1.
Imperial units
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The system of imperial units or the imperial system is the system of units first defined in the British Weights and Measures Act of 1824, which was later refined and reduced. The Imperial units replaced the Winchester Standards, which were in effect from 1588 to 1825, the system came into official use across the British Empire. The imperial system developed from what were first known as English units, the Weights and Measures Act of 1824 was initially scheduled to go into effect on 1 May 1825. However, the Weights and Measures Act of 1825 pushed back the date to 1 January 1826, the 1824 Act allowed the continued use of pre-imperial units provided that they were customary, widely known, and clearly marked with imperial equivalents. Apothecaries units are mentioned neither in the act of 1824 nor 1825, at the time, apothecaries weights and measures were regulated in England, Wales, and Berwick-upon-Tweed by the London College of Physicians, and in Ireland by the Dublin College of Physicians. In Scotland, apothecaries units were unofficially regulated by the Edinburgh College of Physicians, the three colleges published, at infrequent intervals, pharmacopoeiae, the London and Dublin editions having the force of law. The Medical Act of 1858 transferred to The Crown the right to publish the official pharmacopoeia and to regulate apothecaries weights, Metric equivalents in this article usually assume the latest official definition. Before this date, the most precise measurement of the imperial Standard Yard was 0.914398416 metres, in 1824, the various different gallons in use in the British Empire were replaced by the imperial gallon, a unit close in volume to the ale gallon. It was originally defined as the volume of 10 pounds of distilled water weighed in air with brass weights with the standing at 30 inches of mercury at a temperature of 62 °F. The Weights and Measures Act of 1985 switched to a gallon of exactly 4.54609 l and these measurements were in use from 1826, when the new imperial gallon was defined, but were officially abolished in the United Kingdom on 1 January 1971. In the USA, though no longer recommended, the system is still used occasionally in medicine. The troy pound was made the unit of mass by the 1824 Act, however, its use was abolished in the UK on 1 January 1879, with only the troy ounce. The Weights and Measures Act 1855 made the pound the primary unit of mass. In all the systems, the unit is the pound. For the yard, the length of a pendulum beating seconds at the latitude of Greenwich at Mean Sea Level in vacuo was defined as 39.01393 inches, the imperial system is one of many systems of English units. Although most of the units are defined in more than one system, some units were used to a much greater extent, or for different purposes. The distinctions between these systems are not drawn precisely. One such distinction is that between these systems and older British/English units/systems or newer additions, the US customary system is historically derived from the English units that were in use at the time of settlement

2.
Metre
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The metre or meter, is the base unit of length in the International System of Units. The metre is defined as the length of the path travelled by light in a vacuum in 1/299792458 seconds, the metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole. In 1799, it was redefined in terms of a metre bar. In 1960, the metre was redefined in terms of a number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted, the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Measuring devices are spelled -meter in all variants of English, the suffix -meter has the same Greek origin as the unit of length. This range of uses is found in Latin, French, English. Thus calls for measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. In 1668, Wilkins proposed using Christopher Wrens suggestion of defining the metre using a pendulum with a length which produced a half-period of one second, christiaan Huygens had observed that length to be 38 Rijnland inches or 39.26 English inches. This is the equivalent of what is now known to be 997 mm, no official action was taken regarding this suggestion. In the 18th century, there were two approaches to the definition of the unit of length. One favoured Wilkins approach, to define the metre in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the metre as one ten-millionth of the length of a quadrant along the Earths meridian, that is, the distance from the Equator to the North Pole. This means that the quadrant would have defined as exactly 10000000 metres at that time. To establish a universally accepted foundation for the definition of the metre, more measurements of this meridian were needed. This portion of the meridian, assumed to be the length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator

