164-428: Cuisenaire rods are mathematics learning aids for pupils that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors . In the early 1950s, Caleb Gattegno popularised this set of coloured number rods created by Georges Cuisenaire (1891–1975), a Belgian primary school teacher, who called
328-487: A unitary authority providing all local government functions since 1998. There are no civil parishes in the borough. Some of the built-up area's outer suburbs are outside the borough boundaries in West Berkshire and Wokingham . These outer suburbs belong to civil parishes, in some cases with their own town status. Reading has elected at least one Member of Parliament to every Parliament since 1295. Since
492-549: A 2 cm (0.8 in) unit cube rather than 1 cm (0.4 in). She produced various resources to complement the rods, such as trays to arrange the rods in, and tracks to arrange them on. Tony Wing, in producing resources for Numicon, built on many of Stern's ideas, also making trays and tracks available for use with Cuisenaire rods. In 1961, Seton Pollock produced the Colour Factor system, consisting of rods from lengths 1 to 12 cm (0.39 to 4.7 in). Based on
656-1117: A common denominator. This can be achieved by scaling the first number with the denominator of the second number while scaling the second number with the denominator of the first number. For instance, 1 3 + 1 2 = 1 ⋅ 2 3 ⋅ 2 + 1 ⋅ 3 2 ⋅ 3 = 2 6 + 3 6 = 5 6 {\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}} . Two rational numbers are multiplied by multiplying their numerators and their denominators respectively, as in 2 3 ⋅ 2 5 = 2 ⋅ 2 3 ⋅ 5 = 4 15 {\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}} . Dividing one rational number by another can be achieved by multiplying
820-442: A general identity element since 1 is not the neutral element for the base. Exponentiation and logarithm are neither commutative nor associative. Different types of arithmetic systems are discussed in the academic literature. They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects other than numbers. Integer arithmetic
984-410: A limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number. Numeral systems are either positional or non-positional. All early numeral systems were non-positional. For non-positional numeral systems, the value of a digit does not depend on its position in the numeral. The simplest non-positional system
1148-475: A local magistrates' court . Reading was an ancient borough , being described as a borough by the time of the Domesday Book in 1086. The borough was initially controlled by Reading Abbey as its manorial owner. The town gradually gained a degree of independence from the abbey from the 13th century onwards, particularly after the town's merchant guild was granted a royal charter in 1253. Following
1312-863: A month. The old Victorian Corn Exchange now provides an alternative access to a shopping centre. Every year Reading hosts the Reading Festival , which has been running since 1971. The festival takes place on the Friday, Saturday and Sunday of the August bank holiday weekend and is the largest of its kind in the United Kingdom aside from the Glastonbury Festival . Reading Festival takes place at Little Johns Farm in Reading, Richfield Avenue. For some twenty years until 2006, Reading
1476-464: A more complex non-positional numeral system . They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the Roman numeral system . It has
1640-497: A number of educators. The French-Canadian educator Madeleine Goutard in her 1963 Mathematics and Children , wrote: The teacher is not the person who teaches him what he does not know. He is the one who reveals the child to himself by making him more conscious of, and more creative with his own mind. The parents of a little girl of six who was using the Cuisenaire rods at school marveled at her knowledge and asked her: 'Tell us how
1804-639: A number, it is also possible to multiply by its reciprocal . The reciprocal of a number is 1 divided by that number. For instance, 48 ÷ 8 = 48 × 1 8 {\displaystyle 48\div 8=48\times {\tfrac {1}{8}}} . The multiplicative identity element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, 13 × 1 = 13 {\displaystyle 13\times 1=13} and 13 × 1 13 = 1 {\displaystyle 13\times {\tfrac {1}{13}}=1} . Multiplication
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#17327942033691968-490: A plane. Further branches of number theory are probabilistic number theory , which employs methods from probability theory , combinatorial number theory , which relies on the field of combinatorics , computational number theory , which approaches number-theoretic problems with computational methods, and applied number theory, which examines the application of number theory to fields like physics , biology , and cryptography . Influential theorems in number theory include
2132-475: A population density of 4,327 per square kilometre (11,207/sq mi). Meanwhile, the wider urban area had a population of 318,014 in the 2011 census , ranking 23rd in the United Kingdom. This grew to an estimated 337,108 by mid-2018. According to the 2011 census, 74.8% of the borough's population were described as White (65.3% White British ), 9.1% as South Asian , 6.7% as Black , 3.9% Mixed , 4.5% as Chinese and 0.9% as other ethnic group . In 2010, it
2296-553: A positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1. Irrational numbers involve an infinite non-repeating series of decimal digits. Because of this, there is often no simple and accurate way to express the results of arithmetic operations like 2 + π {\displaystyle {\sqrt {2}}+\pi } or e ⋅ 3 {\displaystyle e\cdot {\sqrt {3}}} . In cases where absolute precision
2460-487: A presence in the town, supplying Fibre-to-the-Premises ( FTTP ) broadband internet connections at speeds of up to 1 Gbit/s. Reading School (a state grammar school ), founded in 1125, is the 16th oldest school in England. There are six other state secondary schools and 38 state primary schools within the borough, together with a number of private schools and nurseries. Alfred Sutton Boys' School closed in
2624-507: A range of values if one does not know the precise magnitude, for example, because of measurement errors . Interval arithmetic includes operations like addition and multiplication on intervals, as in [ 1 , 2 ] + [ 3 , 4 ] = [ 4 , 6 ] {\displaystyle [1,2]+[3,4]=[4,6]} and [ 1 , 2 ] × [ 3 , 4 ] = [ 3 , 8 ] {\displaystyle [1,2]\times [3,4]=[3,8]} . It
2788-470: A self-governing, self-funding theatre group and registered charity founded in 1947 that operates and maintains its own 97-seat theatre. Rabble Theatre in Caversham and Reading Rep on London Road offer classic and contemporary performances. Jelly is an artist-led organisation that has been committed to improving access to the arts since 1993. The demonym for a person from Reading is Redingensian , giving
2952-444: A series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since ( 5 × 4 ) × 2 {\displaystyle (5\times 4)\times 2} is the same as 5 × ( 4 × 2 ) {\displaystyle 5\times (4\times 2)} . Addition is an arithmetic operation in which two numbers, called
3116-475: A set of ten rods sawn out of wood, with lengths from 1 cm (0.4 in) to 10 cm (4 in). He painted each length of rod a different colour and began to use these in his teaching of arithmetic. The invention remained almost unknown outside the village of Thuin for about 23 years until, in April 1953, British mathematician and mathematics education specialist Caleb Gattegno was invited to see pupils using
3280-442: A significant presence in the town include Huawei Technologies , Pegasystems , Access IS , CGI Inc. , Agilent Technologies , Cisco , Ericsson , Symantec , Verizon Business , and Commvault . These companies are distributed around Reading or just outside the borough boundary, some in business parks including Thames Valley Park in nearby Earley, Green Park Business Park and Arlington Business Park . Reading town centre
