Useless math that became useful

Solution 1:

Quote from G. H. Hardy¹

The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics.

Just 30 years after his death, the RSA algorithm was introduced which is deeply rooted in number theory and is now important part of sending encrypted information electronically, e.g., over the Internet.

Solution 2:

The Quaternions were considered useless for a long time.

Anyhow, the set of all unit quaternions is a double cover of $SO_3(\mathbb R)$. This allows us to represent any rotation matrix by a quaternion, which is used now in computer games (instead of using 9 parameters to parametrize a rotation matrix, we can use only 3 for the quaternions).

You can read more here.

Solution 3:

The solutions of the closest-packing problem, of how to most densely pack non-overlapping congruent $n$-spheres in $\mathbb R^n,$ have, for some $n>3,$ been found to have applications to error-detecting and error-correcting codes in digital transmissions.

In the book Knots by Kaufman there is an example of an application of knot theory (the study of homeomorphic embeddings of $S^1$ into $\mathbb R^3$) to statistical mechanics.