3.
Hectare
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The hectare is an SI accepted metric system unit of area equal to 100 ares and primarily used in the measurement of land as a metric replacement for the imperial acre. An acre is about 0.405 hectare and one hectare contains about 2.47 acres, in 1795, when the metric system was introduced, the are was defined as 100 square metres and the hectare was thus 100 ares or 1⁄100 km2. When the metric system was further rationalised in 1960, resulting in the International System of Units, the are was not included as a recognised unit. The hectare, however, remains as a non-SI unit accepted for use with the SI units, the metric system of measurement was first given a legal basis in 1795 by the French Revolutionary government. At the first meeting of the CGPM in 1889 when a new standard metre, manufactured by Johnson Matthey & Co of London was adopted, in 1960, when the metric system was updated as the International System of Units, the are did not receive international recognition. The units that were catalogued replicated the recommendations of the CGPM, many farmers, especially older ones, still use the acre for everyday calculations, and convert to hectares only for official paperwork. Farm fields can have long histories which are resistant to change, with names such as the six acre field stretching back hundreds of years. The names centiare, deciare, decare and hectare are derived by adding the standard metric prefixes to the base unit of area. The centiare is a synonym for one square metre, the deciare is ten square metres. The are is a unit of area, equal to 100 square metres and it was defined by older forms of the metric system, but is now outside of the modern International System of Units. It is commonly used to measure real estate, in particular in Indonesia, India, and in French-, Portuguese-, Slovakian-, Serbian-, Czech-, Polish-, Dutch-, in Russia and other former Soviet Union states, the are is called sotka. It is used to describe the size of suburban dacha or allotment garden plots or small city parks where the hectare would be too large, the decare is derived from deka, the prefix for 10 and are, and is equal to 10 ares or 1000 square metres. It is used in Norway and in the former Ottoman areas of the Middle East, the hectare, although not strictly a unit of SI, is the only named unit of area that is accepted for use within the SI. The United Kingdom, United States, Burma, and to some extent Canada instead use the acre, others, such as South Africa, published conversion factors which were to be used particularly when preparing consolidation diagrams by compilation. In many countries, metrication redefined or clarified existing measures in terms of metric units, non-SI units accepted for use with the International System of Units

4.
Yard
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The yard is an English unit of length, in both the British imperial and US customary systems of measurement, that comprises 3 feet or 36 inches. It is by international agreement in 1959 standardized as exactly 0.9144 meters, a metal yardstick originally formed the physical standard from which all other units of length were officially derived in both English systems. In the 19th and 20th centuries, increasingly powerful microscopes and scientific measurement detected variation in these prototype yards which became significant as technology improved. In 1959, the United States, United Kingdom, Australia, New Zealand, the name derives from the Old English gerd, gyrd, &c. which was used for branches, staves, and measuring rods. It is first attested in the late-7th century laws of Ine of Wessex, where the yard of land mentioned is the yardland, an old English unit of tax assessment equal to 1⁄4 hide. Around the same time, the Lindisfarne Gospels account of the messengers from John the Baptist in the Book of Matthew used it for a branch swayed by the wind. In addition to the yardland, Old and Middle English both used their forms of yard to denote the lengths of 15 or 16 1⁄2 ft used in computing acres. A unit of three English feet is attested in a statute of c. 1300 but there it is called an ell, the use of the word yard to describe this length is first attested in Langlands poem on Piers Plowman. The usage seems to derive from the prototype standard rods held by the king, the word yard is a homonym of yard in the sense of an enclosed area of land. This second meaning of yard has a related to the verb to gird and is probably not related. The origin of the measure is uncertain, both the Romans and the Welsh used multiples of a shorter foot, but 2 1⁄2 Roman feet was a step and 3 Welsh feet was a pace. The Proto-Germanic cubit or arms-length has been reconstructed as *alinâ, which developed into the Old English ęln, Middle English elne and this has led some to derive the yard of three English feet from pacing, others from the ell or cubit, others from Henry Is arm standard. Based on the etymology of the yard, others suggest it originally derived from the girth of a persons waist. But the yard was the standard adopted by the early English soverigns. The yard continued till the reign of Henry VII. when the ell was introduced, that being a yard, the ell was borrowed from the Paris drapers. Subsequently, however, Queen Elizabeth re-introduced the yard as the English standard of measure, the earliest record of a prototype measure is the statute II Edgar Cap. 8, which survives in several variant manuscripts, in it, Edgar the Peaceful directed the Witenagemot at Andover that the measure held at Winchester should be observed throughout his realm. The statutes of William I similarly refer to and uphold the measures of his predecessors without naming them