3444-469: A significant role during the Glorious Revolution : the second Battle of Reading was the only substantial military action of the campaign. The 18th century saw the beginning of a major iron works in the town and the growth of the brewing trade for which Reading was to become famous. Reading's trade benefited from better designed turnpike roads which helped it establish its location on
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#17327942033693608-402: A similar role in the sciences , like physics and economics . Arithmetic is present in many aspects of daily life , for example, to calculate change while shopping or to manage personal finances . It is one of the earliest forms of mathematics education that students encounter. Its cognitive and conceptual foundations are studied by psychology and philosophy . The practice of arithmetic
3772-439: A special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to 3 10 {\displaystyle {\tfrac {3}{10}}} , and 25.12 is equal to 2512 100 {\displaystyle {\tfrac {2512}{100}}} . Every rational number corresponds to a finite or a repeating decimal . Irrational numbers are numbers that cannot be expressed through
3936-556: Is exponentiation by squaring . It breaks down the calculation into a number of squaring operations. For example, the exponentiation 3 65 {\displaystyle 3^{65}} can be written as ( ( ( ( ( 3 2 ) 2 ) 2 ) 2 ) 2 ) 2 × 3 {\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3} . By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than
4100-406: Is 0 and the additive inverse of a number is the negative of that number. For instance, 13 + 0 = 13 {\displaystyle 13+0=13} and 13 + ( − 13 ) = 0 {\displaystyle 13+(-13)=0} . Addition is both commutative and associative. Multiplication is an arithmetic operation in which two numbers, called the multiplier and
4264-426: Is 0. 3 . Every repeating decimal expresses a rational number. Real number arithmetic is the branch of arithmetic that deals with the manipulation of both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and π . Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses
4428-661: Is 42 miles (68 km) north of the English south coast. The centre of Reading is on a low ridge between the River Thames and River Kennet , close to their confluence , reflecting the town's history as a river port. Just above the confluence, the Kennet cuts through a narrow steep-sided gap in the hills forming the southern flank of the Thames flood plain . The Kennet, which naturally divided into multiple shallow streams through
4592-590: Is a conversion of a nonconformist chapel dating from 1707. Besides the two major shopping malls, Reading has three smaller shopping arcades, the Bristol and West Arcade, Harris Arcade and The Walk, which contain smaller specialist stores. An older form of retail facility is represented by Union Street , popularly known as Smelly Alley . Reading has no indoor market, but there is a street market in Hosier Street. A farmers' market operates on two Saturdays
4756-450: Is a great mistake. What we ought to do is use these materials to enable children to make for themselves, out of their own experience and discoveries, a solid and growing understanding of the ways in which numbers and the operations of arithmetic work. Our aim must be to build soundly, and if this means that we must build more slowly, so be it. Some things we will be able to do much earlier than we used to, fractions for example. Gattegno formed
4920-808: Is a major shopping centre. In 2007, an independent poll placed Reading 16th in a league table of best performing retail centres in the United Kingdom. The main shopping street is Broad Street , which runs between The Oracle in the east and Broad Street Mall in the west and was pedestrianised in 1995. The smaller Friars Walk in Friar Street is closed and will be demolished if the proposed Station Hill redevelopment project goes ahead. There are three major department stores in Reading: John Lewis & Partners (known as Heelas until 2001), Debenhams (now closed down), and House of Fraser . The Broad Street branch of bookseller Waterstone's
5084-687: Is a museum dedicated to recording the changing face of farming and the countryside in England. It houses designated collections of national importance. It is owned and run by the University of Reading , as are the Ure Museum of Greek Archaeology , the Cole Museum of Zoology and the Harris Botanic Gardens , all of which can be found on the university's Whiteknights Campus . The small Riverside Museum at Blake's Lock tells
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5248-545: Is a prime number that has no other prime factorization. Euclid's theorem states that there are infinitely many prime numbers. Fermat's last theorem is the statement that no positive integer values can be found for a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} , to solve the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} if n {\displaystyle n}
5412-465: Is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound of the approximation error is a more sophisticated approach. In the example, the person's height might be represented as 1.62 ± 0.005 meters or 63.8 ± 0.2 inches . In performing calculations with uncertain quantities, the uncertainty should be propagated to calculated quantities. When adding or subtracting two or more quantities, add
5576-902: Is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number π is rounded to 4 decimal places, the result is 3.142 because the following digit is a 5, so 3.142 is closer to π than 3.141. These methods allow computers to efficiently perform approximate calculations on real numbers. In science and engineering, numbers represent estimates of physical quantities derived from measurement or modeling. Unlike mathematically exact numbers such as π or 2 {\displaystyle {\sqrt {2}}} , scientifically relevant numerical data are inherently inexact, involving some measurement uncertainty . One basic way to express
5740-414: Is about calculations with real numbers , which include both rational and irrational numbers . Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers . Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of
5904-642: Is also a regional retail centre, serving a large area of the Thames Valley with its shopping centres, including the Oracle , the Broad Street Mall, and the pedestrianised area around Broad Street. It is home to the University of Reading . Every year it hosts the Reading Festival , one of England's biggest music festivals. Reading has a professional association football team, Reading F.C. , and participates in many other sports. Reading dates from
6068-507: Is an elementary branch of mathematics that studies numerical operations like addition , subtraction , multiplication , and division . In a wider sense, it also includes exponentiation , extraction of roots , and taking logarithms . Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers . Rational number arithmetic involves operations on fractions of integers. Real number arithmetic
6232-450: Is an inverse of the operation " ∘ {\displaystyle \circ } " if it fulfills the following condition: t ⋆ s = r {\displaystyle t\star s=r} if and only if r ∘ s = t {\displaystyle r\circ s=t} . Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation
6396-436: Is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0. There are not only inverse elements but also inverse operations . In an informal sense, one operation
6560-526: Is at least thousands and possibly tens of thousands of years old. Ancient civilizations like the Egyptians and the Sumerians invented numeral systems to solve practical arithmetic problems in about 3000 BCE. Starting in the 7th and 6th centuries BCE, the ancient Greeks initiated a more abstract study of numbers and introduced the method of rigorous mathematical proofs . The ancient Indians developed
6724-583: Is both commutative and associative. Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in superscript right after the base. Examples are 2 4 = 16 {\displaystyle 2^{4}=16} and 3 {\displaystyle 3} ^ 3 = 27 {\displaystyle 3=27} . If
