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).
Core Idea¶
Integral part is treated here as the recurring mathematicslogicstatistics 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.
Scope of Application¶
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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.
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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.
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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.
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Notation. In some sources, boldface or double brackets are used for floor, and reversed brackets or for ceiling.
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Notation. The fractional part is the sawtooth function, denoted by for real and defined by the formula.
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).
Manages Complexity¶
Integral part compresses multiple mathematicslogicstatistics 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.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics 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.
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.
Neighborhood in Abstraction Space¶
Integral part sits in a crowded region of the domain-specific corpus (34th 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
- Absolute value — 0.90
- Binade — 0.89
- False position method — 0.88
- Filling radius — 0.88
- Two-Element Boolean Algebra — 0.87
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