5.
Foot (unit)
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The foot is a unit of length in the imperial and US customary systems of measurement. Since 1959, both units have been defined by international agreement as equivalent to 0.3048 meters exactly, in both systems, the foot comprises 12 inches and three feet compose a yard. Historically the foot was a part of local systems of units, including the Greek, Roman, Chinese, French. It varied in length from country to country, from city to city and its length was usually between 250 mm and 335 mm and was generally, but not always, subdivided into 12 inches or 16 digits. The United States is the industrialized nation that uses the international foot and the survey foot in preference to the meter in its commercial, engineering. The foot is legally recognized in the United Kingdom, road signs must use imperial units, the measurement of altitude in international aviation is one of the few areas where the foot is widely used outside the English-speaking world. The length of the international foot corresponds to a foot with shoe size of 13,14,15.5 or 46. Historically the human body has been used to provide the basis for units of length. The foot of a male is typically about 15. 3% of his height, giving a person of 160 cm a foot of 245 mm. These figures are less than the used in most cities over time. Archeologists believe that the Egyptians, Ancient Indians and Mesopotamians preferred the cubit while the Romans, under the Harappan linear measures, Indus cities during the Bronze Age used a foot of 13.2 inches and a cubit of 20.8 inches. The Egyptian equivalent of the measure of four palms or 16 digits—was known as the djeser and has been reconstructed as about 30 cm. The Greek foot had a length of 1⁄600 of a stadion, one stadion being about 181.2 m, the standard Roman foot was normally about 295.7 mm, but in the provinces, the pes Drusianus was used, with a length of about 334 mm. Originally both the Greeks and the Romans subdivided the foot into 16 digits, but in later years, after the fall of the Roman Empire, some Roman traditions were continued but others fell into disuse. In AD790 Charlemagne attempted to reform the units of measure in his domains and his units of length were based on the toise and in particular the toise de lÉcritoire, the distance between the fingertips of the outstretched arms of a man. The toise has 6 pieds each of 326.6 mm, at the same time, monastic buildings used the Carolingian foot of 340 mm. The procedure for verification of the foot as described in the 16th century by Jacob Koebel in his book Geometrei, the measures of Iron Age Britain are uncertain and proposed reconstructions such as the Megalithic Yard are controversial. Later Welsh legend credited Dyfnwal Moelmud with the establishment of their units, the Belgic or North German foot of 335 mm was introduced to England either by the Belgic Celts during their invasions prior to the Romans or by the Anglo-Saxons in the 5th & 6th century

6.
Square root
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In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss

7.
Inch
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The inch is a unit of length in the imperial and United States customary systems of measurement now formally equal to 1⁄36 yard but usually understood as 1⁄12 of a foot. Derived from the Roman uncia, inch is also used to translate related units in other measurement systems. The English word inch was a borrowing from Latin uncia not present in other Germanic languages. The vowel change from Latin /u/ to English /ɪ/ is known as umlaut, the consonant change from the Latin /k/ to English /tʃ/ or /ʃ/ is palatalisation. Both were features of Old English phonology, inch is cognate with ounce, whose separate pronunciation and spelling reflect its reborrowing in Middle English from Anglo-Norman unce and ounce. In many other European languages, the word for inch is the same as or derived from the word for thumb, the inch is a commonly used customary unit of length in the United States, Canada, and the United Kingdom. It is also used in Japan for electronic parts, especially display screens, for example, three feet two inches can be written as 3′ 2″. Paragraph LXVII sets out the fine for wounds of various depths, one inch, one shilling, an Anglo-Saxon unit of length was the barleycorn. After 1066,1 inch was equal to 3 barleycorns, which continued to be its legal definition for several centuries, similar definitions are recorded in both English and Welsh medieval law tracts. One, dating from the first half of the 10th century, is contained in the Laws of Hywel Dda which superseded those of Dyfnwal, both definitions, as recorded in Ancient Laws and Institutes of Wales, are that three lengths of a barleycorn is the inch. However, the oldest surviving manuscripts date from the early 14th century, john Bouvier similarly recorded in his 1843 law dictionary that the barleycorn was the fundamental measure. He noted that this process would not perfectly recover the standard, before the adoption of the international yard and pound, various definitions were in use. In the United Kingdom and most countries of the British Commonwealth, the United States adopted the conversion factor 1 metre =39.37 inches by an act in 1866. In 1930, the British Standards Institution adopted an inch of exactly 25.4 mm, the American Standards Association followed suit in 1933. By 1935, industry in 16 countries had adopted the industrial inch as it came to be known, in 1946, the Commonwealth Science Congress recommended a yard of exactly 0.9144 metres for adoption throughout the British Commonwealth. This was adopted by Canada in 1951, the United States on 1 July 1959, Australia in 1961, effective 1 January 1964, and the United Kingdom in 1963, effective on 1 January 1964. The new standards gave an inch of exactly 25.4 mm,1.7 millionths of a longer than the old imperial inch and 2 millionths of an inch shorter than the old US inch. The United States retains the 1/39. 37-metre definition for survey purposes and this is approximately 1/8-inch in a mile