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6888-403: Is closed under division as long as the divisor is not 0. Both integer arithmetic and rational number arithmetic are not closed under exponentiation and logarithm. One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the nth root of the result based on the denominator of
7052-507: Is closely related to affine arithmetic, which aims to give more precise results by performing calculations on affine forms rather than intervals. An affine form is a number together with error terms that describe how the number may deviate from the actual magnitude. Reading, England Reading ( / ˈ r ɛ d ɪ ŋ / RED -ing ) is a town and borough in Berkshire , England. Most of its built-up area lies within
7216-401: Is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, 7 + 9 {\displaystyle 7+9} is the same as 9 + 7 {\displaystyle 9+7} . Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in
7380-620: Is complemented by Palmer Park , a purpose built public park in east Reading gifted to the town by the proprietors of Huntley & Palmers in 1889. A string of open spaces stretch along one or other side of the River Thames throughout its passage through Reading. From west to east these are Thameside Promenade , Caversham Court , Christchurch Meadows , Hills Meadow , View Island and King's Meadow . Reading also has five local nature reserves : Clayfield Copse in Caversham , with
7544-460: Is from Reading, made the film Cemetery Junction , which, although filmed elsewhere in the United Kingdom, is set in 1970s Reading and is named after a busy junction in East Reading . Jasper Fforde 's Nursery Crimes Division novels, The Big Over Easy and The Fourth Bear , are also placed in Reading. The BBC Two sitcom Beautiful People , based on the memoirs of Simon Doonan ,
7708-431: Is greater than 2 {\displaystyle 2} . Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have
7872-439: Is infinite without repeating decimals. The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is R {\displaystyle \mathbb {R} } . Even wider classes of numbers include complex numbers and quaternions . A numeral is a symbol to represent a number and numeral systems are representational frameworks. They usually have
8036-772: Is largely based on Reading. Described with topographical accuracy, it is still possible to follow the steps of the novel's characters in present-day Reading. Reading also appears in the works of Thomas Hardy where it is called 'Aldbrickham'. It features most heavily in his final novel, Jude the Obscure , as the temporary home of Jude Fawley and Sue Bridehead. Oscar Wilde was imprisoned in Reading Gaol from 1895 to 1897. While there, he wrote his letter De Profundis . After his release, he lived in exile in France and wrote The Ballad of Reading Gaol , based on his experience of
8200-427: Is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5. One way to ensure that the result is an integer is to round the result to a whole number. However, this method leads to inaccuracies as the original value is altered. Another method is to perform the division only partially and retain
8364-417: Is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding . For truncation, a certain number of leftmost digits are kept and remaining digits are discarded or replaced by zeros. For example, the number π has an infinite number of digits starting with 3.14159.... If this number is truncated to 4 decimal places, the result is 3.141. Rounding
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#17327942033698528-437: Is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance 14 − 8 = 14 + ( − 8 ) {\displaystyle 14-8=14+(-8)} . This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations. The additive identity element
8692-506: Is only one road across the Kennet flood plain. All other routes between the three built-up areas are in the central area. Like the rest of the United Kingdom, Reading has a maritime climate , with limited seasonal temperature ranges and generally moderate rainfall throughout the year. The nearest official Met Office weather station is located at the Reading University Atmospheric Observatory on
8856-467: Is regulated by the Environment Agency . Reading's water supply is largely derived from underground aquifers , and as a consequence the water is hard . The commercial energy supplier for electricity and gas is at the consumer's choice. SSEN runs the local electricity distribution network, while SGN runs the gas distribution network. A notable part of the local energy infrastructure is
9020-930: Is restricted to the study of integers and focuses on their properties and relationships such as divisibility , factorization , and primality . Traditionally, it is known as higher arithmetic. Numbers are mathematical objects used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different numeral systems to represent them. The main kinds of numbers employed in arithmetic are natural numbers , whole numbers, integers , rational numbers , and real numbers . The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as { 1 , 2 , 3 , 4 , . . . } {\displaystyle \{1,2,3,4,...\}} . The symbol of
9184-707: Is set in Reading in the late 1990s. The Maiwand Lion in Forbury Gardens , an unofficial symbol of Reading, commemorates the 328 officers of the Royal Berkshire Regiment who died in the Battle of Maiwand in 1880. There are a number of other works of public art in Reading. The Blade , a fourteen-storey building completed in 2009, is 86 m (282 ft) tall and can be seen from the surrounding area. Jacksons Corner with its prominent sign, former home of Jacksons department store, occupies
9348-517: Is the unary numeral system . It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers. Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks . Egyptian hieroglyphics had
9512-462: Is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers. Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an addition table or a multiplication table . Other common methods are verbal counting and finger-counting . For operations on numbers with more than one digit, different techniques can be employed to calculate
9676-404: Is the following: The rods are used in teaching a variety of mathematical concepts, and with a wide age range of learners. Topics they are used for include: Though primarily used for mathematics, they have also become popular in language-teaching classrooms, particularly The Silent Way . They can be used: In her first school, and in schools since then, Maria Montessori used coloured rods in
9840-428: Is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in 13 + 4 − 4 = 13 {\displaystyle 13+4-4=13} . Defined more formally, the operation " ⋆ {\displaystyle \star } "
10004-413: Is the inverse of exponentiation. The logarithm of a number x {\displaystyle x} to the base b {\displaystyle b} is the exponent to which b {\displaystyle b} must be raised to produce x {\displaystyle x} . For instance, since 1000 = 10 3 {\displaystyle 1000=10^{3}} ,
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#173279420336910168-580: Is the key concept. A patent is registered in Pollock's name for an "Apparatus for teaching or studying mathematics". The aesthetic and numerically comprehensive Color Factor system was marketed for some years by Seton Pollock's family, before being conveyed to the educational publishing house Edward Arnold . The colors of Pollock's system were named distinctively using, for example, "scarlet" instead of "red", and "amber" instead of "orange". They are listed below. (in centimetres) Arithmetic Arithmetic
10332-626: The Readingas , an Anglo-Saxon tribe whose name means Reada's People in Old English (the Anglo-Saxons often had the same name for a place and its inhabitants). In late 870, an army of Danes invaded the kingdom of Wessex and set up camp at Reading. On 4 January 871, in the first Battle of Reading , King Ethelred and his brother Alfred the Great attempted unsuccessfully to breach
