The history of mathematics is the study into the origins of math. Its uncertain when math originated but it has been found in use since the Ancient Egyptian empire in 2000 BC.
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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.
Only a person who does not understand mathematics is disadvantaged. Mathematics in itself has no disadvantages.
Depending on what type of science, yes mathematics can be science. Like if you're doing physics that can be mathematics.
Mathematics is used extensively in the physical sciences, by mathematics itself is not a physical science.