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Area

Area

:This article explains the meaning of area as a physical quantity. The article area (geometry) is more mathematical. See also area (disambiguation). Area is a quantity expressing the size of a part of a surface. Surface area is the summation of the areas of the exposed sides of an object.

Units

Units for measuring surface area include: :square metre = SI derived unit :are = 100 square metres :hectare = 10,000 square metres :square kilometre = 1,000,000 square metres :square megametre = 10¹² square metres Imperial units, as currently defined from the metre: :square foot (plural square feet) = 0.09290304 square metres. :square yard = 9 square feet = 0.83612736 square metres :square perch = 30.25 square yards = 25.2928526 square metres :acre = 160 square perches or 43,560 square feet = 4046.8564224 square metres :square mile = 640 acres = 2.5899881103 square kilometres Old European area units, still in used in some private matters (e.g. land sale advertisements) :square fathom = 3.5967 square metres :cadastral moon(acre) = 1600� square fathoms = 5755 square metres The article Orders of magnitude links to lists of objects of comparable surface area.

Useful formulas

- Area of a rectangle (and, in particular, a square): length × width
- Area of a triangle: ½ × base × height
- Area of a disk: π × r²
- Area of an ellipse: π × a × b
- Area of a sphere: 4 × π × r² = π × d²
- Area of a trapezoid: If a and b are the two parallel sides and h is the distance (height) between the parallels, the area formula is as below: :

A=\frac(a+b)h

A=\frac

- Total surface area of a right circular cylinder: 2 × π × r × (h + r)
- Lateral surface area of a right circular cylinder: 2 × π × r × h
- Total surface area of a right circular cone: π × r × (l + r)
- Lateral surface area of a right circular cone: π × r × l

External links

- [http://www.unitconversion.org/unit_converter/area.html Online Area Converter - convert between various units of area, such as square meter, hectare, rood, and so on]
- [http://www.unitconversion.org/unit_converter/area-v.html Interactive Area Conversion table - convert selected unit to all other units of area]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- als:Fläche ja:面積 ko:면적 simple:Area th:พื้นที่ zh-min-nan:Bīn-chek

Physical quantity

A physical quantity is either a quantity within physics that can be measured (e.g. mass, volume, etc.), or a result of measurement, and is usually expressed as the product of a numerical value and a physical unit (whereby SI units are usually preferred).

Example

: P = 42.3 x 10³ W = 42.3 kW where : P represents the physical quantity of power : 42.3 x 10³ is the numerical value : k is the SI prefix kilo representing 10³ : W is the symbol for the unit of power, the watt : kW is the kilowatt (= 10³W)

Subscripted variables

Usually, the symbols for physical quantities are chosen to be a single lower case or capital letter of the Latin or Greek alphabet. Often, the symbols are modified by subscripts or superscripts.

Examples

- E_p - potential energy
- c_p - heat capacity at constant pressure

Extensive vs. Intensive

A quantity is called:
- extensive when its magnitude is additive for subsystems (e.g. volume, V, or mass, m)
- intensive when the magnitude is independent of the extent of the system (e.g. temperature, T, or pressure, p) The following variables are neither extensive nor intensive:
- Angular momentum
- Area
- Force
- Length
- Time

Prefixes

Some extensive physical quantities may be prefixed to qualify the meaning:
- specific is added to refer to the quantity divided by its mass
- molar is added to refer to the quantity divided by amount of substance

Examples

- specific volume v = V/m
- molar volume V_m = V/n

Area (disambiguation)

Area, adjective areal, from Latin area, adj. arealis, "a piece of level ground, an open space, threshing floor", English from 1538.
- area - units of measurement
- area (geometry) - the Euclidian geometrical concept
- areal feature in linguistics

Quantity

:For the use in linguistics, see length (phonetics). Quantity is a general term used to refer to any type of quantitative property or attribute, such as mass, length, or time. A particular quantity is a magnitude of a scalar or vector quantity. The term quantity is also often used to refer to denumerable (countable) collections of objects. A given quantity is usually represented either as a number of units, together with the type of those units, or a number of objects with a referent defining the type of object. Thus, scalar quantities such as mass, and vector quantities such as force, are continuous quantities and are usually represented as a multiple of a real number and a unit of continuous quantity, such as a gram or newton. A count of a denumerable collection of entities is represented as an integer and the type of object or entity, such as an apple or a set. A number, including a particular measurement, is not by itself a quantity. Examples are
- 1.76 litres (liters) of milk, which is continuous quantity
-

2 \pi r

metres, where r is the length of a radius of a circle expressed in metres (or meters)
- one apple, two apples, three apples, where the number is an integer representing the count of a denumerable collection of objects (apples)
- 500 people (also involving a count) Where the count is one then the indefinite article may be used (for example, a car) and similar alternatives exist for other particular counts (for example, a brace of pheasant, a dozen eggs). Quantification in its very simplest sense can be found in statements such as "A is greater than B". In the example cited, an expression is made that A has a greater quantity of something (such as volume or charisma) than B; and that if A and B were placed in an ordered set, then A would come after B if the order is arranged on an increasing (rather than decreasing) scale.

Object (philosophy)

In philosophy, an object is a thing, an entity, or a being. This may be taken in several senses. In its weakest sense, the word object is the most all-purpose of nouns, and can replace a noun in any sentence at all. (In ordinary usage, the word has something like this effect, but not as extreme.) Thus objects are things as diverse as the pyramids, Alpha Centauri, the number seven, my belief in predestination, and your mother's fear of dogs. Charles S. Peirce succinctly defines the broad notion of an object as follows: :"By an object, I mean anything that we can think, i.e. anything we can talk about." [http://www.helsinki.fi/science/commens/terms/object.html] In a more restricted sense, an object is something that can have properties and bear relations to other objects. On this account, properties and relations (as well as propositions) are not included among objects, but are explicitly contrasted with them, as falling into a different logical category. Sets and universals are also perhaps not objects on this account. In a further restricted sense, objects do not include anything abstract, but only things located somehow in space and time — minds and bodies, for instance. Numbers, ideas, and the like are out. In further restricted senses, objects are often just the material objects (excluding minds), or even just the inanimate material objects (the protons and electrons we are made of, but not we ourselves). Objects are often treated as types of particulars, but occasionally, philosophers see fit to speak of abstract objects — Platonic forms would be an example. An abstract object is normally referred to something that does not exist physically. It is rational to say that abstract objects exist psychically, as opposed to physically.

