Academics are always being asked to demonstrate the “impact” of their research. (Is it like being hit by a rogue cyclist? Or is it more like a pile-driver, or even an asteroid strike?) But while it is not unreasonable to ask whether a particular piece of academic research is useful, the difficulties in answering the question are extraordinary.

The quality of a piece of research is subjective, and using measures such as the number of peer-reviewed articles published simply outsources the subjective judgment to somebody else. But there is a deeper problem: in a complex world, it is impossible for anyone to judge what the significance of a research breakthrough might eventually be.

Nowhere is this more true than in the field of mathematics. The most famous example is the development of imaginary numbers. The very name conveys the supposed uselessness of the concept. Square the imaginary unit, i, and you get minus one. Baffling.

Imaginary numbers were regarded with great scepticism after they were developed in Bologna in the 16th century as the logical solution to an abstract problem. Eventually, however, they turned out to be essential for, among other applications, electrical engineering – hardly something that could have been imagined by their creators.

So are imaginary numbers typical of the unexpected bounties of pure mathematics – or an unrepresentative poster child? Two recent commentators have tried to expand the number of examples. Professor Caroline Series of the University of Warwick devoted a recent presidential lecture at the British Science Festival to this topic, focusing on the applications of non-Euclidian geometry.

When Euclid originally laid down the axioms of his geometric system 23 centuries ago, one of them seemed less than obvious. For 2,000 years mathematicians tried to derive the “fifth postulate” – equivalent to the claim that the internal angles of a triangle add up to 180 degrees – from more basic building blocks, and failed. Eventually it transpired that the axiom was optional. Consistent systems of geometry were possible in which the internal angles of triangles summed to more than 180 degrees, or even to fewer.

Surfaces on which the sum of angles in a triangle is less than 180 degrees look like leaves of kale. Prof Series points out that the development of kale-like geometric systems, called hyperbolic geometry, initially seemed a curiosity but made possible Einstein’s theory of special relativity. Now, says Prof Series, hyperbolic geometry promises to advance our understanding of the way complex networks such as the internet behave and grow.

Peter Rowlett, a maths educator and historian, recently gathered further examples together in the journal Nature. The “sphere packing problem” – beginning with the conjecture that grocers have found the most efficient way to stack oranges – has been an open area of research for four centuries, but in the 1970s a solution for eight-dimensional “spheres” was used to design efficient modems. This meant that internet access no longer required specialised cables.

Quaternions, which extend imaginary numbers into a further dimension, began to be developed by William Hamilton in Dublin in 1843. They were eclipsed by matrix algebra, before being rediscovered as indispensable for generating 3D computer graphics efficiently. Rowlett’s contributors offered several other examples.

Cost-benefit analysis has its place. But the benefits of academic research can pop up in such unexpected ways, sometimes immediately and sometimes after centuries. We should not set too much store by any bureaucrat’s analysis of “academic impact”.

Also published at ft.com.