NationMaster - Encyclopedia: Floor function

In mathematics, the floor function of a real number x, denoted $lfloor x rfloor$ or floor(x), is the largest integer less than or equal to x (formally, $forall x in mathbb{R}$ $lfloor x rfloor=sup{ninmathbb{Z}mid nle x}$ ). For example, floor(2.9) = 2, floor(−2) = −2 and floor(−2.3) = −3. For nonnegative x, a more traditional name for floor(x) is the integral part or integral value of x. The function $x -lfloor xrfloor$ , also written as x mod 1, or {x}, is called the fractional part of x. Every fraction x can be written as a mixed number, the sum of an integer and a proper fraction. The floor function and fractional part functions extend this decomposition to all real values. Image File history File links Floor_function. ... Image File history File links Floor_function. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ... A negative number is a number that is less than zero, such as −3. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

1 Some properties of the floor function
2 The ceiling function
3 Some properties of the ceiling function
4 The operator (int) in C
5 Uniform distribution mod 1
6 Truncation
7 Typesetting

Some properties of the floor function

One has

$lfloor xrfloor le x < lfloor x rfloor + 1$

with equality on the left if and only if x is an integer.

The floor function is idempotent: $lfloorlfloor xrfloorrfloor=lfloor xrfloor$ .
For any integer k and any real number x,

$lfloor k+x rfloor = k + lfloor xrfloor.$

The ordinary rounding of the number x to the nearest integer can be expressed as floor(x + 0.5).
The floor function is not continuous, but it is upper semi-continuous. Being a piecewise constant function, its derivative is zero where it exists, that is, at all points which are not integers.
If x is a real number and n is an integer, one has n ≤ x if and only if n ≤ floor(x). In fancy language: the floor function is part of a Galois connection; it is the upper adjoint of the function that embeds the integers into the reals.
Using the floor function, one can produce several explicit (yet impractical) formulas for prime numbers.
For real, non-integer x, the floor function has the Fourier series representation

$lfloor xrfloor = x - frac{1}{2} + frac{1}{pi} sum_{k=1}^infty frac{sin(2 pi k x)}{k}.$

If m and n are coprime positive integers, then

$sum_{i=1}^{n-1} lfloor im / n rfloor = (m - 1) (n - 1) / 2$

Beatty's theorem shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.

There are two main definitions of idempotence (IPA , like eye-dem-potent-s) in mathematics. ... // Rounding Rounding means reducing the significant digits in a number while trying to keep its value similar. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematical analysis, semi-continuity (or semicontinuity) is a property of real-valued functions that is weaker than continuity. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... The Formula for Primes, is a mathematical formula underpinning the theory that in mathematics it is known that no non-constant polynomial function P(n) exists that evaluates to a prime number for all integers n (or even almost all n). ... Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... Coprime - Wikipedia /**/ @import /skins-1. ... In mathematics, Beattys theorem states that if p and q are two positive irrational numbers with then the positive integers are all pairwise distinct, and each positive integer occurs precisely once in the list. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

The ceiling function

A closely related mathematical function is the ceiling function, given x, ceiling(x) also denoted by $lceil x rceil$ is the function defined as $lceil x rceil=inf{ninmathbb{Z}/xle n}$ . This is, the smallest integer not less than x. For example, ceiling(2.3) = 3, ceiling(2) = 2 and ceiling(−2.3) = −2. Image File history File links Ceiling_function. ... Image File history File links Ceiling_function. ... Partial plot of a function f. ...

Some properties of the ceiling function

It is easy to show that:

$lceil x rceil = - lfloor - x rfloor$

Also:

$x leq lceil x rceil < x + 1$

For any integer k, we also have the following equality:

$lfloor k / 2 rfloor + lceil k / 2 rceil = k$ .

