NUMBERS
The following terms and their respective definitions describe all types of numbers. A number is a piece of information which non-ambiguously represents exactly one of limitlessly many unique abstract patterns.
To view hidden text inside each of the preformatted text boxes below, scroll horizontally.
ONE: the smallest natural number; the length of the line segment whose endpoints are adjacent integers on some dimensional axis of a Cartesian grid.
ZERO: the quantity which represents the absence of some countable phenomenon; the integer which represents the halfway point between negative one (-1) and positive one (1) on some dimensional axis of a Cartesian grid.
NUMBER: a piece of information which represents exactly one finite quantity.
Depending on how a particular number is represented, a number may or may not be represented using a finite number of digits. For example, the fraction 1/3 is a finite sequence of symbols representing the number 1 divided by the number 3 (and the result of that division is a non-integer number with infinitely many digits (i.e. 1.3333333…) when represented as a base-ten number).
Numbers are abstract phenomena but are communicated and stored using physical resources inside of and between information processing agents.
INFINITY: the (hypothetical or actual) instantiation of limitlessly many copies of exactly one pattern or the (hypothetical or actual) instantiation of limitlessly many unique patterns.
Although infinity is not an actual number, the infinity symbol (∞) is sometimes used in mathematical expressions as though it were a number. For example, infinity is used as the upper bound to the limit which the input value to the function f(x) = 1/x approaches as that input value x increases indefinitely:
// As x approaches infinity, f(x) approaches (but never equates to) the limit 0. // In other words, as x increases indefinitely (assuming x is a positive real number), f(x) shrinks indefinitely (but never to absolute zero). lim f(x) = 0 // f(x) = 1/x. Hence, f(1) = 1, f(2) = 0.5, f(666) = 0.0015015015, and f(999999) = 0.000001. x -> ∞
A limit which is similar to the one defined above swaps the infinity symbol and zero:
// As x approaches zero, f(x) approaches (but never equates to) the limit ∞. // In other words, as x decreases indefinitely (assuming x is a positive real number), f(x) grows indefinitely (but never to ∞). lim f(x) = ∞ // f(x) = 1/x. Hence, f(1) = 1, f(0.5) = 2, f(0.0015015015) = 666, and f(0.000001) = 999999. x -> 0
NATURAL_NUMBER: an element of the infinitely large set whose elements consist exclusively of one and of every unique sum comprised of one being added to itself for some finite number of additions.
"X" -> "One (1) countable instance of the pattern X". "XX" -> "Two (2 (i.e. 1 + 1)) countable instances of the pattern X". "XXX" -> "Three (3 (i.e. 1 + 1 + 1)) countable instances of the pattern X".
INTEGER: an element of the infinitely large set whose elements consist exclusively of each natural number, each natural number multiplied by negative one, and zero.
An integer can be represented by a finite sequence of binary digits (with the negative sign being a particular binary digit).
array_of_integers := [-666, 0, 22, 1, 303, -78, 5, -45]
RATIONAL_NUMBER: an element of the infinitely large set whose elements consist exclusively of each integer and each ratio (A/B) such that A (i.e. the numerator) represents any integer while B (i.e. the denominator) represents any integer other than zero.
is_rational_number(1/3) = true. is_rational_number(1/1) = true. is_rational_number(1/3) = true. is_rational_number(square_root(2)) = false. is_rational_number(square_root(1)) = true. // square_root(1) = 1. is_rational_number(square_root(0)) = true. // square_root(0) = 0. is_rational_number(square_root(-1)) = false. // i := square_root(-1). // i is an imaginary number. Each rational number is a real number. is_rational_number(0/1) = true. // (0/1) = 0. is_rational_number(0/0) = false. // Infinity is not a number. is_rational_number(1/0) = false. // Infinity is not a number.
IRRATIONAL_NUMBER: an element of the infinitely large set whose elements consist exclusively of real numbers which are not rational numbers.
An example of an irrational number is Pi.
REAL_NUMBER an element of the infinitely large set whose elements consist exclusively of numbers which each represents a specific point along some dimensional axis of a Cartesian grid.
IMAGINARY_NUMBER an element of the infinitely large set whose elements consist exclusively of numbers which are each the product of the square root of negative one multiplied by some real number.
i := square_root(-1). // imaginary number (i * i) = -1. // real number ((i * i) * i) := ((-1) * i). // imaginary number
COMPLEX_NUMBER: the sum of a real number and an imaginary number.
(2 * i) + 3. // complex number (2 * i). // imaginary number (1 * i). // imaginary number (0 * i) = 0. // imaginary number to the left of the equal sign and real number to the right of the equal sign (0 * i) - 8 = -8. // complex number to the left of the equal sign and real number to the right of the equal sign (3.33 * i) + 0 = 3.33 * i. // complex number to the left of the equal sign and imaginary number to the right of the equal sign
ALGEBRAIC_REAL_NUMBER: a number which is the root of a non-zero polynomial equation such that the coefficients of that polynomial equation are rational numbers.
An example of an algebraic number is the Golden Ratio.
golden_ratio := (1 + (5 ^ (1/2))) / 2. some_polynomial := y = (x ^ 2) − x − 1. proof that golden_ratio is a root of some_polynomial: 0 = (((1 + (5 ^ (1/2))) / 2)) ^ 2) − ((1 + (5 ^ (1/2))) / 2) − 1.
TRANSCENDENTAL_NUMBER: a real or complex number which is not an algebraic number.
An example of a transcendental number is Euler’s Number.
This web page was last updated on 10_JULY_2025. The content displayed on this web page is licensed as PUBLIC_DOMAIN intellectual property.