
Non-linear relationships are tricky relationships. To understand why, let's first look at a linear relationship.
Imagine you want to buy a piece of string. The string costs 2 pounds per metre, so if you buy
This relationship between the length of string,

The relationship is pretty straightforward. For example, it's easy to see whether you can afford to increase the length of string by
In this example a change in the input to the price function is proportional to the resulting change in the output of the price function (the change in output equals twice the change in input).
Now suppose that, instead of a piece of string, you want to buy a square piece of fabric, priced at 2 pounds per square metre. If the side of the square you want to buy has length
This is a non-linear relationship. When you plot

What if you now want to increase the side length of your square by
The difference between the price
This is quite a complicated expression involving both the original side length
This feature of non-linearity means that it can be difficult making predictions based on non-linear relationships. Indeed non-linear equations can give rise to chaotic systems whose behaviour can be impossible to predict — the weather is an example of a system whose underlying equations are non-linear and exhibit mathematical chaos.
Our equations above were specific examples of linear and non-linear relationships. More generally, the equation of a straight line can be written as
where
This isn't true for polynomial relationships such as
Even more generally, the concepts of linearity and non-linearity also exist for relationships between more than two variables, or when the variables stand, not for numbers, but other types of mathematical objects, such as functions or vectors. The defining feature of linearity is that changes in output are proportional to changes in input, and of non-linearity that this is not the case. Find out more in the Wikipedia article on non-linear systems.
This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.