Semantics

Symbols represent objects; how they do so, the map-territory relation, is the basic problem of semantics.

External links

- [http://plato.stanford.edu/entries/object/ Stanford Encyclopedia of Philosophy entry] Category:Philosophical terminology

SI derived unit

SI derived units are part of the SI system of measurement units and are derived from the seven SI base units.

Dimensionless derived units

The following SI units are actually dimensionless ratios, formed by dividing two identical SI units. They are therefore considered by the BIPM to be derived. Formally, their SI unit is simply the number 1, but they are given these special names, for use whenever the lack of a unit might be confusing.

Derived units with special names

Base units can be put together to derive units of measurement for other quantities. Some have been given names.

Other quantities and units

Conversion between kelvins and degrees Celsius

A change in temperature of 1°C is equal to a change in temperature of 1K. Temperature in °C = Temperature in kelvins - 273.15 Thus, one could think of the Kelvin scale as the same as the Celsius scale, with its zero point moved down to absolute zero. This perspecitive is historically accurate; however, it has become more convenient to fix the standard for the kelvin, and thus the Celsius scale is derived from that standard (i.e., it now depends on absolute zero and the triple point of water with a 0.01 K offset — the boiling point of water no longer has anything to do with the official definition of degrees Celsius). Temperature differences are often measured in degrees Celsius; however, it doesn't matter: differences in temperature are equivalent whether kelvins or degrees Celsius are used. Therefore, a change in temperature (ΔT), when expressed in an equation, can be calculated using either kelvins or degrees celsius so long as one is consistent.

References

- I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC: Quantities, Units and Symbols in Physical Chemistry, 2nd edition (June 1993), Blackwell Science Inc (p. 72) ko:SI 유도 단위 ja:SI組立単位 Category:SI units Category:SI derived units

Are

For the municipality in Sweden, see Åre; for the boroughtown and rural municipality in Estonia see Are; for the language from Papua New Guinea see Are For the ARE see either Arab Republic of Egypt or Association for Research and Enlightenment ---- An are (symbol a) is a unit of area. It is not an SI unit. It is accepted (although discouraged) for use with the SI. It is rarely used. The are is most familiar through its derived unit, the hectare.

Definition

1 are (a) = 100 square metres (m²)

Conversions

One are is equivalent to:
- a square with sides 10 metres long
- 1076.3910 square feet

Multiples

- 100 ares = 1 hectare (written 1 ha)
- 10 ares = 1 decare (written 1 daa)

External link

- [http://www.bipm.org/en/si/si_brochure/chapter4/4-1.html Official SI website: Table 8. Other non-SI units currently accepted for use with the International System Their use is not encouraged.]
- [http://www.projects.ex.ac.uk/trol/scol/index.htm Conversion Calculator for Units of AREA] Category:Units of area als:Ar ja:アール

Imperial unit

This article is about post-1824 Imperial units, please see also English unit, U.S. customary unit or Avoirdupois. ---- The Imperial units or the Imperial system is a collection of English units, first defined in the Weights and Measures Act of 1824, later refined (until 1959) and reduced. The units were introduced in the United Kingdom and its colonies, including Commonwealth countries, but excluding the then already independent United States.

Relation to other systems

The distinction between this imperial system and the U.S. customary units (also called standard units there) or older British/English units/systems and newer additions is often not drawn precisely. Most length units are shared among the Imperial and U.S. systems, albeit partially and temporally defined slightly differently. Capacity measures differ the most due to the introduction of the Imperial gallon and the unification of wet and dry measures. The avoirdupois system only applies to weights; it has a long flavour and a short flavour for the hundredweight and ton. The term imperial should not be applied to English units that were outlawed in Weights and Measures Act of 1824 or earlier, or which had fallen out of use by that time, nor to post-imperial inventions such as the slug or poundal. Although most of the units are defined in more than one system, some subsidiary units were used to a much greater extent, or for different purposes, in one area rather than the other.

Measures of length

poundal.]] After the 1 July 1959 deadline, agreed upon in 1958, the U.S. and the British yard were defined identically (0.9144 m) to the international yard. Metric equivalents in this article usually assume this latest official definition. Before this date, the most precise measurement of the Imperial Standard Yard was 0.914398416 m (Sears et al. 1928. Phil Trans A 227:281).
- The pole is also called rod or perch. Until the adoption of the international definition of 1852 metres in 1970, the British nautical mile was defined as 6080 feet (1.85318 km). It was not readily expressible in terms of any of the intermediate units, because it was derived from the circumference of the Earth (like the original metre). Depth of water at sea was expressed in fathoms (6 feet = 1.8288 m).

Measures of area

Measures of volume

In 1824, Britain adopted a close approximation to the ale gallon known as the Imperial gallon. The Imperial gallon was based on the volume of 10 lb of distilled water weighed in air with brass weights with the barometer standing at 30 in and at a temperature of 62 °F. In 1963, this definition was refined as the space occupied by 10 lb of distilled water of density 0.998 859 g/mL weighed in air of density 0.001 217 g/mL against weights of density 8.136 g/mL. This works out to exactly 4.545 964 591 L, or 277.420 in³. The Weights and Measures Act of 1985 finally switched to a gallon of exactly 4.546 09 L (approximately 277.419 43 cu in) [http://www.sizes.com/units/gallon_imperial.htm]. The full table of British apothecaries' measure is as follows: For a comparison to the U.S. customary system see the article on Comparison of the Imperial and US customary systems.

