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”?

**Answer**

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

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