10496-587: The 2024 general election , the borough of Reading has been divided between the parliamentary constituencies of Reading Central , Reading West and Mid Berkshire (which also covers part of West Berkshire), and Earley and Woodley (which also covers part of the borough of Wokingham). Reading is the site of venues for both the Crown Court , administering criminal justice, and the County Court , responsible for civil cases. Lesser matters are dealt with in
10660-585: The Borough of Reading , although some outer suburbs are parts of neighbouring local authority areas. Located in the Thames Valley at the confluence of the rivers Thames and Kennet , Reading is 40 miles (64 km) east of Swindon , 28 miles (45 km) south of Oxford , 40 miles (64 km) west of London and 16 miles (26 km) north of Basingstoke . Reading is a major commercial centre, especially for information technology and insurance. It
10824-555: The English Civil War , with a major siege and loss of trade, but played a pivotal role in the Glorious Revolution , whose only significant military action was fought on its streets. The 18th century saw the beginning of a major ironworks in the town and the growth of the brewing trade for which Reading was to become famous. The 19th century saw the coming of the Great Western Railway and the development of
10988-591: The Hindu–Arabic numeral system , the radix is 10. This means that the first digit is multiplied by 10 0 {\displaystyle 10^{0}} , the next digit is multiplied by 10 1 {\displaystyle 10^{1}} , and so on. For example, the decimal numeral 532 stands for 5 ⋅ 10 2 + 3 ⋅ 10 1 + 2 ⋅ 10 0 {\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}} . Because of
11152-541: The M4 motorway , but it was ultimately decided to leave Reading's boundaries unchanged. Reading's boundaries south of the Thames therefore have not changed since 1911, despite the urban area having now expanded well beyond the borough boundaries. Cross-boundary working between the borough council and the neighbouring councils which cover the suburban and adjoining rural areas is sometimes criticised, particularly over matters such as transport and school catchment areas. Prior to
11316-641: The South Eastern Railway in 1849 and the London and South Western Railway in 1856. The Summer Assizes were moved from Abingdon to Reading in 1867, effectively making Reading the sole county town of Berkshire, a decision that was officially approved by the privy council in 1869. The town became county borough under the Local Government Act 1888 . In the 19th and 20th centuries, the town's three largest industries were known as
11480-552: The Three Bs : beer (1785–2010, H & G Simonds ), bulbs (1837–1974, Suttons Seeds ), and biscuits (1822–1976, Huntley and Palmers ). The town continued to expand in the 20th century, annexing Caversham across the River Thames in Oxfordshire in 1911, as well as most of Tilehurst to the west at the same time. Compared to many other English towns and cities, Reading suffered little physical damage during either of
11644-584: The Whiteknights Campus , which has recorded atmospheric measurements and meteorological observations since 1970. The local absolute maximum temperature of 37.6 °C (99.7 °F) was recorded on 19 July 2022 and the local absolute minimum temperature of −14.5 °C (5.9 °F) was recorded in January 1982. In mid-2018, the area covered by the Borough of Reading had 174,820 inhabitants and
11808-428: The absolute uncertainties of each summand together to obtain the absolute uncertainty of the sum. When multiplying or dividing two or more quantities, add the relative uncertainties of each factor together to obtain the relative uncertainty of the product. When representing uncertainty by significant digits, uncertainty can be coarsely propagated by rounding the result of adding or subtracting two or more quantities to
11972-536: The central branch of the library was relocated to a new building on King's Road in 1985. The Reading Museum opened in 1883 in the town's municipal buildings . It contains galleries relating to the history of Reading and to the excavations of Calleva Atrebatum , together with a full-size bowdlerised replica of the Bayeux Tapestry , an art collection, and galleries relating to Huntley and Palmers . The Museum of English Rural Life , in East Reading ,
12136-425: The dissolution of the abbey in 1538 the borough was granted a new charter in 1542. The borough boundaries were then set out in a subsequent charter from Elizabeth I in 1560. The borough covered the whole of the parish of St Laurence and parts of the parishes of St Giles and St Mary . The part of St Giles' parish outside the borough was known as the hamlet of Whitley , and the part of St Mary's parish outside
12300-494: The fundamental theorem of arithmetic , Euclid's theorem , and Fermat's last theorem . According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the number 18 is not a prime number and can be represented as 2 × 3 × 3 {\displaystyle 2\times 3\times 3} , all of which are prime numbers. The number 19 , by contrast,
12464-589: The lattice method . Computer science is interested in multiplication algorithms with a low computational complexity to be able to efficiently multiply very large integers, such as the Karatsuba algorithm , the Schönhage–Strassen algorithm , and the Toom–Cook algorithm . A common technique used for division is called long division . Other methods include short division and chunking . Integer arithmetic
12628-431: The quotient . The symbols of division are ÷ {\displaystyle \div } and / {\displaystyle /} . Examples are 48 ÷ 8 = 6 {\displaystyle 48\div 8=6} and 29.4 / 1.4 = 21 {\displaystyle 29.4/1.4=21} . Division is often treated as a special case of multiplication: instead of dividing by
12792-568: The remainder . For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions. A simple method to calculate exponentiation is by repeated multiplication. For instance, the exponentiation of 3 4 {\displaystyle 3^{4}} can be calculated as 3 × 3 × 3 × 3 {\displaystyle 3\times 3\times 3\times 3} . A more efficient technique used for large exponents
12956-594: The 16th century, civic administration for the town of Reading was situated in the Yield Hall , a guild hall situated by the River Kennet near today's Yield Hall Lane. After a brief stay in what later became Greyfriars Church , the town council created a new town hall by inserting an upper floor into the refectory of the Hospitium of St John , the former hospitium of Reading Abbey . For some 400 years up to
13120-413: The 1970s, this was to remain the site of Reading's civic administration through the successive rebuilds that eventually created today's Town Hall . In 1976, Reading Borough Council moved to the new Civic Centre . In 2014, they moved again to civic offices in a refurbished existing office building on Bridge Street , in order to facilitate the demolition and redevelopment of the previous site. Reading
13284-537: The 1974 reforms came into effect, a more limited review of the borough's boundaries north of the Thames was carried out, which saw the Caversham Park area and part of the parish of Mapledurham on the western side of Caversham transferred into the borough of Reading in 1977. The borough council became a unitary authority in 1998, when the county council was abolished under the Banham Review , which saw
13448-452: The 64 operations required for regular repeated multiplication. Methods to calculate logarithms include the Taylor series and continued fractions . Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer. Number theory studies the structure and properties of integers as well as
13612-521: The 8th century. It was a trading and ecclesiastical centre in the Middle Ages , the site of Reading Abbey , one of the largest and richest monasteries of medieval England with royal connections, of which the 12th-century abbey gateway and significant ancient ruins remain. By 1525, Reading was the largest town in Berkshire, and tenth in England for taxable wealth. The town was seriously affected by