Measures of weight and mass

Britain has made some use of three different weight systems, troy weight, used for precious metals, avoirdupois weight, used for most other purposes, and apothecaries' weight, now virtually unused since the metric system is used for all scientific purposes. The use of the troy pound (373.241 721 6 g) was abolished in Britain on January 6, 1879, with only the troy ounce (31.103 476 8 g) and its decimal subdivisions retained. In all the systems, the fundamental unit is the pound, and all other units are defined as fractions or multiples of it. Note that the British ton is 2240 pounds (the long ton), which is very close to a metric tonne, whereas the ton generally used in the United States is the "short ton" of 2000 pounds (907.184 74 kg), both are 20 hundredweights. For more on Commonwealth-U.S. differences see Comparison of the Imperial and US customary systems.

Current use of Imperial units

British law now defines each Imperial unit entirely in terms of the metric equivalent. See the [http://www.hmso.gov.uk/si/si1995/Uksi_19951804_en_2.htm Units of Measurement Regulations 1995]. This regulation effectively outlaws their usage in retail and trading except in previously established exceptions. This has now been proved by in court against the so called 'Metric Martyrs', a small group of market traders. Despite this, many small market traders still use the customary measures, citing customer preference especially among the older population. In the United States and in a few Caribbean countries, the U.S. customary units, which are similar to Imperial units based upon older English units and in part share definitions, are still in common use. English units have been replaced elsewhere by the SI (metric) system. Most Commonwealth countries have switched entirely to the international system of units. The United Kingdom completed its legal transition to SI units in 1995, but a few such units are still in official use: draught beer must still be sold in pints, most roadsign distances are still in yards and miles, and speed limits are in miles per hour, therefore interfaces in cars must have miles, and even though the troy pound was outlawed in Great Britain in the Weights and Measures Act of 1878, the troy ounce still may be used for the weight of precious stones and metals. The use of SI units is increasingly mandated by law for the retail sale of food and other commodities, but many British people still use Imperial units in colloquial discussion of distance (miles and yards), speed (miles per hour), weight (stones and pounds), liquid (pints and gallons) and height (feet and inches). In Canada, the government's efforts to implement the metric system were more extensive: pretty much any agency, institution, or thing provided by the government will use SI units exclusively. Imperial units were eliminated from all road signs, although both systems of measurement will still be found on privately-owned signs (such as the height warnings at the entrance of a multi-storey parking facility). Temperatures in degrees Fahrenheit will occasionally be heard on English Canadian commercial radio stations, but only those that cater to older listeners. The law requires that measured products (such as fuel and meat) be priced in metric units, although there is leniency in regards to fruits and vegetables. Traditional units persist in ordinary conversation and may be experiencing a resurgence due to the reduction in trade barriers with the United States. Few Canadians would use SI units to describe their weight and height, although driver's licences use SI units. In livestock auction markets, cattle are sold in dollars per hundredweight (short, of course), whereas hogs are sold in dollars per hundred kilograms. Land is surveyed and registered in metric units, but imperial units still dominate in construction, house renovation and gardening talk (although "two-by-fours" don't actually measure 2×4", for example).

References

- Appendices B and C of [http://ts.nist.gov/ts/htdocs/230/235/h442003.htm NIST Handbook 44]
- Barry N. Taylor's [http://physics.nist.gov/Pubs/SP811/ NIST Special Publication 811], also available as [http://physics.nist.gov/Document/sp811.pdf a PDF file]

External links

- [http://www.metric.org.uk/ The UK Metric Association]
- [http://www.bwmaonline.com/ British Weights And Measures Association]
- [http://www.metric4us.com Metric4us.com]
- [http://laws.justice.gc.ca/en/w-6/109089.html Canada - Weights and Measures Act 1970-71-72]
- [http://193.120.124.98/gen531996a.html Ireland - Metrology Act 1996]
- [http://www.hmso.gov.uk/si/si1995/Uksi_19951804_en_2.htm UK - Units of Measurement Regulations 1995]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- Jacques J. Proot's [http://users.aol.com/jackproot/met/spvolas.html Anglo-Saxon weights & measures] page.
- Category:Systems of units ja:ヤード・ポンド法

Foot (unit of length)

:For other uses, see Foot (disambiguation). A foot (plural: feet) is a non-SI unit of distance or length, measuring around a third of a metre. There are twelve inches in one foot and three feet in one yard. The international standard symbol for feet is ft (see ISO 31-1, Annex A). The standardization of weights and measures has left several different standard foot measures. The most commonly used foot today is the English foot, used in the United Kingdom and the United States and elsewhere, which is defined to be exactly 0.3048 metre. This unit is sometimes denoted with a prime (e.g. 30′ means 30 feet), often approximated by an apostrophe. Similarly, inches can be denoted by a double prime (often approximated by a quotation mark), so 6′ 2″ means 6 feet 2 inches. In addition to the current standard international foot, there is also a slightly different U.S. survey foot, used only in connection with surveys by the U.S. Coast and Geodetic Survey, it is defined as exactly 1200/3937 m (610 nm greater than 0.3048 m).[http://www.ngs.noaa.gov/PUBS_LIB/FedRegister/FRdoc59-5442.pdf] The foot as a measure was used in almost all cultures. The first known standard foot measure was from Sumeria, where a definition is given in a statue of Gudea of Lagash from around 2575 BC. The imperial foot was adapted from an Egyptian measure by the Greeks, with a subsequent larger foot being adopted by the Romans.

Etymology

The popular belief is that original standard was the length of a man's foot. The original measurement was from King Henry I, who had a foot 12 inches long; he wished to standardise the unit of measurement in England. The average foot length is about 9.4 inches (240 mm) for current Europeans. Approximately 996 out of 1000 British men have a foot that is less than 12 inches long. A plausible explanation for the missing inches is that the measure did not refer to a naked foot, but to the length of footwear. This is consistent with the measure being convenient for practical purposes such as on building sites etc. People almost always pace out lengths whilst wearing shoes or boots, rather than removing them and pacing barefoot.

