For more rigorous and precise analysis, the numerical approximations must be used.

For some subjects a numerical geographical subdivision follows the classification letters on the first line.

For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences ; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science ; they are used in calculus and numerical analysis to approximate other functions.

For a great many functions and practical applications, the Riemann integral can also be readily evaluated by using the fundamental theorem of calculus or ( approximately ) by numerical integration.

For large volume scientific datasets, such as satellite data and output of numerical climate, weather, or ocean models, specific binary serialization standards have been developed, e. g. HDF, netCDF and the older GRIB.

For any two empirical thermometers, this does not require that the relation between their numerical scale readings be linear, but it does require that relation to be strictly monotonic.

For the sake of economy, a particular genus, species, or variety may share a numerical reporting code and symbol with another similar cloud type.

For this reason, methods based on introductory linear algebra texts are generally not suitable for implementation in software ; rather, one should consult contemporary numerical analysis sources for an algorithm like the one below, which does not amplify rounding errors unnecessarily.

For modeling of scattering in cases where the Rayleigh and Mie models do not apply such as irregularly shaped particles, there are many numerical methods that can be used.

For numerical reasons, atmospheric models such as general circulation models ( GCMs ) usually predict the nondimensional logarithm of surface pressure.

For example, if a 1 kHz signal is spread to 100 kHz, the process gain expressed as a numerical ratio would be 100, 000 / 1, 000

For example, the iPhone ( which by its name implies that its primary function is that of a mobile phone ) can perform many different tasks, but does not feature a traditional numerical pad to make phone calls.

For any two empirical thermometers, this does not require that the relation between their numerical scale readings be linear, but it does require that relation to be strictly monotonic.

For instance, the following algebraic data type describes a simple language representing numerical expressions:

For complex geometries, a numerical solution method such as the finite element method is necessary.

For example, math is taught through reading and discussing Euclid and Galileo, rather than actually completing numerical problem sets.

For example, Phoenician trades – people invented numerical and linguistic pictographic inscriptions out of the need to record transactions.

For three days the outcome of the battle hung in the balance, but the Soviet ' numerical superiority proved decisive and by 7 July Polish forces were in full retreat along the entire front.

For a given numerical aperture ( NA ), the resolution of microscopy for flat objects under coherent illumination can be improved using interferometric microscopy.

For the arithmetic Asian option in Lévy models one can rely on numerical methods or on analytic bounds

For file names containing numbers Windows Explorer now tries to sort based on numerical value rather than just comparing each number digit by digit for every character position in the file name.

For example, the SI, the international system of units, is such a system of units solely defined as convenient to human use and the numerical values of dimensional physical constants have no natural significance, only in a manner that relates to the human experience.

For more on numerical analysis and the role of the Rossby number, see:

For 1989, the " Delta " name was dropped, as was the hood ornament, leaving the model to simply become the " Eighty Eight ," now spelled out instead of using the " 88 " in numerical form, and deleted the amber color from the taillights.

For calculations of OBP before 1954, or where sacrifice flies are not explicitly listed, the number of sacrifice flies should be assumed to be zero.

For example, most ab initio calculations make the Born – Oppenheimer approximation, which greatly simplifies the underlying Schrödinger equation by assuming that the nuclei remain in place during the calculation.

For example, some include battle deaths in their calculations.

For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions.

( For a large user population, the Engset and the Erlang-B calculations give the same result.

For precise calculations, errors due to rounding and the number of significant digits of constants must be considered.

An alternative representation uses degrees and minutes, where parts of a minute are expressed in decimal notation with a fraction, thus: 23 ° 27. 500 ′ E. Degrees may also be expressed as a decimal fraction: 23. 45833 ° E. For calculations, the angular measure may be converted to radians, so longitude may also be expressed in this manner as a signed fraction of π ( pi ), or an unsigned fraction of 2π.

For calculations, the West / East suffix is replaced by a negative sign in the western hemisphere.

For network solids, the term formula unit is used in stoichiometric calculations.

For IBM, the earliest Gibson Mix calculations shown are the 1954 IBM 650 at 0. 06 kIPS and 1956 IBM 705 at 0. 5 kIPS.

For this reason the research literature on atomic calculations quotes the ground state configuration of nickel as 4s < sup > 1 </ sup > 3d < sup > 9 </ sup >.

For example, the growing use of packaged software, such as spreadsheet and database management software packages, allows users to write simple programs to access data and perform calculations.

For example, calculations show that the Titan II first stage, launched on its own, would have a 25-to-1 ratio of fuel to vehicle hardware.

For monthly calculations, national surveys or national registers from employment offices are used in conjunction with quarterly EU-LFS data.

For true ( in vacuo ) specific gravity calculations air pressure must be considered ( see below ).

* For more on ramjet math calculations see The Star Flight Handbook ( ISBN 0-471-61912-4 ).

For these calculations it is essential to know the latitude of the event.

For example, the part of an arithmetic logic unit, or ALU, that does mathematical calculations is constructed using combinational logic.

For precise calculations, computer programs that can produce a model of the magnetic field using the finite element method are employed.

For calculations which use floating point and arrays heavily, Maxima offers the possibility of generating code in other programming languages ( notably Fortran ) which may execute more efficiently.

For example, a ' walk around ' stock intake done just prior to the MRP calculations can be a practical solution for a small inventory ( especially if it is an " open store ").

For example, when c is very small, loss of significance can occur in either of the root calculations, depending on the sign of b.

For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the ( complex ) elliptic integral ( Brent, 1976 ).

For calculations involving annuities, you must decide whether the payments are made at the end of each period ( known as an ordinary annuity ), or at the beginning of each period ( known as an annuity due ).