional factors that we need to include.

Improve the model

The last thing we should do is to improve the model where necessary. In the case of our cooling cup of coffee, we assumed that the coffee is all the same temperature. This seems reasonable, but if we have a very tall and thin cup then the temperature differences will become important and we will need to add this additional information into our mathematical model.

In reality, it’s very rare to find a perfect model the first time. Instead, applied mathematicians will continuously check their models with real data and refine the assumptions that have been made until the model is fit for purpose.

Applied mathematics in the real world

Many of the examples of applied mathematics that we learn about at school involve mechanical systems — predicting the trajectory of a ball flying through the air or working out the forces acting on a ladder leaning against the wall — but applied mathematics underpins much of our modern life.

Finance

Financial services are one of the biggest employers of applied mathematicians. They build models describing markets and financial risk. For example, the amount that you pay for your car insurance will be determined by a model that helps the insurance company decide how risky (read “expensive for them”) you will be to insure!

Medicine

We’ve already mentioned that medicine is using applied mathematics to make diagnoses using image recognition, but this is just one of hundreds of applications. Applied mathematicians have developed models that describe how diseases spread through a population as well as predicting methods for preventing outbreaks within hospitals. They have also succeeded in creating models that describe how our bodies work, from a model of the brain that enables personalised treatment of depression, to a virtual model that will optimise the treatment of wounds.

Engineering

Applied mathematics is also hugely important in engineering. It is the underpinning language of engineers, describing how structures and machines behave before they are built, helping to ensure that products are safe, reliable and efficient. Beyond the design of safe buildings and efficient vehicles, applied mathematics can support the development of clean energy technologies and even predict how materials behave at an atomic level.

Climate science

We’ve mentioned a couple of times throughout this blog that one of the typical uses of applied mathematics is in predicting the weather, but it also has far deeper links to climate science. The climate is too complicated for us to experiment on directly, so applied mathematicians can build models that describe the world around us instead. By comparing these models to real-world observations, we can predict how sea-levels might rise or even identify the leak of greenhouse gases. Applied mathematicians are also taking care of our local environment. For example, modelling the behaviour of invasive species helps us make decisions to keep our crops and woodlands safe.

Final thoughts

Applied mathematics is an incredibly diverse field, with applications spanning engineering, the physical sciences and the life sciences. It has the ability to drive new scientific discoveries, and to keep our world running in the way we expect.

Whether we’re developing safer aircraft, predicting disease outbreaks, or understanding climate change, applied mathematics helps us understand the complex world around us. With rapid advances in computing power and AI, the opportunities to use applied mathematics to improve our lives continue to expand.

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