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A lot of things. For example: partial differential equations are used to find weak points in aircraft before they fly, group theory is used to encrypt data over the internet, multi-particle stochastic processes, can tell us how diseases will spread, and how traffic behaves on roads, as well as how an economy works.

Maths is also used in all sciences - everything scientists do they could not do without mathematics.

On a lower level, it allows us to count how many things we have, and to decide for instance that 3 loaves of bread is better than 2.

Q: What is mathematics for?

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Pure Mathematics is the branch of mathematics that deals only with mathematics and how it works - it is the HOW of mathematics. It is abstracted from the real world and provides the "tool box" of mathematics; it includes things like calculus. Applied mathematics is the branch of mathematics which applies the techniques of Pure Mathematics to the real world - it is the WHERE of mathematics; it includes things like mechanics. Pure Mathematics teaches you HOW to integrate, Applied mathematics teaches you WHERE to use integration.

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