External link

- http://www.onlineconversion.com/ from feet to international system
- http://www.knowledgedoor.com/1/Library_of_Units_and_Constants/Group_Index/foot_group.htm Foot Foot Foot Category:Human-based units of measure ja:フィート

Perch

:For the perch as a unit of length or area, see Rod (unit). Perca flavescens (Yellow perch)
Perca fluviatilis (European perch)
Perca schrenkii (Balkhash perch)
A perch is a freshwater bony fish belonging to the family Osteichthyes. Perch, of which there are three species, lend their name to the largest order of vertebrates: the Perciformes, from the Greek perke meaning perch, and the Latin forma meaning shape. All perciform fish share the perch's general morphology. The European perch (Perca fluviatilis) is found in Europe and northern Asia. It is 15-60 cm long, and may weigh up to 10.4 kg. It is usually dark green with red fins. It has been successfully introduced in New Zealand and Australia where it is called redfin perch. The Balkhash perch (Perca schrenkii) is found in Kazakhstan; in Lake Balkhash and Lake Alakol. It is very similar to the European perch, and grows to a comparable size. In the United States and Canada there is the smaller (10-25 cm long, 1.4-4.5 kg in weight) and wider-mouthed species, the yellow perch (Perca flavescens). It is paler and yellowish and its fins are not as red; although recognized as a distinct species[http://www.fishbase.org/Summary/SpeciesSummary.cfm?genusname=Perca&speciesname=flavescens], the yellow perch may be a subspecies of the European perch (in which case its binomial name would be Perca fluviatilis flavescens). This view is supported by successful cross-breeding of the two species, which has generated faster growing offspring[http://www.fundp.ac.be/recherche/projets/en/99275103.html]. However, this may be an example of interspecies hybrid vigor and it is unclear whether or not these hybrids are viable. Perch have ctenoid scales. When looking through a microscope, the scale look like a plate with growth rings and spikes on the top edges. Externally the anatomy of perch is simple enough. On the dorsal side of the fish, there consists a upper maxilla and lower mandible for the mouth, a pair of nostrils, and two lidless eyes. On the posterior sides are the operculum, which are used to protect the gills. Also there is the lateral line system which is sensitive to vibrations in the water. They have a pair of pectoral and pelvic fins. On the anterior end of the fish, there are two dorsal fins. The first one is spiny and the second is soft. There is also a caudal fin and anal fin. Also there is a cloacal opening right behind the anal fin. The perch spawns at the end of April or beginning of May, depositing it upon weeds, or the branches of trees or shrubs that have become immersed in the water; it does not come into condition again until July. The best time for fishing for perch is from September to February; it haunts the neighborhood of heavy deep eddies, camp sheathings, beds of weeds, with sharp streams near, and trees or bushes growing in or overhanging the water. The baits for perch are, minnows, red, marsh, brandling or lob worms, shrimps and artificial lures. The tackle should be fine but strong, as with a fish bait a trout or pike may frequently be hooked. Perch, unlike fish of prey, are gregarious, and in the winter months, when the frosts and floods have destroyed and carried away the beds of weeds, congregate together in the pools and eddies, and are then to be angled for with greatest success from 10 to 4 o'clock, from the edge of a stream eddy.

References

- Gilberson, Lance, Zoology Lab Manual 4th edition. Primis Custom Publishing. 1999. Category:Percidae

Square mile

:This article is about the unit of measure. The Square Mile is a traditional name for the City of London in the United Kingdom. A square mile is the area equal to a square with sides each 1 mile long. It is not an SI unit. The SI unit of area is the square metre.

Symbol

There is no universally agreed symbol but the following are used:
- square mile
- sq mile
- sq mi
- sq m (this can be confused with square metre)
- mile²
- mi²

Conversions

1 square mile is equivalent to:
- 27 878 400 square feet
- 640 acres
- 2 589 988.11 square metres
- 2.589 988 11 square kilometres In the Public Land Survey System of the US and the Dominion Land Survey of Canada, the size of a standard section of land is one square mile.

Fathom

---- A fathom is a non-SI unit of length. It was often used as a measure of depth using a lead-weighted sounding line. By extension, "to fathom", has come to mean "to measure", "to get to the bottom of" or "to understand" something.

Definition

::1 English fathom = 6 feet = 2 yards = 1.8288 meters ::1 Wiener fathom = 1.8964838 meters ::1 Greek fathom = 6 feet 1 inch = 1.8542 meters

Origin

The name derives from the Old English word fæthm meaning 'outstretched arms' which was the original definition of the unit's measure. In Middle English it was fathme.

Explanation

The length is equivalent to 2 yards. The fathom was first used for land measurement but is now restricted to nautical uses, especially the measurement of the depth of water or the length of nautical rope or cable. Civilian maps in English-speaking countries used to have depths commonly marked in fathoms, but this has changed to metres generally, even in US maps. Nautical charts have changed on a separate schedule.

Sounding

It is easy to measure a length of line or rope as a rough number of fathoms by repeatedly stretching the rope between the two outstretched arms. Water depths have traditionally been measured this way by a "leadsman" using a sounding line. The word fathom can be used as a verb to describe this process. On the Mississippi river in the 1850s, the leadsmen also used old-fashioned words for some of the numbers; for example instead of "two" they would say "twain". Thus when there was only two fathoms left under the boat they would call "by the mark twain!". The American writer Samuel Clemens, a former river pilot, took his pen name, "Mark Twain", from this cry. A Greek fathom is 6 ft 1 in.

Analogous units

The fathom is a generic unit and an analogous measure can be found in many cultures. Some are listed below.