13776-825: The Bulmershe College of Higher Education , a teacher training college , in 1989, becoming Bulmershe Court Campus . The Henley Management College , situated in Buckinghamshire and about 10 miles (16 km) from Reading, was taken over in 2008, becoming Greenlands Campus . The University of West London maintains a presence in the town for its higher education students, principally in nursing , but has now divested itself of its previous ownership of Reading College and its further education students. The Reading Borough Libraries service dates back to 1877. Initially housed in Reading Town Hall ,
13940-985: The Cuisenaire Company in Reading , England, in 1954, and by the end of the 1950s, Cuisenaire rods had been adopted by teachers in 10,000 schools in more than a hundred countries. The rods received wide use in the 1960s and 1970s. In 2000, the United States-based company Educational Teaching Aids (ETA) acquired the US Cuisenaire Company and formed ETA/Cuisenaire to sell Cuisenaire rods-related material. In 2004, Cuisenaire rods were featured in an exhibition of paintings and sculptures by New Zealand artist Michael Parekowhai . Another arrangement, common in Eastern Europe, extended by two large (> 10 cm or 4 in) sizes of rods,
14104-709: The Danes' defences. The battle is described in the Anglo-Saxon Chronicle , and that account provides the earliest known written record of the existence of Reading. The Danes remained in Reading until late in 871, when they retreated to their winter quarters in London . After the Battle of Hastings and the Norman conquest of England , William the Conqueror gave land in and around Reading to his foundation of Battle Abbey . In its 1086 Domesday Book listing,
14268-466: The Greater Reading area. Reading has over 100 parks and playgrounds, including 5 miles (8 km) of riverside paths. In the town centre is Forbury Gardens , a public park built on the site of the outer court of Reading Abbey . The largest public park in Reading is Prospect Park , an estate in west Reading previously owned by Frances Kendrick but acquired by Reading Corporation in 1901. This
14432-658: The Latin term " arithmetica " which derives from the Ancient Greek words ἀριθμός (arithmos), meaning "number", and ἀριθμητική τέχνη (arithmetike tekhne), meaning "the art of counting". There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with natural numbers . However, the more common view is to include operations on integers , rational numbers , real numbers , and sometimes also complex numbers in its scope. Some definitions restrict arithmetic to
14596-773: The Reading Electric Supply Company Limited, then from 1933 by the Reading Corporation until the nationalisation of the British electricity supply industry in 1948. The dialling code for fixed-line telephones in Reading is 0118. BT provides fixed-line telephone coverage throughout the town and ADSL broadband internet connection to most areas. Parts of Reading are cabled by Virgin Media , supplying cable television, telephone and broadband internet connections. Hyperoptic also has
14760-497: The River Thames. In 1312, King Edward II directed that its bridges should be kept in good order. It is not known how badly Reading was affected by the Black Death that swept through England in the 14th century, but it is known that the abbot, Henry of Appleford , was one of its victims in 1361, and that nearby Henley lost 60% of its population. The Abbey was largely destroyed in 1538 during Henry VIII's dissolution of
14924-547: The Thames from Oxfordshire (except the Caversham Park area, which was transferred to the parish of Eye and Dunsden ), and most of the parish of Tilehurst (including the main village at Tilehurst Triangle and the area around the parish church at Churchend) to the west. Local government was reformed in 1974 under the Local Government Act 1972 , which saw Reading redesignated as a non-metropolitan district , with Berkshire County Council providing county-level services in
15088-538: The United Kingdom offices of foreign multinationals, as well as being a major retail centre. Whilst located close enough to London to be sometimes regarded as part of the London commuter belt , Reading is a net inward destination for commuters. During the morning peak period, there are some 30,000 inward arrivals in the town, compared to 24,000 departures. Major companies Microsoft , Oracle and Hibu (formerly Yell Group) have their headquarters in Reading. The insurance company Prudential has an administration centre in
15252-799: The Warren. Reading has a local newspaper, the Reading Chronicle , published on Thursdays. The town's other local newspaper, the Reading Post , ceased publication on paper in December 2014, in order to transition to an online only format under the title getreading . As of 2018, getreading joined the InYourArea local news network. A local publishing company, the Two Rivers Press , has published over 70 book titles, many on
15416-457: The accuracy and speed with which arithmetic calculations could be performed. Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition , subtraction , multiplication , and division . In a wider sense, it also includes exponentiation , extraction of roots , and logarithm . The term "arithmetic" has its root in
15580-435: The addends, are combined into a single number, called the sum. The symbol of addition is + {\displaystyle +} . Examples are 2 + 2 = 4 {\displaystyle 2+2=4} and 6.3 + 1.26 = 7.56 {\displaystyle 6.3+1.26=7.56} . The term summation is used if several additions are performed in a row. Counting is a type of repeated addition in which
15744-510: The base can be understood from context. So, the previous example can be written log 10 1000 = 3 {\displaystyle \log _{10}1000=3} . Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in 14 1 = 14 {\displaystyle 14^{1}=14} . However, exponentiation does not have
15908-401: The basic numerals 0 and 1. Computer arithmetic deals with the specificities of the implementation of binary arithmetic on computers . Some arithmetic systems operate on mathematical objects other than numbers, such as interval arithmetic and matrix arithmetic. Arithmetic operations form the basis of many branches of mathematics, such as algebra , calculus , and statistics . They play
16072-492: The borough council take over county-level functions, effectively restoring the council to the powers it had held when Reading was a county borough prior to 1974. As part of those reforms, the Local Government Commission had initially recommended expanding Reading's boundaries to include Earley, Tilehurst parish, Purley on Thames and the parts of the parishes of Shinfield , Burghfield and Theale north of
16236-539: The borough for the first time. Ahead of those reforms, the borough council campaigned to have Reading's boundaries enlarged to take in Earley, Woodley , Purley on Thames , the residual Tilehurst parish (covering the parts of Tilehurst which had not been transferred into the borough in 1911), and the eastern part of the parish of Theale . The government decided to make no change to Reading's boundaries, leaving them as they had been since last reviewed in 1911. Shortly after
16400-546: The borough was known as the tithing of Southcote . The borough was reformed in 1836 to become a municipal borough under the Municipal Corporations Act 1835 , which standardised how most boroughs operated across the country. The borough boundaries, which had not been changed since 1560, were enlarged in 1887 to take in Southcote, Whitley, the north-western parts of Earley , and the eastern end of
16564-532: The centre of Reading, was embanked as part of the construction of the Kennet and Avon Canal in the 18th century, allowing the development of wharves . The floodplains adjoining Reading's two rivers are subject to occasional flooding. As Reading has grown, its suburbs have spread: to the west between the two rivers into the foothills of the Berkshire Downs as far as Calcot , Tilehurst and Purley ; to
16728-440: The claim that every even number is a sum of two prime numbers . Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of fields and rings , as in algebraic number fields like the ring of integers . Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in
