Useful Notes On Mathematics

Give Useful Notes On Mathematics here.

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Mathematics is one of the oldest and well-developed disciplines, due to having some features of science: there are multiple branches that all build on each other; a theory never falls out of prominence but just gets absorbed into a "more correct" or "more general" theory, and very rigorous notions of what is true and what can be claimed to be correct. It also skirts the major drawbacks of science, the biggest two being that it is a natural thing that you can develop on your own and you don't need a lot of background to learn stuff, and there is no large money requirement to perform experiments as all you need is pencil, paper, chalk and a blackboard to do research.

It is also hated by roughly 99.9% of all people. It is the major reason that there aren't as many scientists, doctors, engineers, and Wall-Street workers. This is probably due to the way it is taught: 6 to 8 years of number crunching that a four function calculator can do followed by a massive, unmotivated generalization. It doesn't help that elementary school teachers are not trained in mathematics and very few schools have a special math teacher. This system is guaranteed to lose everyone that might have been interested in mathematics, see this pretty equation: Boredom + Steep difficulty curve + Blind leading the blind =/= good feelings towards mathematics.

There are four major disciplines in mathematics: Algebra, Analysis, Geometry, and Number Theory. Algebra is the study of structures that have operations like addition and multiplication, functions between these structures, and how to understand large structures by looking at the small structures that you can use to make them. Analysis is calculus done rigorously, with a deep understanding of the real line and of sufficiently "nice," for suitable definitions of nice functions on it. There is also Complex Analysis, where you do everything that you did in calculus in the complex numbers. You probably have a good idea of what Geometry is, so I'll just say that it is the study of n-dimensional shapes, and the global properties of them. Number Theory is the understanding of the integers, and other structures that are very similar to integers. One of the main problems is to understand the distribution of the primes, and the most famous unsolved problem in all of mathematics is related to the distribution of the prime numbers (the Riemann Hypothesis).

There is also applied mathematics, which is mainly probability and statistics, differential equations and dynamical systems, and combinatorics. The idea of probability and statistics is that if we give you a theoretical model, can you tell us what results we would expect and vice-versa? Differential equations and dynamical systems is related mainly to chemistry and physics: I give you very general rules that a system follows, and you tell me what sort of solutions can happen. Combinatorics is counting things. It is also murderously hard (how many ways are there to write 4953 as the sum of smaller numbers if we allow repetition and order doesn't matter?).

Most math researchers work in a subfield of the areas listed above, and have a special type of problem that they work on. The biggest field is probably Algebraic Geometry, as there are many ways to get at it, and there are many theories that can be useful in solving problems there.

The most common depiction of mathematics in media is of dry lectures in high-school, where the teacher is 100% unaware of the class and assigns tons of homework. If the setting is college, then the professor is insane, but the lectures are only a little more interesting. The homework is a lot less but very hard, and it still is an unenjoyable experience. If an ad wants to show off some math on a blackboard, there is stuff from physics with a lot of integral signs and Greek letters. They don't use actual math, as most people don't believe that there are that many words in actual math research.