Geometry/Chapter 11

Section 11.1 Number Sets and Fields

You are familiar with many or all of the terms natural numbers, integers, rational numbers as well as irrational numbers, real numbers, complex numbers. Geometry uses abbreviation and set notation for these. A set is a number of unique objects such as numbers. The following proposition describes number sets abbreviated thus: natural numbers are $\mathbb {N}$ , integers are $\mathbb {Z}$ (from the German word zahlen [ˈtsaːlən]), rationals are $\mathbb {Q}$ (quotients,) reals are $\mathbb {R}$ , and complexes are $\mathbb {C}$ , and each set on the left is a subset, i.e. set contained within a set, of all sets on the right. That is indicated by a symbol similar to $<$ , because each set on the right is either greater than all on the left, or seems to be greater than all on the left, though in some cases they are equal. It is fine if you know how certain ones are equal, but mainly note that the ones on the right have greater complexity than all to their left.

proposition 11.0

$\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C}$

Let us review the definitions of these sets with set builder notation. Following are the natural numbers; there are two ways of defining them. All objects within braces are elements, i.e. members, of the same set.

counting numbers

$\{1,2,3,\ldots \}$

whole numbers

$\{0,1,2,3,\ldots \}$

Integers are counting numbers and negative whole numbers: clearly any counting number can be called an integer.

whole numbers

$\mathbb {Z} =\{\ldots ,-3,-2,-1,0,1,2,3,\ldots \}$

Each number set represents a number line, and a set with a subscript means numbers from 0 to 1 less than subscript, for example $\mathbb {Z} _{3}=\{0,1,2\}$ . That is a reason counting numbers start at 0.

The following statement uses the symbols |, which means 'such that the following is true' (people usually just say 'such that,') and $\in$ (Greek letter eta). The object to the left of eta is an element of the set to the right of eta. Any integer can be written as a rational, but rational numbers include fractions/decimals.

proposition 11.1

$\mathbb {Q} =\left\{{\frac {p}{q}}:p,q\in \mathbb {Z} ,q\neq 0\right\}$

The rational, real, and complex numbers are fields, which are higher categories than sets. One property of fields is that they are ordered in a certain more complex way; it is good if you noticed the way to define rationals uses some more complex order.

proposition 11.2

$\mathbb {R}$ eal numbers are simply sums of rationals for which they equal infinite decimals, such as $3.141592\ldots$

Any rational number can be written as a real one (such as with infinite zeroes afterwards,) but reals include irrational numbers. There are more complicated ways to define the reals which you do not need to know yet, but if you want to, read about Cantor's snake.

$\mathbb {C}$ omplex numbers include imaginary numbers, and any non-imaginary number can be written as a complex one. These are not used much in geometry except advanced topics, but there is something you might find interesting about them. A number set with a superscript n means the set in n dimensions, defining n orthogonal number lines.

proposition 11.3

$\mathbb {R} ^{2}=\mathbb {C}$

Section 11.2 Dimensional Sets in Real Space

A ray represents $\mathbb {N}$ , and lines represent $\mathbb {Z}$ , $\mathbb {Q}$ , but the ray and lines are more continuous ('appearing solid') than the sets.

proposition 11.4

$\mathbb {R}$ represents the real number line, which is infinite.

A number set with a sign superscripted means the set in that sign, and objects on the left and right of a $\cup$ symbol are defined to be in the same set. For example $\mathbb {R} ^{+}\cup 0$ is a ray from 0 to infinity in real space. Recall that $\mathbb {R}$ is a line, and note that it is a real (continuous; appearing 'solid') line, and recall that $\mathbb {R} ^{2}$ is 2 orthogonal lines.

proposition 11.5

$\mathbb {R} ^{2}$ represents the Cartesian real plane, which is infinite.

proposition 11.6

$\mathbb {R} ^{3}$ represents a 3-dimensional Euclidean space, which is infinite.

optional proposition (for enthusiasm/fun) 11.7

$\mathbb {R} ^{4}$ represents a 4-dimensional Euclidean space, a continuum or part of one, which is infinite.

Points in $\mathbb {R} ^{4}$ are defined $(x,y,z,t)$ and it is used to described temporal geometric operations such as rotations, translations, and transformations, which are the main things you might use it for in geometry, but the set also defines 4-dimensional objects.