Finally, the fractional part of 43 / 7 is the reciprocal of 7, so its approximation in this scheme, 7, is exact ( 7 / 1 = 7 + 0 / 1 ) and produces the exact expression for 415 / 93.

The commonly used fractional approximation is also observed in the two “ drops of a seventh ” that follow exactly after the first eleven notes – 11 x 2 / 7, or 22 / 7.

:* Ledesma S. and Liu D. ( 2000 ) " Synthesis of fractional Gaussian noise using linear approximation for generating self-similar network traffic ", Computer Communication Review, 30, 4 – 17.

He also used He Chengtian's interpolation method for approximating irrational number with fraction in his astronomy and mathematical works, he obtained as a good fraction approximate for pi ; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, " the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe " Along with his son, Zu Geng, Zu Chongzhi used the Cavalieri Method to find an accurate solution for calculating the volume of the sphere.

From the definition above, for α = 0, there will be no change after applying fractional Fourier transform, and for α = π / 2, fractional Fourier transform becomes a Fourier transform, which rotates the time frequency distribution with π / 2.

In this equation, λ is the wavelength c / f ; Δφ is the part of the phase delay that does not fulfill π ( that is, φ modulo π ); N is the integer number of wave half-cycles of the round-trip and ΔN the remaining fractional part.

Kerosene, a thin, clear liquid formed from hydrocarbons, with a density of 0. 78 – 0. 81 g / cm < sup > 3 </ sup >, is obtained from the fractional distillation of petroleum between 150 ° C and 275 ° C, resulting in a mixture of carbon chains that typically contain between six and 16 carbon atoms per molecule.

Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e < sup > 2 </ sup >/ h to nearly one part in a billion.

Define a map from K < sup >×</ sup > to the set of all nonzero fractional ideals of R by sending every element to the principal ( fractional ) ideal it generates.

The 64-bit timestamps used by NTP consist of a 32-bit part for seconds and a 32-bit part for fractional second, giving NTP a time scale that rolls over every 2 < sup > 32 </ sup > seconds ( 136 years ) and a theoretical resolution of 2 < sup >− 32 </ sup > seconds ( 233 picoseconds ).

Now the integral part of the logarithm is simply n, and the fractional part is lb ( 2 < sup >− n </ sup > x ).

Leonhard Euler also searched for this number, but failed to find it, but did find a fractional number that meets the other conditions, 777480 / 2879 < sup > 2 </ sup >.

These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of I < sup > α </ sup > ƒ.

Extending work by John Wallis who calculated such areas for y = ( 1 − x < sup > 2 </ sup >)< sup > n </ sup > with n = 0, 1, 2, 3, ... he considered fractional exponents.

The Boltzmann distribution for the fractional number of particles N < sub > i </ sub > / N occupying a set of states i possessing energy E < sub > i </ sub > is:

This is called gating error and causes an average error in the calculated frequency of Δf = 1 /( 2 T < sub > m </ sub >), or a fractional error of Δf / f = 1 /( 2 f T < sub > m </ sub >) where T < sub > m </ sub > is the timing interval and f is the measured frequency.

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of n. The simple formula for the factorial, n < nowiki >!</ nowiki > = 1 × 2 × … × n, cannot be used directly for fractional values of n since it is only valid when n is a natural number ( i. e., a positive integer ).

The fractional frequency y ( t ) is the normalized delta from the nominal frequency v < sub > n </ sub >, thus

This generally leads to a BUN: Cr ratio > 20 and a fractional excretion of Na of < 1 % and an elevated urine osmolarity.

The Kisliuk adsorption isotherm is written as follows, where Θ < sub >( t )</ sub > is fractional coverage of the adsorbent with adsorbate, and t is immersion time:

For fractional arguments between 0 and 1, and for a > 1:

is the fractional derivative ( if q > 0 ) or fractional integral ( if q < 0 ).

For the analysis of fractional distillation, the two independent variables are instead considered to be liquid-phase composition ( x < sub > 1L </ sub >) and pressure.

Differentiation of volcanic rocks tends to increase the silica ( SiO < sub > 2 </ sub >) content mainly by fractional crystallization.

This has the effect of keeping the middle bits ; the < var > I </ var >- number of least significant integer bits, and the < var > Q </ var >- number of most significant fractional bits.

According to Freeman Clarke < sub > 2 </ sub > ( UK-based provider of fractional CIOs and IT Directors ) fractional CIOs are generally used by organisations with a turnover of under £ 50M.

The fractional chromatic number χ < sub > f </ sub >( G ) is defined to be

The fractional chromatic number χ < sub > f </ sub >( G ) of a graph G can be obtained as a solution to a linear program.

This radiogenic helium is trapped with natural gas in concentrations up to 7 % by volume, from which it is extracted commercially by a low-temperature separation process called fractional distillation.

The fractional part of 93 / 43 is the reciprocal of 43 / 7 which is about 6. 1429.

*− 7 / 8 measures how much of the signal has a fractional frequency of + 7 / 8

The fractional contribution of each to the current energy density of the Universe is given by the density parameters Ω < sub > m </ sub >, Ω < sub > r </ sub >, and Ω < sub > Λ </ sub >.

The larger one, I < sub > m </ sub >, is the group of all fractional ideals relatively prime to m ( which means these fractional ideals do not involve any prime ideal appearing in m < sub > f </ sub >).

The smaller one, P < sub > m </ sub >, is the group of principal fractional ideals ( u / v ) where u and v are nonzero elements of O < sub > K </ sub > which are prime to m < sub > f </ sub >, u ≡ v mod m < sub > f </ sub >, and u / v > 0 in each of the orderings of m < sub >∞</ sub >.

The weight-average molecular weight, M < sub > w </ sub >, is also related to the fractional monomer conversion, p, in step-growth polymerization as per Carothers ' equation:

Define the inverse different or Dedekind's complementary module as the set of x ∈ K such that tr ( xy ) is an integer for all y in O < sub > K </ sup >, then I is a fractional ideal of K containing O < sub > K </ sup >.

In polycondensation, in order to achieve a high degree of polymerization ( and hence molecular weight ), X < sub > n </ sub >, a high fractional monomer conversion, p, is required, as per Carothers ' equation: X < sub > n </ sub >

This property also makes it straightforward to represent a timestamp as a fractional day, so that. 534 can be interpreted as five decimal hours and 34 decimal minutes after the start of that date, or 0. 534 ( 53. 4 %) of a day ( or 12 hours 48 minutes and 36 seconds ) in that date.

For example, six fractional sexagesimal places of a day was used in their specification of the length of the year, although they were unable to measure such a small fraction of a day in real time.

Therefore any attempt to divide a month into days or a year into days will leave a fractional remainder of a partial day.

This is an artificial harmonization that, after the fractional remainders have sufficiently accumulated, adds a whole day ( or month ) into the cycle.

The most distinctive aspect of Swatch Internet Time is its notation ; as an example, "@ 248 " would indicate a time 248. beats after midnight, equivalent to a fractional day of 0. 248 CET, or 04: 57: 07. 2 UTC.

It differs from the planet's solar day, which includes an extra fractional rotation needed to accommodate the portion of the planet's orbital period during one day.

Zeller observed that, by starting each year on March 1, the day of the week for each succeeding month progressed by multiplying the month by a constant value and discarding the fractional remainder.