Math notes / Geometry and its relatives

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Geometry concerns itself with shapes, sizes, relative positions, orientations and such, basic transformations applied to points and shapes, and such.

It is taught for its general usefulness in various fields.

Particularly that practical side mostly focuses on euclidean space, though geometry courses tend to briefly go into polar coordinates, and sometimes basic cases of curves and manifolds.

Parts of geometry cross over into algebra, and into other fields.

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:

• http://en.wikipedia.org/wiki/Interval_(mathematics)