In maths an invariant of an object is a quantity that stays the same when you make some sort of a change to the object. 

As an example, think about a triangle drawn on a flat piece of paper. When you rotate the triangle, reflect it in an axis, or just move it along (translate it), its angles stay the same. The angles of a triangle are therefore invariants for each of the three types of transformation: rotations, reflections and translations. (The same is true for the length of each of its three sides. It stays the same when you rotate, reflect, or translate the triangle.)

As another example, think of two numbers $m$ and $n$ with $m>n$. The difference between them is $m-n$. Now add the same amount of $c$ to each of the original numbers to get $m+c$ and $n+c$. Their difference is

$$(m+c)-(n+c)=m-n,$$

which is the same as the difference between the original two numbers. Therefore, the difference between two numbers is an invariant for the operation of adding a constant $c$ to both.

Finally, here's an example of an invariant involving an equation. Suppose the equation is

$$2x-4=0,$$

The solution is

$$x=2.$$

Now multiply each coefficient of the equation by some constant number $c$ to get

$$2cx-4c=0.$$

The solution now is

$$4c/2c=2,$$

which is the same as the solution to the original equation. This is true for any linear equation of the form

$$ax+b=0.$$

Its solution is $-b/a,$ which is the same as that of the equation

$$acx+bc=0.$$

Solutions of linear equations are therefore invariants under the operation of multiplying all coefficients by a constant $c.$ (The same is true for all polynomial equations and many others too.)

Why do we care about invariants?

Knowing the invariants of an object for a given type of transformation is incredibly useful. For starters, it tells you that you don't need to re-measure or re-calculate the invariant quantity after you've applied the transformation. You don't need to re-measure the angles of a triangle after you have rotated, reflected or translated it. And you don't need to recalculate the solution of a linear equation after you have multiplied each of its coefficients by the same constant.

Invariants can also help you identify objects of a certain type. When we think about triangles we usually don't care about where exactly they are located in space, or about the orientation they have been laid out in. What we care about is their intrinsic shape. This intrinsic shape is determined uniquely by three pieces of information. If you know either two sides of the triangle and the angle between them, or two angles of the triangle and the side between them, or the three sides,  then you can construct a triangle of this shape unambiguously. A suitable combination of invariants identifies the triangle. 

Similarly, the linear equations $ax+b=0$ and $acx+bc=0$ are two distinct objects. If, however, you are only interested in the solution of a linear equation, then you might not want to distinguish between the two. In this case, the invariant $-b/c$  uniquely determines the class of linear equations you're interested in.

Loosely speaking, you can think of mathematical invariants in terms of catching a criminal. Knowing the criminal's clothing or hair colour might not get you very far as these attributes can easily be changed. Knowing the criminal's fingerprints or DNA is much better — both of these are invariants under any transformation that changes the criminal's appearance.

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 our 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.