The operator `(int)` in C

The (int) function

C and related programming languages have a feature called type casting which allows to turn a floating point value into an integer by prefixing it with (int). This operation is a mixture of the floor and ceiling function: for positive or 0 x it returns floor(x), and for negative x it returns ceiling(x). Image File history File links Int_function. ... Image File history File links Int_function. ... Wikibooks has a book on the topic of C Programming The C programming language (often, just C) is a general-purpose, procedural, imperative computer programming language developed in the early 1970s by Dennis Ritchie for use on the Unix operating system. ... A programming language is an artificial language that can be used to control the behavior of a machine (often a computer). ... This article needs cleanup. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

This operation loses significant data, and can therefore magnify rounding errors with disastrous consequences. For instance, (int)(0.6/0.2) will return 2 in most implementations of C, even though 0.6/0.2 = 3. The reason is that computers work internally with the binary numeral system, and it is not possible to represent the numbers 0.6 and 0.2 by a finite binary string. So some rounding errors occur, and the result is computed as 2.999999999999999555910790149937 which the (int) operator will convert to 2. A round-off error is the difference between the calculated approximation of a number and its exact mathematical value. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ...

Many other languages, such as Java (tested with Sun JDK version 1.5.0_05) and Perl (as of version 5.8.0) behave similarly and experiment rounding errors, as does the POSIX floor() function, every time floating point types are used to store real values. Java refers to a number of computer software products and specifications from Sun Microsystems (the Javaâ„¢ technology) that together provide a system for developing and deploying cross-platform applications. ... Perl, also Practical Extraction and Report Language (a backronym, see below) is a dynamic procedural programming language designed by Larry Wall and first released in 1987. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Because of issues like these, most modern calculators use the decimal numeral system internally, but this only solves a small part of possible problems, and not the fact that (int)((4.0/9.0) * 9.0) will still return 3 instead of 4, even when using decimal floating point representation at any datatype precision (The effective result comes from the truncation of 0.44444...4444 * 9.0 = 3.99999...9996, for which either a binary or decimal internal base has no influence on the precision of the result and the error, even when correct rounding to the nearest representable number is used throughout the calculation). A calculator is a device for performing calculations. ... Binary-coded decimal (BCD) is, after character encodings, the most common way of encoding decimal digits in computing and in electronic systems. ...

A more elegant solution is to use better numeric datatypes to allow representing rational values exactly, or to avoid aggravating rounding errors which will only appear when non polynomial functions are used in the calculation. For example, a pair can be used for storing a floating point numerator and a positive integer divisor (a solution used in the Calculator accessory in Windows): all intermediate calculation and storage use this datatype, and the decimal floating point value is computed only for the final display of the calculator, with correct rounding up to the displayed precision.

Uniform distribution mod 1

If x is an irrational number, then the fractional parts nx mod 1, where n runs through the positive integers, are uniformly distributed in the open interval (0,1). This can be made precise in various ways, one of which states In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

$int_0^1 f(t); dt = lim_{Ntoinfty} frac{1}{N}sum_{n=1}^N f(nx ;operatorname{mod}; 1)$

for every real-valued continuous function $f:[0,1]tomathbb{R}$ (see limit (mathematics), integration and equidistribution theorem). In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the unit interval, when a is an irrational number. ...

According to a general principle of diophantine approximation discovered by Hermann Weyl, that property is equivalent to something much easier to check in this case: namely that sums In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ...

$sumlimits_{n=0}^N e^{2 pi i k n x}$

for $kinmathbb{N}$ have estimates O(N). Because these are geometric progressions, that can be proved rather directly. The condition that x be irrational comes out to be that It has been suggested that this article or section be merged into Asymptotic notation. ... In mathematics, a geometric progression (also known as a geometric sequence, and, inaccurately, as a geometric series; see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

$sin pi k x ne 0.$

Truncation

While the floor function only outputs natural numbers, truncation allows "cutting off the numbers" at a specified position. In mathematics, truncation is the term used for reducing the number of digits right of the decimal point, by discarding the least significant ones. ...

Typesetting

The floor and ceiling function are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing, and, e.g., in the LaTeX typesetting system these symbols can be constructed with the lfloor, rfloor, lceil and rceil commands. The LaTeX logo, typeset with LaTeX , written as LaTeX in plain text, is a document preparation system for the (TeX) typesetting program. ...

Category: Elementary special functions

Results from FactBites:

Floor function - Wikipedia, the free encyclopedia (716 words)
	Being a piecewise constant function, its derivative is zero where it exists, that is, at all points which are not integers.
	In fancy language: the floor function is part of a Galois connection; it is the upper adjoint of the function that embeds the integers into the reals.
	The floor and ceiling function are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing, and, e.g., in the LaTeX typesetting system these symbols can be constructed with the \lfloor, \rfloor, \lceil and \rceil commands.

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