Can you provide me historical examples of pure mathematics becoming "useful"? I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go.

Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that according to the history of mathematics, the math that is useful today was once pure mathematics (I'm not so sure but I guess that when the calculus was invented, it hadn't a practical application).

Also, I guess that the development of pure mathematics is important because it allows us to think about non-intuitive objects before encountering some phenomena that is similar to these mathematical non-intuitive objects, with this in mind can you provide me historical examples of pure mathematics becoming "useful"?
Here are few such examples


Public-key cryptosystems based on elliptic curves, factorization, trapdoor functions, lattices and hyperelliptic curves. 
Use of algebraic topology  in distributed computing and sensor networks. Topology has uses in various other branches engineering as well.
Use of differential geometry for computer graphics, computer vision algorithms, robotics and general relativity.
Error-correcting codes based on algebraic geometry. Algebraic geometry
has also applications in robotics.
Lattice theory is used in program analysis and verfication.
Group theory is used in chemistry as well as physics
Tropical geometry, a branch of algebraic geometry, has applications in mathematical biology.
Digital electronics is impossible without Boolean algebra.
Topos theory has been applied to music theory
This may be a little late but I think the most basic and pure proof and yet the most astonishing was Euclid's

Wikipedia Entry on Euclid

There are uses in Digital Media, Cryptography, Physics and Engineering.

A lot of Pure math is knowing how to apply it, most theories have a specific problem set they are known to solve because they are designed that way or were found to solve that problem set that way but when you apply theories in ways which are not typical of the solution you incite innovation and expand your horizons.