Integral part¶

The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations).

Version: v1 · 2026-09-28 · History
Domain-specific #: 10079
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Elementary Functions, Number Theory → Mathematics

Core Idea¶

Integral part is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations).

In mathematics, the floor function is the function that takes a real number as input and returns the greatest integer less than or equal to , written or. Similarly, the ceiling function returns the least integer greater than or equal to , written or. For example, for floor: , , and for ceiling: , and.

The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). However, the term "integer part" is ambiguous, as it can also mean truncation towards zero, which differs from the floor function for negative numbers. Although and are equal for non-integer values of , and thus produce graphs that appear exactly alike, they differ when is an integer.

For Integral part, the abstraction is narrower than the article's general subject matter: a positive case must preserve The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The integral part or integer part of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula.
Constitutive relation — The fractional part is the sawtooth function, denoted by for real and defined by the formula.
Operating condition — Given real numbers and , integers and and the set of integers \mathbb{Z} , floor and ceiling may be defined by the equations.
Recognition evidence — Division by positive integers gives rise to an interesting and sometimes useful property.
Admissible variation — For an integer and a positive integer , the modulo operation, denoted by , gives the value of the remainder when is divided by.
Characteristic consequence — Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.
Failure boundary — The exponent of the highest power of that divides is given by a version of Legendre's formula.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations).
Not an over-broad reading. The above are never true if is not an integer; however, for every and , the following inequalities hold.
Not an over-broad reading. At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for fixed and a multiple of the Fourier series given converges to , rather than to.
Not an over-broad reading. (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.
Not automatically Arithmetic function. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Integral part applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Notation. (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.
For in the critical strip ,. In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.
Beatty sequence. This formula is valid for all with real part greater than −1, (except , where there is a pole) and combined with the Fourier expansion for can be used to extend the zeta function to the entire complex plane and to prove its functional equation.
Notation. In some sources, boldface or double brackets are used for floor, and reversed brackets or for ceiling.
Notation. The fractional part is the sawtooth function, denoted by for real and defined by the formula.
For all x,. LaTeX has supported UTF-8 since 2018, so the Unicode characters can now be used directly.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Integral part names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). The strongest recognition evidence in the frozen account is: Division by positive integers gives rise to an interesting and sometimes useful property. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The above are never true if is not an integer; however, for every and , the following inequalities hold. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Integral part compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—the fractional part is the sawtooth function, denoted by for real and defined by the formula.—and the practical consequence—gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations).
Check operation and conditions. Given real numbers and , integers and and the set of integers \mathbb{Z} , floor and ceiling may be defined by the equations.
Demand recognition evidence. Division by positive integers gives rise to an interesting and sometimes useful property.
Test variation. Change an implementation or setting while preserving for an integer and a positive integer , the modulo operation, denoted by , gives the value of the remainder when is divided by.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Integral part transfers literally when a new case preserves the same carrier type, relation, and recognition test. (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article. In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.

Beyond the home domain. No canonical parent is asserted for Integral part. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Since the right-hand side of the general case is symmetrical in and , this implies that. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations); recognition evidence → Division by positive integers gives rise to an interesting and sometimes useful property

Applied / In Practice¶

is the number of letters in the alphabet (e.g., 26 in English). The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Number of strings without repeated characters; invariant → The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations); boundary → the case exits the class when the above are never true if is not an integer; however, for every and , the following inequalities hold

Structural Tensions¶

T1 — Stable identity versus admissible variation. The above are never true if is not an integer; however, for every and , the following inequalities hold. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for fixed and a multiple of the Fourier series given converges to , rather than to. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Since floor and ceiling are not periodic, they do not have uniformly convergent Fourier series expansions. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The integral part or integer part of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Integral part literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. The fractional part is the sawtooth function, denoted by for real and defined by the formula. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Integral part distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Integral part is structural-leaning. Its structural side is the repeatable organization summarized by The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Given real numbers and , integers and and the set of integers \mathbb{Z} , floor and ceiling may be defined by the equations. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The integral part or integer part of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. The fractional part is the sawtooth function, denoted by for real and defined by the formula. It further constrains recognition and variation through: Given real numbers and , integers and and the set of integers \mathbb{Z} , floor and ceiling may be defined by the equations. Division by positive integers gives rise to an interesting and sometimes useful property.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Integral part literal. Its documented scope includes the condition that (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article. Another bounded application condition is that In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—For an integer and a positive integer , the modulo operation, denoted by , gives the value of the remainder when is divided by.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Integral part. The reviewed identity is: The floor of is also called the integral part, integer part, greatest integer, or entier of, and was historically denoted (among other notations). The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Integral part sits in a crowded region of the domain-specific corpus (34^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Number Systems & Symmetry (8 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations)?
Arithmetic function. A function defined on positive integers, usually with complex values, that encodes a number-theoretic property of each integer. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Irreducible Floor. A quantity of interest has a structural lower (or upper) bound that the available proximate levers cannot push past without inducing pathology elsewhere, because the floor is a consequence of the system's generating mechanism rather than a target the operator chose. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Floor Area. Assign a purpose-qualified area to a building or part by selecting level boundaries, inclusions, exclusions, and allocation rules under a named measurement standard. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Integral part remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Floor_and_ceiling_functions (revision 1369276806).
Preserved source candidate: http://www.mathwords.com/f/floor_function.htm
Preserved source candidate: http://www.mathwords.com/c/ceiling_function.htm
Preserved source candidate: https://www.latex-project.org/news/latex2e-news/ltnews28.pdf
Preserved source candidate: http://math.colgate.edu/~integers/w33/w33.pdf
Preserved source candidate: http://en.cppreference.com/w/cpp/numeric/math/floor
Preserved source candidate: http://en.cppreference.com/w/cpp/numeric/math/ceil
Preserved source candidate: https://docs.microsoft.com/en-us/dotnet/api/system.math.floor
Preserved source candidate: https://docs.microsoft.com/en-us/dotnet/api/system.math.ceiling

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.