16892-437: The classroom to teach concepts of both mathematics and length. This is possibly the first instance of coloured rods being used in the classroom for this purpose. Catherine Stern also devised a set of coloured rods produced by staining wood with aesthetically pleasing colours, and published books on their use at around the same time as Cuisenaire and Gattegno. Her rods were different colours to Cuisenaire's, and also larger, with
17056-502: The concept of zero and the decimal system , which Arab mathematicians further refined and spread to the Western world during the medieval period. The first mechanical calculators were invented in the 17th century. The 18th and 19th centuries saw the development of modern number theory and the formulation of axiomatic foundations of arithmetic. In the 20th century, the emergence of electronic calculators and computers revolutionized
17220-544: The corner of Kings Road and High Street, just south of the Market Place. Reading has six Grade I listed buildings, 22 Grade II* and 853 Grade II buildings, in a wide variety of architectural styles that range from the medieval to the 21st century. The Grade I listed buildings are Reading Abbey , the Abbey Gateway , Greyfriars Church , St Laurence's Church , Reading Minster , and the barn at Chazey Farmhouse on
17384-488: The decimal fraction notation. Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions. Not all rational numbers have a finite representation in the decimal notation. For example, the rational number 1 3 {\displaystyle {\tfrac {1}{3}}} corresponds to 0.333... with an infinite number of 3s. The shortened notation for this type of repeating decimal
17548-522: The degree of certainty about each number's value and avoid false precision is to round each measurement to a certain number of digits, called significant digits , which are implied to be accurate. For example, a person's height measured with a tape measure might only be precisely known to the nearest centimeter, so should be presented as 1.62 meters rather than 1.6217 meters. If converted to imperial units, this quantity should be rounded to 64 inches or 63.8 inches rather than 63.7795 inches, to clearly convey
17712-576: The desired level of accuracy. The Taylor series or the continued fraction method can be utilized to calculate logarithms. The decimal fraction notation is a special way of representing rational numbers whose denominator is a power of 10. For instance, the rational numbers 1 10 {\displaystyle {\tfrac {1}{10}}} , 371 100 {\displaystyle {\tfrac {371}{100}}} , and 44 10000 {\displaystyle {\tfrac {44}{10000}}} are written as 0.1, 3.71, and 0.0044 in
17876-652: The distinction between the natural and the whole numbers by including 0 in the set of natural numbers. The set of integers encompasses both positive and negative whole numbers. It has the symbol Z {\displaystyle \mathbb {Z} } and can be expressed as { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } {\displaystyle \{...,-2,-1,0,1,2,...\}} . Based on how natural and whole numbers are used, they can be distinguished into cardinal and ordinal numbers . Cardinal numbers, like one, two, and three, are numbers that express
18040-475: The effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits. Another positional numeral system used extensively in computer arithmetic is the binary system , which has a radix of 2. This means that the first digit is multiplied by 2 0 {\displaystyle 2^{0}} , the next digit by 2 1 {\displaystyle 2^{1}} , and so on. For example,
18204-533: The execution of Charles Wooldridge , carried out in Reading Gaol whilst he was imprisoned there. In March 2021, street artist Banksy claimed responsibility for a painting on the wall of the jail. It depicted an inmate escaping with bedsheets and a typewriter, said to resemble Oscar Wilde . Reading was the location of the world's first commercial studio for photograph printing, which was set up by William Henry Fox Talbot in 1844. Ricky Gervais , who
18368-408: The exponent is a natural number then exponentiation is the same as repeated multiplication, as in 2 4 = 2 × 2 × 2 × 2 {\displaystyle 2^{4}=2\times 2\times 2\times 2} . Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to
18532-458: The exponent. For example, 5 2 3 = 5 2 3 {\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}} . The first operation can be completed using methods like repeated multiplication or exponentiation by squaring. One way to get an approximate result for the second operation is to employ Newton's method , which uses a series of steps to gradually refine an initial guess until it reaches
18696-421: The field of numerical calculations. When understood in a wider sense, it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations. Arithmetic is closely related to number theory and some authors use the terms as synonyms. However, in a more specific sense, number theory
18860-483: The first number with the reciprocal of the second number. This means that the numerator and the denominator of the second number change position. For example, 3 5 : 2 7 = 3 5 ⋅ 7 2 = 21 10 {\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}} . Unlike integer arithmetic, rational number arithmetic
19024-414: The fortune made by local merchant John Kendrick . Reading played a role during the English Civil War . Despite its fortifications, it had a Royalist garrison imposed on it in 1642. The subsequent Siege of Reading by Parliamentary forces succeeded in April 1643. The town's cloth trade was especially badly damaged, and the town's economy did not fully recover until the 20th century. Reading played
19188-502: The integer 1, called the numerator, by the integer 2, called the denominator. Other examples are 3 4 {\displaystyle {\tfrac {3}{4}}} and 281 3 {\displaystyle {\tfrac {281}{3}}} . The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is Q {\displaystyle \mathbb {Q} } . Decimal fractions like 0.3 and 25.12 are
19352-703: The largest urban areas in the United Kingdom to be without city status , Reading has unsuccessfully bid for city status on four recent occasions – in 2000 to celebrate the new millennium; in 2002 to celebrate the Golden Jubilee of Queen Elizabeth II ; in 2012 for the Diamond Jubilee ; and in 2022 to mark the Platinum Jubilee . Local government for the borough is provided by Reading Borough Council , which has been
19516-493: The left. This process is repeated until all digits have been added. Other methods used for integer additions are the number line method, the partial sum method, and the compensation method. A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative. A basic technique of integer multiplication employs repeated addition. For example,
19680-458: The leftmost last significant decimal place among the summands, and by rounding the result of multiplying or dividing two or more quantities to the least number of significant digits among the factors. (See Significant figures § Arithmetic .) More sophisticated methods of dealing with uncertain values include interval arithmetic and affine arithmetic . Interval arithmetic describes operations on intervals . Intervals can be used to represent
19844-492: The logarithm base 10 of 1000 is 3. The logarithm of x {\displaystyle x} to base b {\displaystyle b} is denoted as log b ( x ) {\displaystyle \log _{b}(x)} , or without parentheses, log b x {\displaystyle \log _{b}x} , or even without the explicit base, log x {\displaystyle \log x} , when
20008-626: The major coaching routes from London to Oxford and the West Country . In 1723, despite considerable local opposition, the Kennet Navigation opened the River Kennet to boats as far as Newbury . Opposition stopped when it became apparent that the new route benefited the town. After the opening of the Kennet and Avon Canal in 1810, one could go by barge from Reading to the Bristol Channel . From 1714, and probably earlier,
20172-560: The mid-1980s. Reading College has provided further education in Reading since 1955, with over 8,500 local learners on over 900 courses. English language schools in Reading include Gateway Languages, the English Language Centre, ELC London Street and Eurospeak Language School . The University of Reading was established in 1892 as an affiliate of Oxford University . It moved to its London Road Campus in 1904 and to its new Whiteknights Campus in 1947. It took over
