We can easily think of numbers as the symbolic referents for and the addition of points in space and time. A point is any [PoW] part of the world that can be separated from all other [PoW]s, and a point can be a concept that has no spatial or temporal existence in empirical reality.
"One" or "1" is the name for one point. "Two" or "2" is the name for two points. "Zero" or "0" is the absence of points or, in other words, a symbol that represents no point or points. This is a page about the ways mathematicians have grouped and categorized the interesting sets of points that make up a particular type of numbers.
There are no numbers in empirical reality but only things which numbers can be applied to.
There are three apples on a table with three apples, but the application of the property "3" or "three" is the application of a mental construct to empirical or mental reality. Three apples can exist as ideas in the mind as well as they can exist on a kitchen table.
Numbers are the basic elements or objects that are used in mathematics. There have been numerous important divisions concerning the types of numbers that exist.
In this description of numbers, the term set is used. A separate area of study, related to mathematics and logic, is set theory. A set in math and logic is made up of members or elements of that set. A set is a collection of objects (members or elements) that are referred to as well-defined. Well-defined means defined logically or mathematically using axioms. Axioms are assumed principles.
For example, the set of all stars in our solar system includes one member, the sun. This could be written, if "S" stood for "The set of all stars in our solar system", as S = {sun}.
Members or elements of sets can also themselves be sets called subsets. For example, the set of animals "A" includes the subset of reptiles "R" which itself includes the subset of snakes "S" and so on. It is common to label the set and write it as equivalent to a bracketed list, such as A = {reptiles, mammals, amphibians, ...}, R = {snakes, alligators, crocodiles, ...}, S = {cobras, rattlesnakes, coral snakes, ...}.
The following are commonly used number types:
Natural numbers are the set of all positive integers, written as N = {1, 2, 3, 4, 5, ...}. "N" stands for the set of natural numbers. These are often thought of as the counting numbers, as we do not count with zero or negative numbers.
Whole numbers are the set of all positive integers and zero (or the set of all non-negative integers), written as {0, 1 , 2, 3, 4, ...}.
Integers are the set of all negative and positive integers and zero, written as Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}. "Z" stands for the set of integers. Natural numbers and whole numbers are both subsets of integers.
Real numbers are the set of any number that can occur on a number line, which means it includes the subsets of rational numbers and irrational numbers. Real numbers are written as R = {x, x is on the number line}. "R" stands for the set of real numbers.
The number line below consists of the descending negative numbers in orange, the origin at zero in white, and the ascending positive numbers in green. If any number x is to the left of any number y on the number line, then x < y and, if any number x is to the right of any number y on the number line, then x > y.
The absolute value of a number is the distance that number is from the origin or zero. Therefore, because absolute value measures distance, it is always non-negative. Absolute value is written as a number, x, with a bar on either side, as in | x |. Therefore, | x | = x and | -x | = x.
Rational numbers are the set of all numbers that can be written as one integer over another except where the denominator is zero, written as Q = {x / y, where x and y are integers, and where y 
0}. "Q" stands for the set of rational numbers and refers to the quotient, or the result of dividing the numerator or dividend, x, by the denominator or divisor, y. If x = 10 and y = 2, then the numerator or dividend is 10, the denominator or divisor is 2, and the result or quotient is 5.
Rational numbers are repeating or terminating in terms of their decimal remainder.
Irrational numbers are the set of all real numbers that are not rational, written as I = {x, where x is a real number that is not rational}. "I" stands for the set of real numbers that are not rational.
An irrational number can not be written or represented as one integer divided by another integer. For example, 
(the symbol for pi, which is the ratio of any circle's circumference to its diameter, or = circumference / diameter = C / d) is an irrational number that is a repeating decimal, rounded to 3.14159. Irrational numbers are non-repeating, non-terminating in terms of their decimal remainder.
Fundamentally, there is nothing reasonable or unreasonable (rational or irrational) about numbers that don't repeat but naming them in this way is helpful in distinguishing them.
Complex numbers are any number of the form [ a + bi ] where [ a ] and [ b ] are real numbers and [ i ] is the imaginary unit.
The imaginary unit, [ i ], is a number whose square is [ -1 ]. Because no number squared is equal to [ -1 ], the special case of the imaginary unit is used.
When [ b = 0 ], the complex number is a real number because anything times zero is zero, leaving only the [ a ] in the expression [ a + bi ]. When [ b 0 ], then the complex number is called any imaginary number because it is multiplied times the imaginary unit, [ i ]. The [ a ] part of a complex number, of the form, [ a + bi ] is called the real part and the [ b ] or the [ bi ] is called the imaginary part.
Even numbers are any number that is divisible by [2] without a remainder, like [10], which, when divided by [2] is [5]. Odd numbers are numbers that are not divisible by two without a remainder like [5], which, when divided by [2] is [2.5].
Prime numbers are numbers that have only the factors of one and itself. Factors are the numbers that, when multiplied, are equal to the number being factorized.
A prime number, such as 2, has the factors 2 and 1. A prime number, such as 3, has the factors 3 and 1. 4 is not a prime number, because it can be factored in two different ways, [2 x 2] and [1 x 4].
The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The number of prime numbers is infinite. A prime factorization is the list of all prime numbers that, when multiplied, equal the number being factorized. The prime factorization of 24, then, would be [12 x 2] = [3 x 4 x 2] = [3 x 2 x 2 x 2]. Prime factorization is important because it allows fractions to be reduced to their lowest common denominators.
There are infinite ways to categorize and label numbers, and many more number types are used in various mathematical fields. For example, perfect numbers are the numbers that are the sum of their positive divisors not including themselves. 6 is the first perfect number, as 1 + 2 + 3 = 6. The next is 28 because 1 + 2 + 4 + 7 + 14 = 28.
In the next section [1.02] about lines, we examine the space between two points as it is referred to as a line.
