# How does mathematics apply to driverless cars?

Driverless cars rely on a full range of mathematical concepts.

Core Mathematics: In order to drive in the most efficient manner, driverless cars can exploit software based on the methods of Calculus to determine the most appropriate speeds to drive, to minimise energy consumption. By enabling driverless cars to connect with traffic signal controllers, Calculus mathematics can also be used to optimise traffic flows.

Mechanics: The definition of a driverless car (compared to cars with driver assist modes) is one that can perform the task of controlling both the speed and direction of the vehicle without the need for a driver (or tele-operative) to be involved. This requires the car, through computer programming, to have a very good understanding of not only the dynamics of the vehicle, but the dynamics of all other objects that might cause an obstruction or accident. The human brain, over time, has been hard-wired to understand the laws of mechanics, and to make complex calculations that predict where multiple moving objects will be in a few seconds time. This is how driver’s are able to avoid accidents. Without the benefit of a human-brain, driverless cars need to be hard-wired to understand these things, through computer programming, based on mechanical mathematics.

Statistics: How will driverless cars be insured in the future. This is a big question, taxing the car insurance industry today. All insurance products are based upon statistical models, that assess the probabilities of accidents occurring, and identifying appropriate prices to apply to insurance products. The statistical work for Car insurance products has evolved over many years of car insurance companies gathering data about how human driver behaviour impacts the likelihood of having an accident. With driverless cars, and the added complexities of driverless cars interacting with driven cars, all historic assumptions are ‘out of the window’. Furthermore, with the use of telematics devices and improved sensors, there is so much more data available nowadays, than what existed even 10 years ago. Imagine how much more data will be available in another 10 years time. All this ‘big’ data requires analysis, using statistical techniques, to build new statistical models both  the insurance industry AND the driverless cars themselves – where there is the potential for driverless cars to learn from this data, to improve their abilities to drive safely – all requiring statistical techniques that underpin an area known as ‘Artificial Intelligence’.

Decision Mathematics: Inside the ‘brain’ of a driverless car, we will need to be able to replicate (and improve upon!) all of the decision making that drivers undertake. Some of these tasks have already been automated with Decision Mathematics, through the identification of the best route to take, using SatNavs, GoogleDirections and apps like Waze. The most crucial decisions, that the driverless cars will need lightning responses to, will be in the event of potential hazards and accidents with other vehicles…such as an event where a pedestrian walks out in front of the car, at the same time the driverless car being aware that the driven car behind is driving too fast to avoid crashing into the driverless car, if it applies its brakes to avoid the pedestrian…

Pure Maths: Not only does Pure Maths help scientists create complex, and increasingly accurate, models of how the Universe works, including the mysteries of ‘dark matter’, the practice of ‘abstract thinking’ is highly applicable to the most sophisticated software languages, which are used to programme driverless cars.