Benson Farb answered this beautifully in his University of Chicago commencement address. Here is an excerpt:

Since I am a pure mathematician, Dean Hefley suggested as a possible topic for this talk: “Why the square root of negative 1 is necessary”. I could take up this challenge of justifying pure science on its vast applicability; indeed the square root of negative 1, the basic “imaginary number”, underlies a huge swath of modern technology, from the design of circuits, airplanes and skyscrapers, to the construction of economic and financial models, to robotics. I have decided, however, to take the opposite point of view. I want to defend the value of basic science for its own sake...

...the purpose of pure mathematics, of basic science, is not the quick harvest. It is nothing less than an attempt to bring human thought and understanding to a higher level. It is an attempt to change not just what we think about the world, but how we think about it. The importance of this for human evolution is incalculable. As British physicist JJ Thomson said: “Research in applied science leads to reforms, research in pure science leads to revolutions.”

Benson Farb, 2012