20336-484: The monasteries . The last abbot , Hugh Faringdon , was subsequently tried and convicted of high treason and hanged, drawn and quartered in front of the Abbey Church. By 1525, Reading was the largest town in Berkshire and the tenth largest town in England when measured by taxable wealth reported in tax returns. By 1611, it had a population of over 5,000 and had grown rich on its trade in cloth, as instanced by
20500-407: The multiplicand is a natural number then multiplication is the same as repeated addition, as in 2 × 3 = 2 + 2 + 2 {\displaystyle 2\times 3=2+2+2} . Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called
20664-433: The multiplicand, are combined into a single number called the product . The symbols of multiplication are × {\displaystyle \times } , ⋅ {\displaystyle \cdot } , and *. Examples are 2 × 3 = 6 {\displaystyle 2\times 3=6} and 0.3 ⋅ 5 = 1.5 {\displaystyle 0.3\cdot 5=1.5} . If
20828-516: The name of the local rugby team Redingensians , based in Sonning , and of former members of Reading School . Jane Austen attended Reading Ladies Boarding School , based in the Abbey Gateway , in 1784–1786. Mary Russell Mitford lived in Reading for a number of years and then spent the rest of her life just outside the town at Three Mile Cross and Swallowfield . The fictional Belford Regis of her eponymous novel, first published in 1835,
20992-484: The natural numbers is N {\displaystyle \mathbb {N} } . The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as { 0 , 1 , 2 , 3 , 4 , . . . } {\displaystyle \{0,1,2,3,4,...\}} and have the symbol N 0 {\displaystyle \mathbb {N} _{0}} . Some mathematicians do not draw
21156-512: The number 1 is continuously added. Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is − {\displaystyle -} . Examples are 14 − 8 = 6 {\displaystyle 14-8=6} and 45 − 1.7 = 43.3 {\displaystyle 45-1.7=43.3} . Subtraction
21320-430: The number 13 is written as 1101 in the binary notation, which stands for 1 ⋅ 2 3 + 1 ⋅ 2 2 + 0 ⋅ 2 1 + 1 ⋅ 2 0 {\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}} . In computing, each digit in the binary notation corresponds to one bit . The earliest positional system
21484-655: The other four McIlroy Park , Blundells Copse , Lousehill Copse and Round Copse all in Tilehurst The principal National Health Service (NHS) hospital in Reading is the Royal Berkshire Hospital , founded in 1839 and much enlarged and rebuilt since. A second major NHS general hospital , the Battle Hospital , closed in 2005. Berkshire Healthcare NHS Foundation Trust runs a NHS hospital, Prospect Park Hospital, which specialises in
21648-409: The parish of Tilehurst . When elected county councils were established in 1889 under the Local Government Act 1888 , Reading was considered large enough for its existing borough council to provide county-level services, and so Reading was made a county borough , independent from Berkshire County Council . The borough boundaries were enlarged again in 1911 to take in Caversham on the north bank of
21812-547: The power of 1 2 {\displaystyle {\tfrac {1}{2}}} and the cube root of a number is the same as raising the number to the power of 1 3 {\displaystyle {\tfrac {1}{3}}} . Examples are 4 = 4 1 2 = 2 {\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2} and 27 3 = 27 1 3 = 3 {\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3} . Logarithm
21976-418: The precision of the measurement. When a number is written using ordinary decimal notation, leading zeros are not significant, and trailing zeros of numbers not written with a decimal point are implicitly considered to be non-significant. For example, the numbers 0.056 and 1200 each have only 2 significant digits, but the number 40.00 has 4 significant digits. Representing uncertainty using only significant digits
22140-488: The presence of a 2 megawatt (peak) Enercon wind turbine at Green Park Business Park , with the potential to produce 2.7 million kWh of electricity a year, enough to power over a thousand homes. Additionally, Reading Hydro runs a micro hydroelectric power station on the Thames. Reading had its own power station in Vastern Road from 1895 to the 1960s. The power station was initially owned and operated by
22304-414: The product of 3 × 4 {\displaystyle 3\times 4} can be calculated as 3 + 3 + 3 + 3 {\displaystyle 3+3+3+3} . A common technique for multiplication with larger numbers is called long multiplication . This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with
22468-772: The provision of care for people with mental health and learning disabilities. Reading has three private hospitals: the Berkshire Independent Hospital in Coley Park , the Dunedin Hospital situated on the main A4 Bath Road, and the Circle Hospital at Kennet Island. Mains water and sewerage services are provided by Thames Water Utilities Limited , a private sector water supply company, whilst water abstraction and disposal
22632-411: The quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?". A number is rational if it can be represented as the ratio of two integers. For instance, the rational number 1 2 {\displaystyle {\tfrac {1}{2}}} is formed by dividing
22796-441: The ratio of two integers. They are often required to describe geometric magnitudes. For example, if a right triangle has legs of the length 1 then the length of its hypotenuse is given by the irrational number 2 {\displaystyle {\sqrt {2}}} . π is another irrational number and describes the ratio of a circle 's circumference to its diameter . The decimal representation of an irrational number
22960-534: The relations and laws between them. Some of the main branches of modern number theory include elementary number theory , analytic number theory , algebraic number theory , and geometric number theory . Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include divisibility , factorization , and primality . Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like how prime numbers are distributed and
23124-404: The result by using several one-digit operations in a row. For example, in the method addition with carries , the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to
23288-406: The rightmost digit of the multiplicand and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplicand and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two multi-digit numbers. Other techniques used for multiplication are the grid method and
23452-426: The rods réglettes . According to Gattegno, "Georges Cuisenaire showed in the early 1950s that pupils who had been taught traditionally, and were rated 'weak', took huge strides when they shifted to using the material. They became 'very good' at traditional arithmetic when they were allowed to manipulate the rods." The educationalists Maria Montessori and Friedrich Fröbel had used rods to represent numbers, but it
23616-523: The rods allowed pupils "to expand on their latent mathematical abilities in a creative and enjoyable fashion", Gattegno's pedagogy shifted radically as he began to stand back and allow pupils to take a leading role: Cuisenaire's gift of the rods led me to teach by non-interference making it necessary to watch and listen for the signs of truth that are made, but rarely recognized. While the material has found an important place in myriad teacher-centered lessons, Gattegno's student-centered practice also inspired
23780-737: The rods in Thuin. At this point he had already founded the International Commission for the Study and Improvement of Mathematics Education (CIEAEM) and the Association of Teachers of Mathematics , but this marked a turning point in his understanding: Then, Cuisenaire took us to a table in one corner of the room where pupils were standing round a pile of colored sticks and doing sums which seemed to me to be unusually hard for children of that age. At this sight, all other impressions of
