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This one is probably a bit unusual for you. How would you explain what abstract algebra is to a bunch of curious family who haven't been to college. i may as well be in martian as a way of explaining where they are in maths knowledge.

Why are Z,Q and R considered reserved variables for number sets but not i ?

There are many ways to explain abstract algebra, and given the name "algebra" there is some hope that someone who only knows high-school level (regular) algebra would be able to understand this.

Regular algebra is the study of a particular type of numbers (the usual type, rational or real numbers) which obey particular rules. These rules include the commutative, distributive, and associative laws and various other laws.

(The reals/rationals have other properties too, relating to other types of mathematics. They have particular ordinal and topological properties.)

Abstract algebra is the study of other types of numbers, where these laws do not apply -- other laws apply instead. The study of groups (Lie groups, Coxeter groups, braid groups, semi-direct products etc.) gives a law for commuting other than ab=ba. Some algebraic structures don't even obey the associative law.

So, in some sense, abstract algebra is just the natural generalization of normal arithmetic. What happens when the "rules of the game" change? What system(s) of numbers do we get?

As for notation like Z, Q, R for the integers, rationals, and reals (respectively), and the letter i for the imaginary unit, these are mostly historical. You can read a little bit about this on Wikipedia.

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