Overview of Mathematics
Mathematics makes use of symbols and names somewhat liberally, allowing for a certain amount of conciseness and expressiveness, but not always disambiguation. Notion can be context sensitive, be careful.
Also, this is not a guide, or a lesson. I’m mostly trying to catalog useful notation and expressions.
Reader Be Warned
- Operations & Expressions
- Linear Algebra
- Set/Group/Category Theory
- Real Analysis*
- Graphics and Plotting
- Abstraction & Application
Logic is the foundation of all reasoning and thought. Though, not all logics are created equal.
Formal logic denotes a collection of symbols and rules for these symbols. Used carefully, one can start here and build whole models of mathematics and thus anything.
“Not” is written as (or by programmers). For example, all roads do not lead to Rome. Let’s call this proposition .
We could say that all roads do in fact lead to Rome, and that would be . This could also be written as (or even in Kleene’s “Mathematical Logic”).
“And” and “Or” are written as , and , respectively. We obviously can’t claim is true.
(also in Kleene’s “Mathematical Logic”)
We can now formally state in our observer language, , for example.
Approximation: , ,
Somewhere along the way to Rome I might ask, “How long has it been since we left camp?”, which, unless someone or something was perfectly counting, could only be met with hours. It might also be useful to write . Looking over at the travelers along side us, we recognize a few from hours ago when we left. We could say our distance () and their distance () are related as , or even .
Numbers are the primary objects of math. You can use numbers to count, for example, , Go! You can use numbers for measures of units, for example, the current temperature .
Fractions are for frogs, , ribbit.
Operations & Expressions
Math is full of various operations you can write into valid (or invalid) mathematical notation. Values can be computed, like , or .
All hail the mighty equal sign, . Simply put, the equal sign says two things are identical, or equal. We call this an equation.
A really obvious equation might be , while we can represent something as complex as the concept of a limit as simply as or .
Many equations have letters in them, which can allow a fixed value to change with the variable, for example grows quickly, but even faster. Some variables are used to represent physical, or conceptual properties, for example describes the edge lengths of a triangle, where and and the sides, and is the hypotenuse.
Summation and Production
Vectors and Matrices
Vectors come in a many forms. For example as a or as three (in this case) coefficients along with component unit vectors starting with .
Vectors may also be written in matrix form as or .
Matrices are just vectors of vectors.
A is a way to talk about the convergence of something.
Integral inside text looks nice too.
This is where things can get “fun”.