23944-516: The role of county town of Berkshire was shared between Reading and Abingdon . In the eighteenth and nineteenth centuries it was one of the southern termini of the Hatfield and Reading Turnpike that allowed travellers from the north to continue their journey to the west without going through the congestion of London. During the 19th century, the town grew rapidly as a manufacturing centre. The Great Western Railway arrived in 1841, followed by
24108-418: The same denominator then they can be added by adding their numerators and keeping the common denominator. For example, 2 7 + 3 7 = 5 7 {\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}} . A similar procedure is used for subtraction. If the two numbers do not have the same denominator then they must be transformed to find
24272-522: The south and south-east on the south side of the River Kennet as far as Whitley Wood and Lower Earley and as far north of the Thames into the Chiltern Hills as far as Caversham Heights , Emmer Green and Caversham Park Village . Outside the central area, the floors of the valleys containing the two rivers remain largely unimproved floodplain . Apart from the M4 curving to the south there
24436-545: The story of Reading's two rivers. The Museum of Berkshire Aviation has a collection of aircraft and other artefacts relating to the aircraft industry in the town. Reading's location in the Thames Valley to the west of London has made the town a significant element in the nation's transport system. The town grew up as a river port at the confluence of the River Thames and the River Kennet . Both of these rivers are navigable, and Caversham Lock , Blake's Lock , County Lock , Fobney Lock and Southcote Lock are all within
24600-415: The surrounding vanished, to be replaced by a growing excitement. After listening to Cuisenaire asking his first and second grade pupils questions and hearing their answers immediately and with complete self-assurance and accuracy, the excitement then turned into irrepressible enthusiasm and a sense of illumination. Gattegno named the rods "Cuisenaire rods" and began trialing and popularizing them. Seeing that
24764-437: The symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000. A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called
24928-461: The teacher teaches you all this', to which the little girl replied: 'The teacher teaches us nothing. We find everything out for ourselves.' John Holt , in his 1964 How Children Fail , wrote: This work has changed most of my ideas about the way to use Cuisenaire rods and other materials. It seemed to me at first that we could use them as devices for packing in recipes much faster than before, and many teachers seem to be using them this way. But this
25092-426: The topic of local history and art. Three local radio stations broadcast from Reading: BBC Radio Berkshire , Heart South and Greatest Hits Radio Berkshire and North Hampshire . Local news and television programmes are provided by BBC South and ITV Meridian , BBC London & ITV London can also be received. Reading has one local television station, That's Thames Valley , which broadcasts local news throughout
25256-585: The town was explicitly described as a borough . The presence of six mills is recorded: four on land belonging to the king and two on the land given to Battle Abbey. Reading Abbey was founded in 1121 by Henry I , who is buried within the Abbey grounds. As part of his endowments, he gave the abbey his lands in Reading, along with land at Cholsey . The town grew around a crossing of the River Kennet, about 1 mile (1.6 km) upstream from its confluence with
25420-460: The town's brewing, baking and seed growing businesses, and the town grew rapidly as a manufacturing centre. Reading is also the county town of Berkshire. Occupation at the site of Reading may date back to the Roman period , possibly in the form of a trading port for Calleva Atrebatum . However, the first clear evidence for Reading as a settlement dates from the 8th century, when the town came to be known as Readingas . The name probably comes from
25584-422: The town. PepsiCo and Wrigley have offices. Global pharmaceutical giant Bayer Life Sciences relocated to Reading's Green Park Business Park in 2016. Reading has a significant historical involvement in the information technology industry, largely as a result of the early presence in the town of sites of International Computers Limited and Digital Equipment Corporation . Other technology companies with
25748-476: The two world wars that afflicted the 20th century, although many citizens were killed or injured in the conflicts. One significant air raid occurred on 10 February 1943, when a single Luftwaffe plane machine-gunned and bombed the town centre, resulting in 41 deaths and over 100 injuries. The Lower Earley development, begun in 1977, was one of the largest private housing developments in Europe . It extended
25912-511: The urban area of Reading as far as the M4 Motorway . Further housing developments have increased the number of modern houses and hypermarkets in the outskirts of Reading. A major town-centre shopping centre, The Oracle , opened in 1999, is named after the 17th-century Oracle workhouse , which once occupied a small part of the site. It provides three storeys of shopping space and boosted the local economy by providing 4,000 jobs. As one of
26076-402: The work of Cuisenaire and Gattegno, he had invented a unified system for logically assigning a color to any number. After white (1), the primary colors red, blue and yellow are assigned to the first three primes (2, 3 and 5). Higher primes (7, 11 etc.) are associated with darkening shades of grey. The colors of non-prime numbers are obtained by mixing the colors associated with their factors – this
26240-621: Was Georges Cuisenaire who introduced the rods that were to be used across the world from the 1950s onwards. In 1952, he published Les nombres en couleurs , Numbers in Color, which outlined their use. Cuisenaire, a violin player, taught music as well as arithmetic in the primary school in Thuin . He wondered why children found it easy and enjoyable to pick up a tune and yet found mathematics neither easy nor enjoyable. These comparisons with music and its representation led Cuisenaire to experiment in 1931 with
26404-665: Was also known for its WOMAD Festival until it moved to Charlton Park in Malmesbury , Wiltshire . The Reading Beer Festival was first held in 1994 and has now grown to one of the largest beer festivals in the United Kingdom. It is held at King's Meadow for the five days immediately preceding the May Day bank holiday every year. Reading also holds Reading Pride , an annual LGBT festival in Kings Meadow. The Frank Matcham -designed Royal County Theatre, built in 1895,
26568-735: Was developed by ancient Babylonians and had a radix of 60. Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output. The most important operations in arithmetic are addition , subtraction , multiplication , and division . Further operations include exponentiation , extraction of roots , and logarithm . If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations . Two important concepts in relation to arithmetic operations are identity elements and inverse elements . The identity element or neutral element of an operation does not cause any change if it
26732-416: Was located on the south side of Friar Street. It burned down in 1937. Within the town hall is a 700-seat concert hall that houses a Father Willis organ. Reading theatre venues include The Hexagon and South Street Arts Centre. Reading Repertory Theatre is based at Reading College : its Royal Patron is Prince Edward, Duke of Edinburgh . Amateur theatre venues in Reading include Progress Theatre ,
26896-540: Was reported that Reading had 150 different spoken languages within its population. Reading has a large Polish community, which dates back over 30 years, and in October 2006 the Reading Chronicle printed 5,000 copies of a Polish edition called the Kronika Reading . Reading is a commercial centre in the Thames Valley and Southern England . The town hosts the headquarters of several British companies and
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