8.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T

9.
Square kilometre
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Square kilometre or square kilometer, symbol km2, is a multiple of the square metre, the SI unit of area or surface area. For example,3 km2 is equal to 3×2 =3,000,000 m2, topographical map grids are worked out in metres, with the grid lines being 1,000 metres apart. 1,100,000 maps are divided into squares representing 1 km2, each square on the map being one square centimetre in area, for 1,50,000 maps, the grid lines are 2 cm apart. Each square on the map is 2 cm by 2 cm, for 1,25,000 maps, the grid lines are 4 cm apart. Each square on the map is 4 cm by 4 cm, in each case, the grid lines enclose one square kilometre. The area enclosed by the walls of many European medieval cities were about one square kilometre, the approximate area of the old walled cities can often be worked out by fitting the course of the wall to a rectangle or an oval. Examples include Delft, Netherlands 52°0′54″N 4°21′34″E The walled city of Delft was approximately rectangular, the approximate length of rectangle was about 1.30 kilometres. The approximate width of the rectangle was about 0.75 kilometres, a perfect rectangle with these measurements has an area of 1. 30×0.75 =0.9 km2 Lucca 43°50′38″N 10°30′2″E The medieval city is roughly rectangular with rounded north-east and north-west corners. The maximum distance from east to west is 1.36 kilometres, the maximum distance from north to south is 0.80 kilometres. A perfect rectangle of these dimensions would be 1. 36×0.80 =1.088 km2, Brugge 51°12′39″N 3°13′28″E The medieval city of Brugge, a major centre in Flanders, was roughly oval or elliptical in shape with the longer or semi-major axis running north and south. The maximum distance from north to south is 2.53 kilometres, the maximum distance from east to west is 1.81 kilometres. A perfect ellipse of these dimensions would be 2.53 ×1.81 × =3.597 km2. Chester United Kingdom 53°12′1″N 2°52′45″W Chester is one of the smaller English cities that has a city wall. The distance from Northgate to Watergate is about 855 metres. The distance from Eastgate to Westgate is about 589 metres, a perfect rectangle of these dimensions would be × =0.504 km2. Parks come in all sizes, a few are almost exactly one kilometre in area. Here are some examples, Riverside Country Park, UK. Brierley Forest Park, rio de Los Angeles State Park, California, USA Jones County Central Park, Iowa, USA. Using the figures published by golf course architects Crafter and Mogford, assuming a 6,000 metres 18-hole course, an area of 80 hectares needs to be allocated for the course itself

10.
Metrication in Canada
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Until the 1970s, Canada traditionally used the Imperial measurement system, labelled as Canadian units of measurements under Schedule II, Section 4 of the Weights and Measures Act. These units have the name and, with the exception of capacity measures such as the gallon. For example, before metrication in Canada, gasoline was sold by the imperial gallon, in cross-border transactions, it was often unclear whether values quoted in gallons, etc. were referring to the US values or the imperial values of these units. By the mid-1970s, metric product labelling was introduced, in 1972, the provinces agreed to make all road signs metric by 1977. There was some resistance to metrication, especially as the sectors of the economy where the federal Weights, the city of Peterborough, Ontario, was a noted hotbed of opposition to metrication, having been one of the governments three test centres for the metrication process. Bill Domm, a Member of Parliament representing the riding of Peterborough, was one of the countrys most outspoken opponents of metrication, during this period, a few government employees lost their jobs for their opposition to metrication. Neil Fraser, an official with Revenue Canada who publicly opposed mandatory metric conversion, was dismissed for conduct unacceptable for a public servant. Since 1976, the law requires that all prepackaged food products must declare their mass or their volume in metric units, milk has been thoroughly metric since 1980. In April 1975, Fahrenheit temperatures were replaced by Celsius, in September 1975, rainfall was first measured in millimetres and snow in centimetres. Since April 1976, wind speed, visibility, and barometric pressure have been in SI units, in September 1977, every speed-limit sign in the country was changed from miles per hour to kilometres per hour. The election of the Progressive Conservative government of Brian Mulroney in 1984 resulted in the abolition of the Metric Commission on March 31,1985 and this ended the process of affirmative metrication in Canada, and some regulations requiring metric measurements either have been repealed or are no longer enforced. Training on metric conversion was not universal, poor metrication training was a contributing factor to Air Canada Flight 143, the so-called Gimli Glider, running out of fuel mid-flight on 23 July 1983. Notwithstanding the end of officially sanctioned metrication in Canada, most laws, regulations, however, imperial measures still have legal definitions in Canada and can be used alongside metric units. Cars have metric speedometers and odometers, although some speedometers include smaller figures in miles per hour, fuel efficiency for new vehicles is published by Natural Resources Canada in litres per 100 kilometres and miles per gallon. Window stickers in dealer showrooms often include miles per gallon conversions, the railways of Canada continue to measure their trackage in miles, and speed limits in mph. Canadian railcars show weight figures in both imperial and metric, today, Canadians typically use a mix of metric and imperial measurements in their daily lives. However the use of the metric and imperial systems varies according to generations, newborns are measured in SI at hospitals, but the birth weight and length is also announced to family and friends in imperial units. Among the broader population, imperial units are sometimes used to indicate height and weight

11.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter

12.
Square (algebra)
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In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2