References

- [http://www.navyandmarine.org/planspatterns/soundingline.htm An explanation of the fathom marks used at sea] (retrieved Sept 2005). Category:Units of length Category:Human-based units of measure

Orders of magnitude

An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. The ratios most commonly used are 1000, 10, 2, 1024 or e (Euler's number, a transcendental number approximately equal to 2.71828182846 that is used as the base for natural logarithms). Usually, orders of magnitude refers to a series of powers of ten; this article discusses the decimal scale. Orders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value. The order of magnitude of a number is, intuitively speaking, the number of powers of 10 contained in the number. More precisely, the order of magnitude of a number can be defined in terms of the decimal logarithm, usually as the integer part of the logarithm. For example, 4,000,000 has a logarithm of 6.602; its order of magnitude is 6. Thus, an order of magnitude is an approximate position on a logarithmic scale. An order of magnitude estimate of a variable whose precise value is unknown is an estimate rounded to the nearest power of ten. For example, an order of magnitude estimate for a variable between about 3 billion and 30 billion (such as the human population of the Earth) is 10 billion. An order of magnitude estimate is sometimes also called a zeroth order approximation. The pages in the table at right contain lists of items that are of the same order of magnitude in various units of measurement. This is useful for getting an intuitive sense of the comparative scale of familiar objects. SI units are used together with SI prefixes, which were devised with orders of magnitude in mind.

Extremely large numbers

For extremely large numbers, a generalized order of magnitude can be based on their double logarithm or super-logarithm. Rounding these downward to an integer gives categories between very "round numbers", rounding them to the nearest integer and applying the inverse function gives the "nearest" round number. The first gives rise to the categories :..., 1.023-1.26, 1.26-10, 10-1e10, 1e10-1e100, 1e100-1e1000, etc. (the first two mentioned, and the extension to the left, may not be very useful, the two just demonstrate how the sequence mathematically continues to the left). The second gives rise to the categories :negative numbers, 0-1, 1-10, 10-1e10, 1e10-10^1e10, 10^1e10-10^^4, 10^^4-10^^5, etc. (see tetration). The "midpoints" which determine which round number is nearer are in the first case: :1.076, 2.071, 1453, 4.20e31, 1.69e316,... and, depending on the interpolation method, in the second case :-.301, .5, 3.162, 1453, 1e1453, 10^1e1453, 10^^2@1e1453,... (See notation of extremely large numbers.) For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but of course the generalized order of magnitude of the reciprocal can be considered. Similar to the logarithmic scale one can have a double logarithmic and super-logarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the generalised f-mean with f(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (e.g. 2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log x (geometric mean, 2 and 8 giving 4), but unlike in the case of log log log x (4 and 65536 giving 16 if the base is 2, but different otherwise).

External links

- [http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html Powers of 10], a graphic animated illustration that starts with a view of the Milky Way at 10²³ meters and ends with subatomic particles at 10^-16 meters.
- [http://www.alcyone.com/max/physics/orders/metre.html Orders of Magnitude - Distance]
- [http://www.vendian.org/envelope/TemporaryURL/what_is_oom.html What is Order of Magnitude?]
- ko:규모의 비교 ja:数量の比較

Rectangle

In geometry, a rectangle is defined as a quadrilateral polygon in which all four angles are right angles. From this definition, it follows that a rectangle has two pairs of opposite sides of equal length; that is, a rectangle is a parallelogram. A square is a special kind of rectangle where all four sides have equal length; that is, a square is both a rectangle and a rhombus. A rectangle that is not a square is colloquially known as an oblong. rhombus Of the two opposite pairs of sides in a rectangle, the length of the longer side is called the length of the rectangle, and the length of the shorter side is called the width. The area of a rectangle is the product of its length and its width; in symbols, A = lw. For example, the area of a rectangle with a length of 5 and a width of 4 would be 20, because 5 × 4 = 20. See the picture above right. In calculus, the Riemann integral can be thought of as a limit of sums of the areas of arbitrarily thin rectangles.

Oblong

The word oblong was once commonly used as an alternate name for a rectangle. In his translation of Euclid's Elements, Sir Thomas Heath translates the Greek word ετερομηκες [hetero mekes – literally, "different lengths"] in Book One, Definition 22 as oblong. "Of Quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right angled but not equilateral...".

References

- Heath, Sir Thomas L. The Thirteen Books of Euclid's Elements. 2nd ed. 3 vols. 1926; rpt. New York: Dover Publications, Inc., 1956. Category:Quadrilaterals

Square (geometry)

In plane geometry, a square is a polygon with four equal sides and equal angles. Those angles are then necessarily right angles. Squares are regular quadrilaterals, rectangles, rhombi, kites, parallelograms, and isosceles trapezoids/isosceles trapezia. The diagonals of a square are equal and conversely, if the diagonals of a rhombus are proven to be equal, then that rhombus must be a square. The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x₀, x₁) with -1 < x_i < 1.

External links

- [http://agutie.homestead.com/files/triangle_square0.htm Triangle with two squares ] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas" Category:Quadrilaterals ko:정사각형 ja:正方形 simple:Square th:รูปสี่เหลี่ยมจัตุรัส

Length

:This article is about the concept and measurement of distance. For usage in cricket, see line and length. In general English usage, length (symbols: l, L) is but one particular instance of distance – an object's length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with "distance". Height is vertical distance; width (or breadth) is a lateral distance; an object's width is less than its length. No one speaks of "the length from here to Alpha Centauri", but rather of "the distance from here to Alpha Centauri," but when one speaks of distance more abstractly, one says "A kilometre or a mile, is a unit of length" or "...of distance", and the two statements are synonymous. Likewise, a mountain might be a mile in height. Length is the metric of one dimension of space. The metric of space itself is volume, or (length)³. Length is commonly considered to be one of the fundamental units, meaning that it cannot be defined in terms of other dimensions. However, a set of units can be constructed where units of length can be derived from fundamental physical constants - see Planck units and Planck length. Colloquially length sometimes refers to duration, especially when used in context of music.

Units of length(SI)

The SI unit of Length is the metre (U.S. spelling: meter), from which can be derived:from the regular basis of the foundation of the whole world
- centimetre
- kilometre

Other units of length

The Imperial and US customary units of length

- inch
- foot
- yard
- mile

Units are used in astronomy

- Astronomical unit
- Light year
- Parsec

External links

- [http://www.unitconversion.org/unit_converter/length.html Length Converter: convert between units of length, such as meter, yard, mile, and so on]
- [http://www.unitconversion.org/unit_converter/length-v.html Length Conversion table: convert selected unit to all other units of length]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- Category:Norm ko:길이 ja:長さ

Triangle

:For alternative meanings, such as the musical instrument, see triangle (disambiguation). A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.

