You treat the complex numbers as polynomials, and add or subtract the real and imaginary part separately. For multiplication, remember that i2 = -1. Division is a little more complicated - but if you need it, I suggest you ask a more specific question about that.
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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.
'Math(s)' is the shortened word for 'Mathematics'. The word 'mathematics' comes from Classical Greece, and means 'to learn'.
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.