>>10471795

>All elementary math in just one post

Basic math, just look at the axioms (sometimes called rules) of whatever field you're working in.

A good start is the nine field axioms of Q, which you can lookup anywhere. The axiom a*a^-1 = 1 gives us fractions. The nine field axioms encapsulate the arithmetic of numbers, high school algebra, and many other algebraic systems. (Axiom systems also exist for geometry, number theory, and for mathematics as a whole, though they aren't totally complete, you can still get a solid overview by looking at them.) A system satisfying the nine field axioms has the structure of a field. The first field that we all meet is the system Q of rational numbers, or fractions, but there are many more fields. If we drop the axiom about a^−1, we get eight axioms for a more general structure called a ring. The first ring that we all meet is the system Z of integers. Learn the axioms, now go practice solving some equations in that field, using said axioms. Not very hard is it? There's also an algebraic number field Q(α) to deal with irrationals, it has it's own field axioms. In Geometry, you can use vector spaces over fields to construct new ones. Q(α) is a vector space over Q whose dimension equals the degree of α. See how this is all fitting together? Moving on to more axioms, we also have the theory of finite sets, or axiom of foundation. Finite set theory provides objects like graphs. Finally, Zermelo-Fraenkel axioms (ZF) gives us infinity. When we include the Infinity axiom we get the set N of natural numbers, and its power set P(N), which is effectively the set R of real numbers. From R we can build the concepts of geometry and analysis, and virtually all of classical mathematics. Thus, ZF encapsulates a vast amount of mathematics. This can all be learned in any analysis intro text, like Tao's books, how to build fields (structures), their axioms and how to can derive proofs in that field using the axioms, ect.