Types of triangles

Triangles can be classified according to the relative lengths of their sides:
- In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon
- In an isosceles triangle two sides are of equal length. An isosceles triangle also has two equal internal angles.
- In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.

Equilateral triangle	Isosceles triangle	Scalene triangle
Equilateral	Isosceles	Scalene

Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.
- A right triangle (or right-angled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs of the triangle.
- An obtuse triangle has one internal angle larger than 90° (an obtuse angle).
- An acute triangle has internal angles that are all smaller than 90° (three acute angles).

Right triangle	Obtuse triangle	Acute triangle
Right	Obtuse	Acute

Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry. In the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides. trigonometry In Euclidean geometry, the sum of the angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known. trigonometry A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as :

c^2 = a^2 + b^2 \,

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third—something unique to right triangles. The Pythagorean theorem can be generalized to the law of cosines: :

c^2 = a^2 + b^2 - 2ab \cos\gamma \,

which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known. The law of sines states :

\fraca=\fracb=\fracc=\frac1d

where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions. There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is :

;1:1:\sqrt

. The "30-60-90 triangle" has sides in the ratio of

1:\sqrt:2

Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained. Menelaus' theorem A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above. Thales' theorem states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. Thales' theorem An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is not obtuse. The three vertices together with the orthocenter are said to form an orthocentric system. orthocentric system An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.
orthocentric system A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. This is also the triangle's center of gravity: if the triangle were made out of wood, say, you could balance it on its centroid, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side. center of gravity The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.
excircle The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler's line. If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.

Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.

Using geometry

The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction. area To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a parallelogram. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is bh, the area of the given triangle must be ½bh. parallelogram

Using vectors

The area of a parallelogram can also be calculated by the use of vectors. If AB and AC are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |AB × AC|, the magnitude of the cross product of vectors AB and AC. |AB × AC| is also equal to |h × AC|, where h represents the altitude h as a vector. The area of triangle ABC is half of this, or S = ½|AB × AC|. cross product

Using trigonometry

The altitude of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as S = ½ab sin γ. It is of course no coincidence that the area of a parallelogram is ab sin γ.

Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x₁, y₁) and C = (x₂, y₂), then the area S can be computed as 1/2 times the absolute value of the determinant :

\beginx_1 & x_2 \\ y_1 & y_2 \end

or S = ½ |x₁y₂ − x₂y₁|.

Using Heron's formula

Yet another way to compute S is Heron's formula: :

S = \sqrt

where s = ½ (a + b + c) is the semiperimeter, or one half of the triangle's perimeter.

Using the side lengths and a numerically stable formula

Heron's formula is numerically unstable for triangles with a very small angle. A stable alternative involves arranging the lengths of the sides so that: :a ≥ b ≥ c and computing :

S = \frac\sqrt

The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

Non-planar triangles

If any four of a triangle's elements (vertices, and/or elements of its sides) are plane to each other, the triangle is called plane. Geometers also study non-planar triangles in noneuclidean geometries, such as spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, he would find that its angles would be greater than 180°.

External links

- [http://ostermiller.org/calc/triangle.html Triangle Calculator] - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.
- [http://agutie.homestead.com/files/Napoleon0.htm Napoleon's theorem] A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- William Kahan: [http://http.cs.berkeley.edu/~wkahan/Triangle.pdf Miscalculating Area and Angles of a Needle-like Triangle].
- Clark Kimberling: [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]. Lists some 1600 interesting points associated with any triangle.
- Christian Obrecht: [http://perso.wanadoo.fr/obrecht/ Eukleides]. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.
- [http://www.cut-the-knot.org/triangle Triangle constructions, remarkable points and lines, and metric relations in a triangle] at cut-the-knot
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1469&CurriculumID=4 Printable Worksheet on Types of Triangles]
- [http://www.vias.org/comp_geometry/geom_triangle.html Compendium Geometry] Analytical Geometry of Triangles Category:Polygons ko:삼각형 ja:三角形 th:รูปสามเหลี่ยม

Base

A base is:
- In computing:
- database is an organized collection of data.
- it is the name of the database for OpenOffice.org
- Google Base is an online database service from google.
- In mathematics:
- A number that is raised to a power; the base of an exponential function. This finds common use, for example, in the depiction of numbers, for instance, 10 is the base used in the decimal system, whereas 2 is the base in the binary numeral system. See also numeral system, radix and table of bases.
- The base of a logarithmic function.
- One of the parallel sides of a trapezoid or the unequal side of an isosceles triangle.
- In topology, a base (topology) for a topology is a set of open sets such that every element of the topology is a union of the base sets. See also subbase.
- Base of a transform in mathematics
- In politics:
- base is a political party's core group of voters.
- As modified by the word tax, it refers to how much income and assets one has, earns, spends, inherits, etcetera as used in the formula to decide owed tax (example: owed_tax = (base times tax_percent) minus deductible).
- In warfare, a military base is a logistics point such as a supply dump and a concentrated facility for storing and repairing military equipment such as an air force base.
- In 2001, the catchphrase "All your base are belong to us" swept across the Internet.
- In sport
- baseball, a base is one of 4 bags or plates placed at corners of the infield diamond that a player has to run to after hitting the ball.
- BASE jumping is an extreme variation on skydiving.
- In a transistor the base is the controlling connection to the junction.
- In Marxism describes the material equipment and material relations of human society, as distinct from the 'superstructural' forms of society - laws, customs, ideas, beliefs, etc.
- The name of the terrorist group Al-Qaida translates as "the base."
- In chemistry, a base is the reactive complement to an acid. See Acid-base reaction theories.
- With reference to drugs, base is a colloquialism for amphetamine or freebase cocaine.
- In genetics, a base pair consists of two complimentary DNA or RNA nucleotides joined by hydrogen bonds.
- In linguistics, a base is a synonym for root word.
- In telecommunications, BASE is a mobile telephony company in Belgium, a subsidiary of the Dutch telecommunications company KPN.
- An isolated settlement in inhospitable conditions that must rely on outside help in order to survive, such as Antarctica base or Moon base.
- In architecture, a base is the part of a column between the bottom of the shaft and the top of the pedestal.
- An easily-confused homonym for Bass (disambiguation), referring to various usages of bass and to basso, the vocal range.

