A rapidly evolving field of mathematics called fractional calculus can reveal the finest details of physical processes, allowing engineers to improve everything from anaesthesia to batteries

IT IS the ultimate “your life in their hands” moment. The anaesthetist is counting down from 10. You are about to lose the ability to feel, to breathe independently. From the instant you lose consciousness, you are relying almost entirely on that anaesthetist to keep you alive and stop you waking mid-surgery. Almost, because human judgements on how best to regulate the flow of drugs are themselves reliant on mathematical models underlying the monitoring systems that anaesthetists use.

At the heart of those models is calculus, the branch of mathematics that lets us explain and predict how change happens. This ability is absolutely fundamental to science, which calculus has crucially underpinned since its invention in its modern form a little over 300 years ago.

Now we could be moving to the next level. Conventional calculus has its limits when we try to model complex situations. Patient response in anaesthesia is one – hence why there is always an anaesthetist in the room. But a radical, rapidly evolving form of calculus developed in the past few years is giving us a host of mathematical tools that promise to let us understand the finest details of physical processes with unprecedented precision.

It isn’t just drug delivery that could see concrete benefits – it could help us solve all manner of problems, from detecting cancer to preventing the spread of pollution to making more efficient batteries. “I can’t count the number of ways in which this can be applied,” says Abdon Atangana at the University of the Free State in Bloemfontein, South Africa, who discovered some of the key maths behind the breakthrough.

The …