Sequence in the context of Timekeeping

Time is the continuous progression of existence that occurs in an apparently irreversible succession from the past, through the present, and into the future. Time dictates all forms of action, age, and causality, being a component quantity of various measurements used to sequence events, to compare the duration of events (or the intervals between them), and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions.

Time is primarily measured in linear spans or periods, ordered from shortest to longest. Practical, human-scale measurements of time are performed using clocks and calendars, reflecting a 24-hour day collected into a 365-day year linked to the astronomical motion of the Earth. Scientific measurements of time instead vary from Planck time at the shortest to billions of years at the longest. Measurable time is believed to have effectively begun with the Big Bang 13.8 billion years ago, encompassed by the chronology of the universe. Modern physics understands time to be inextricable from space within the concept of spacetime described by general relativity. Time can therefore be dilated by velocity and matter to pass faster or slower for an external observer, though this is considered negligible outside of extreme conditions, namely relativistic speeds or the gravitational pulls of black holes.

The method of exhaustion (Latin: methodus exhaustionis) is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too.

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.

The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence.

In mathematics, a rational number is a number that can be expressed as the quotient or fraction ⁠${\displaystyle {\tfrac {p}{q}}}$⁠ of two integers, a numerator p and a non-zero denominator q. For example, ⁠${\displaystyle {\tfrac {3}{7}}}$⁠ is a rational number, as is every integer (for example, ${\displaystyle -5={\tfrac {-5}{1}}}$). The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also calledthe field of rationals or the field of rational numbers. It is usually denoted by boldface Q, or blackboard bold ⁠${\displaystyle \mathbb {Q} .}$⁠

A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).

In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.

Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy & Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs.

In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant). Two sequences are inversely proportional if corresponding elements have a constant product.

Two functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are proportional if their ratio ${\textstyle {\frac {f(x)}{g(x)}}}$ is a constant function.

In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain ⁠${\displaystyle P}$⁠ is a curve specified by a sequence of points ${\displaystyle (A_{1},A_{2},\dots ,A_{n})}$ called its vertices. The curve itself consists of the line segments connecting the consecutive vertices.
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In mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences".

Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning.

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff".

In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation). A string is often implemented as an array data structure of bytes (or words) that stores a sequence of elements, typically characters, using some character encoding. More general, string may also denote a sequence (or list) of data other than just characters.

Depending on the programming language and precise data type used, a variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements.

In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.

In linguistics, a numeral in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of grammar use the word "numeral" to refer to cardinal numbers that act as a determiner that specify the quantity of a noun, for example the "two" in "two hats". Some theories of grammar do not include determiners as a part of speech and consider "two" in this example to be an adjective. Some theories consider "numeral" to be a synonym for "number" and assign all numbers (including ordinal numbers like "first") to a part of speech called "numerals". Numerals in the broad sense can also be analyzed as a noun ("three is a small number"), as a pronoun ("the two went to town"), or for a small number of words as an adverb ("I rode the slide twice").

Numerals can express relationships like quantity (cardinal numbers), sequence (ordinal numbers), frequency (once, twice), and part (fraction).

In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.

Bertrand Russell used serial relations in The Principles of Mathematics (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.

A bitstream (or bit stream), also known as binary sequence, is a sequence of bits.A bytestream is a sequence of bytes. Typically, each byte is an 8-bit quantity, and so the term octet stream is sometimes used interchangeably. An octet may be encoded as a sequence of 8 bits in multiple different ways (see bit numbering) so there is no unique and direct translation between bytestreams and bitstreams.

Bitstreams and bytestreams are used extensively in telecommunications and computing. For example, synchronous bitstreams are carried by SONET, and Transmission Control Protocol transports an asynchronous bytestream.

A computer program is a sequence or set of instructions in a programming language for a computer to execute. It is one component of software, which also includes documentation and other intangible components.

A computer program in its human-readable form is called source code. Source code needs another computer program to execute because computers can only execute their native machine instructions. Therefore, source code may be translated to machine instructions using a compiler written for the language. (Assembly language programs are translated using an assembler.) The resulting file is called an executable. Alternatively, source code may execute within an interpreter written for the language.

In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set.

There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements.

In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinitieth" item in a sequence.

Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.

In computer science, a list or sequence is a collection of items that are finite in number and in a particular order. An instance of a list is a computer representation of the mathematical concept of a tuple or finite sequence.

A list may contain the same value more than once, and each occurrence is considered a distinct item.