Disk (mathematics)

In geometry, a disk is the region in a plane contained inside of a circle. A disk is said to be closed or open according to whether it contains or not the circle which is its boundary. A ball is a disk in a space with more than two dimensions. See ball (mathematics). Sometimes however, one uses the word "disk" synonymously with "ball", allowing it to have other dimensions.

RADIUS

RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. It is intended to work in both local and roaming situations. When you connect to an ISP using a modem, DSL, cable or wireless connection, you must enter your username and password. This information is passed to a Network Access Server (NAS) device over the Point-to-Point Protocol (PPP), then to a RADIUS server over the RADIUS protocol. The RADIUS server checks that the information is correct using authentication schemes like PAP, CHAP or EAP. If accepted, the server will then authorize access to the ISP system and select an IP address, L2TP parameters, etc. The RADIUS server will also be notified when the session starts and stops, so that the user can be billed accordingly; or the data can be used for statistical purposes. RADIUS was originally developed by Livingston Enterprises for their PortMaster series of Network Access Servers, but later (1997) published as RFC 2058 and RFC 2059 (current versions are RFC 2865 and RFC 2866). Now, several commercial and open-source RADIUS servers exist. Features can vary, but most can look up the users in text files, LDAP servers, various databases, etc. Accounting tickets can be written to text files, various databases, forwarded to external servers, etc. SNMP is often used for remote monitoring. RADIUS proxy servers are used for centralized administration and can rewrite RADIUS packets on the fly (for security reasons, or to convert between vendor dialects). RADIUS is extensible; most vendors of RADIUS hardware and software implement their own dialects. The DIAMETER protocol is the planned replacement for RADIUS, but is still backwards compatible.

Standards

The RADIUS protocol is currently defined in:
- RFC 2865 Remote Authentication Dial In User Service (RADIUS)
- RFC 2866 RADIUS Accounting Other relevant RFCs are:
- RFC 2548 Microsoft Vendor-specific RADIUS Attributes
- RFC 2607 Proxy Chaining and Policy Implementation in Roaming
- RFC 2618 RADIUS Authentication Client MIB
- RFC 2619 RADIUS Authentication Server MIB
- RFC 2620 RADIUS Accounting Client MIB
- RFC 2621 RADIUS Accounting Server MIB
- RFC 2809 Implementation of L2TP Compulsory Tunneling via RADIUS
- RFC 2867 RADIUS Accounting Modifications for Tunnel Protocol Support
- RFC 2868 RADIUS Attributes for Tunnel Protocol Support
- RFC 2869 RADIUS Extensions
- RFC 2882 Network Access Servers Requirements: Extended RADIUS Practices
- RFC 3162 RADIUS and IPv6
- RFC 3576 Dynamic Authorization Extensions to RADIUS

External links

- [http://www.untruth.org/~josh/security/radius/radius-auth.html An Analysis of the RADIUS Authentication Protocol]
- [http://www.freeradius.org/rfc/attributes.html List of RADIUS attributes] Category:Authentication methods Category:Internet protocols Category:Internet standards ja:RADIUS

Pi

The mathematical constant π is the ratio of a circle's circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. The name of the Greek letter π is pi (pronounced pie), and this spelling can be used in typographical contexts where the Greek letter is not available. π is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number. In Euclidean plane geometry, π may be defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. Advanced textbooks define π analytically using trigonometric functions, for example as the smallest positive x for which sin(x) = 0, or as twice the smallest positive x for which cos(x) = 0. All these definitions are equivalent. The numerical value of π truncated to 50 decimal places is: :3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 Although this precision is more than sufficient for use in engineering and science, much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, in addition to supercomputer calculations that have determined over 1 trillion digits of π, no pattern in the digits has ever been found. Digits of π are available from multiple resources on the Internet, and a regular personal computer can compute billions of digits with available software.

Properties

π is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert. π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to square the circle, that is, it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle.

Formulae involving π

Geometry

\pi

appears in many formulae in geometry involving circles and spheres. (All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.) Also, the angle measure of 180° (degrees) is equal to π radians.

Analysis

Many formulae in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.
- François Viète, 1593 (proof): :

\frac2\pi=
\frac2
\frac2
\frac2\ldots

- Leibniz' formula (proof): :

\frac - \frac + \frac - \frac + \frac - \cdots = \frac

:This commonly cited infinite series is usually written as above, but is more technically expressed as: :

\sum_^ \frac = \frac

- Wallis product (proof): :

\frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots = \frac

\prod_^ \frac = \prod_^ \frac \cdot \frac = \frac

- Bailey-Borwein-Plouffe algorithm (See Bailey, 1997 and [http://www.nersc.gov/~dhbailey/ Bailey web page]) :

\pi=\sum_^\infty\frac\left [ \frac  - \frac  - \frac  - \frac \right ]

- An integral formula from calculus (see also Error function and Normal distribution): :

\int_^ e^\,dx = \sqrt

- Basel problem, first solved by Euler (see also Riemann zeta function): :

\zeta(2) = \frac + \frac + \frac + \frac + \cdots = \frac

\zeta(4)= \frac + \frac + \frac + \frac + \cdots = \frac

:and generally,

\zeta(2n)

is a rational multiple of

\pi^

for positive integer n
- Gamma function evaluated at 1/2: :

\Gamma\left(\right)=\sqrt

- Stirling's approximation: :

n! \sim \sqrt \left(\frac\right)^n

- Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"): :

e^ + 1 = 0\;

- Property of Euler's totient function (see also Farey sequence): :

\sum_^ \phi (k) \sim 3 n^2 / \pi^2

- Area of one quarter of the unit circle: :

\int_0^1 \sqrt\,dx =

- An application of the residue theorem :

\oint\frac=2\pi i ,

:where the path of integration is a circle around the origin, traversed in the standard (anti-clockwise) direction.

