Math notes / Geometry and its relatives

From Helpful
Jump to: navigation, search
This is more for overview of my own than for teaching or exercise.

Overview of the areas

Arithmetic · 'elementary mathematics' and similar concepts
Set theory, Category theory
Geometry and its relatives · Topology
Elementary algebra - Linear algebra - Abstract algebra
Calculus and analysis
Logic
Semi-sorted
 : Information theory · Probability theory · Number theory · Decision theory, game theory · Recreational mathematics · Dynamical systems · Unsorted or hard to sort


Math on data:

  • Statistics as a field
some introduction · areas of statistics
types of data · on random variables, distributions
Virtues and shortcomings of...
on sampling · probability
glossary · references, unsorted
Footnotes on various analyses

Other data analysis, data summarization, learning

Data massage
Data clustering · Dimensionality reduction · Fuzzy coding, decisions, learning · Optimization theory, control theory
Connectionism, neural nets · Evolutionary computing



Intervals

Intervals are ranges within the real number set.

They are set definitions which include all values between the mentioned numbers. Whether the specified numbers themselves are included depends on whether they are mentioned/denoted to be open or not:

  • open intervals exclude both, e.g. 1 < x < 2
  • closed intevals include both, e.g. 1 ≤ x ≤ 2
  • half-closed/half-open intervals include one and not the other, e.g. 1 < x ≤ 2 or 1 ≤ x < 2
  • both open and closed intervals, as in the special cases of the empty/null set (∅), -∞ < x < ∞,


Notation and conventions

There are two different (and sometimes confusable) shorthand notations, one using round and square brackets (set builder notation), the other using only square brackets (ISO notation). Just stick to the one you are used to, but it can be handy to know about both.

For example:

  • 1<x<2 would be denoted as (1,2) and ]1,2[, respectively
  • 1≤x≤2 would be denoted as [1,2] and [1,2], respectively
  • 1<x≤2 would be denoted as (1,2] and ]1,2], respectively


Notes:

  • When the second number is larger than the first, it is usually agreed this refers to the empty set
  • (a,a), [a,a), and (a,a] are agreed to refer to the empty set (while [a,a] refers to a one-member set {a})
  • when mentioning infinity, inclusion does not make sense.


You are likely to meet intervals in definitions within calculus and analysis, thereby usually also in some pre-calc type class, and also in more theoretical mathematics.

See also:

Support

Trigonometry

Common trigonometric functions and identities

Note on geometries