Continued fractions

π has many continued fractions representations, including: :

\frac = 1 + \frac

(Other representations are available at [http://functions.wolfram.com/Constants/Pi/10/ The Wolfram Functions Site].)

Number theory

Some results from number theory:
- The probability that two randomly chosen integers are coprime is 6/π².
- The probability that a randomly chosen integer is square-free is 6/π².
- The average number of ways to write a positive integer as the sum of two perfect squares (order matters) is π/4.
- The product of (1-1/p²) over the primes, p, is 6/π².

\prod_ \left(1-\frac   \right) = \frac

Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers , and then take the limit as N approaches infinity. The fact (note the order to which the number approaches an integer) that :

e^ = 262537412640768743.99999999999925007...

or equivalently, :

e^ = 640320^3+743.99999999999925007...

can be explained by the theory of complex multiplication.

Dynamical systems and ergodic theory

Consider the recurrence relation :

x_ = 4 x_i (1 - x_i) \,

Then for almost every initial value x₀ in the unit interval [0,1], :

\lim_ \frac \sum_^ \sqrt = \frac

This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.

Physics

The number π appears routinely in equations describing fundamental principles of the universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems.
- The cosmological constant: :

\Lambda =  \rho

- Heisenberg's uncertainty principle: :

\Delta x \Delta p \ge \frac

- Einstein's field equation of general relativity: :

R_ -  + \Lambda g_ =  T_

- Coulomb's law for the electric force: :

F = \frac

- Magnetic permeability of free space: :

\mu_0 = 4 \pi \times 10^\,\mathrm\,

Probability and statistics

In probability and statistics, there are many distributions whose formulae contain π, including:
- probability density function (pdf) for the normal distribution with mean μ and standard deviation σ: :

f(x) = \,e^

- pdf for the (standard) Cauchy distribution: :

f(x) = \frac

Note that since

\int_^ f(x)\,dx = 1

, for any pdf f(x), the above formulae can be used to produce other integral formulae for π. An interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using: :

\pi \approx \frac

Another approximation of π is to [http://www.statisticool.com/pi.htm throw points randomly] into a quarter of a circle with radius 1 that is inscribed in a square of length 1. Pi, the area of a unit circle, is then approximated as 4
- (points in the quarter circle)/(total points).

History of π

Main article: History of Pi. π has been known in some form since antiquity. References to measurements of a circular basin in the Bible give a corresponding value of 3 for π: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." — 1 Kings 7:23; KJV. Nehemiah, a late antique Jewish rabbi and mathematician explained this apparent lack of precision in π, by considering the thickness of the basin, and assuming that the thirty cubits was the inner circumference, while the ten cubits was the diameter of the outside of the basin.

Numerical approximations of π

Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. Therefore numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. An Egyptian scribe named Ahmes wrote the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period—though Ahmes stated that he copied a Middle Kingdom papyrus—and describes the value in such a way that the result obtained comes out to 256 divided by 81 or 3.160. The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation. The Indian mathematician and astronomer Aryabhata gave an accurate approximation for π. He wrote "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words (4+100)×8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π = 62832/20000 = 3.1416, correct when rounded off to four decimal places. The Chinese mathematician and astronomer Zu Chongzhi computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the 5th century. The Iranian mathematician and astronomer, Ghyath ad-din Jamshid Kashani, 1350-1439, computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as: :2 π = 6.2831853071795865 The German mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his tombstone. The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today. None of the formulae given above can serve as an efficient way of approximating π. For fast calculations, one may use formulae such as Machin's: :

\frac = 4 \arctan\frac - \arctan\frac

together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with :

(5+i)^4\cdot(-239+i)=-114244-114244i.

Formulae of this kind are known as Machin-like formulae. Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used in the past. The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulae were used for this: :

\frac = 12 \arctan\frac + 32 \arctan\frac - 5 \arctan\frac + 12 \arctan\frac

:K. Takano (1982). :

\frac = 44 \arctan\frac + 7 \arctan\frac - 12 \arctan\frac + 24 \arctan\frac

:F. C. W. Störmer (1896). These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers and (obviously) for establishing new π calculation records. In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series: :

\pi = \sum_^ \frac
\left( \frac - \frac - \frac - \frac\right)

This formula permits one to easily compute the k^th binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. [http://www.nersc.gov/~dhbailey/ Bailey's website] contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0). Other formulae that have been used to compute estimates of π include: :

\frac=
\sum_^\infty\frac=
1+\frac\left(1+\frac\left(1+\frac\left(1+\frac(1+...)\right)\right)\right)

:Newton. :

\frac = \frac \sum^\infty_ \frac

:Ramanujan. This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π. :

\frac = 12 \sum^\infty_ \frac

:David Chudnovsky and Gregory Chudnovsky. :

= 20 \arctan\frac + 8 \arctan\frac

:Euler.

Miscellaneous formulae

In base 60, π can be approximated to eight significant figures as :

3 + \frac + \frac + \frac

In addition, the following expressions can be used to estimate π
- accurate to 9 digits: :

(63/25)((17+15\sqrt 5)/(7+15\sqrt5))

- accurate to 17 digits: :

3 + \frac

- accurate to 3 digits: :

\sqrt + \sqrt

:Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of right triangles which are either isosceles or halves of equilateral triangles.

Less accurate approximations

In 1897, a physician and amateur mathematician from Indiana named Edward J. Goodwin believed that the transcendental value of π was wrong. He proposed a bill to Indiana Representative T. I. Record which expressed the "new mathematical truth" in several ways: :The ratio of the diameter of a circle to its circumference is 5/4 to 4. (π = 3.2) :The ratio of the length of a 90 degree arc to the length of a segment connecting the arc's two endpoints is 8 to 7. (π ≈ 3.23...) :The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. (π = 4) :It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side. (π ≈ 9.24 if rectangle is emended to triangle; if not, as above.) The bill also recites Goodwin's previous accomplishments: "his solutions of the