Arithmetic tables for children, Lausanne, 1835

Arithmetic or arithmetics (from the Greek word ἀριθμός, arithmos “number”) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. Professional mathematicians sometimes use the term (higher) arithmetic^[1] when referring to more advanced results related to number theory, but this should not be confused with elementary arithmetic.

History[link]

The prehistory of arithmetic is limited to a small number of artifacts which may indicate conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC although its interpretation is disputed.^[2]

The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board or the Roman abacus to obtain the results.

Early number systems that included positional notation were not decimal, including the sexagesimal (base 60) system for Babylonian numerals and the vigesimal(base 20) system that defined Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation.

The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic.

Greek numerals, derived from the hieratic Egyptian system, also lacked positional notation, and therefore imposed the same complexity on the basic operations of arithmetic. For example, the ancient mathematician Archimedes devoted his entire work The Sand Reckoner merely to devising a notation for a certain large integer.

The gradual development of Hindu-Arabic numerals independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing zero. This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta established the use of zero as a separate number and determined the results for multiplication, division, addition and subtraction of zero and all other numbers, except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht described the excellence of this system as "...valuable methods of calculation which surpass description". The Arabs also learned this new method and called it hesab.

Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations.

Although the Codex Vigilanus described an early form of Arabic numerals (omitting zero) by 976 AD, Fibonacci was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He considered the significance of this "new" representation of numbers, which he styled the "Method of the Indians" (Latin Modus Indorum), so fundamental that all related mathematical foundations, including the results of Pythagoras and the algorism describing the methods for performing actual calculations, were "almost a mistake" in comparison.

In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.

Various types of tools exist to assist in numeric calculations. Examples include slide rules (for multiplication, division, and trigonometry) and nomographs in addition to the electrical calculator.

Arithmetic operations[link]

The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions. Arithmetic is performed according to an order of operations. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws, is called a field.

Addition (+)[link]

Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers.

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.

Addition is commutative and associative so the order the terms are added in does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number yields that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself yields the additive identity, 0. For example, the opposite of 7 is −7, so 7 + (−7) = 0.

Addition can be given geometrically as in the following example:

If we have two sticks of lengths 2 and 5, then if we place the sticks one after the other, the length of the stick thus formed is 2 + 5 = 7.

Subtraction (−)[link]

Subtraction is the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference is positive; if the minuend is smaller than the subtrahend, the difference is negative; if they are equal, the difference is zero.

Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.

There are several methods for calculating results, some of which are particularly advantageous to machine calculation. For example, digital computers employ the method of two's complement. Of great importance is the counting up method by which change is made. Suppose an amount P is given to pay the required amount Q, with P greater than Q. Rather than performing the subtraction P − Q and counting out that amount in change, money is counted out starting at Q and continuing until reaching P. Although the amount counted out must equal the result of the subtraction P − Q, the subtraction was never really done and the value of P − Q might still be unknown to the change-maker.

Multiplication (× or · or *)[link]

Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, sometimes both simply called factors.

Multiplication is best viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number, say Failed to parse (Missing texvc executable; please see math/README to configure.): x , greater than 1 is the same as stretching everything away from zero uniformly, in such a way that the number 1 itself is stretched to where Failed to parse (Missing texvc executable; please see math/README to configure.): x

was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards zero. (Again, in such a way that 1 goes to the multiplicand.)

Multiplication is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 yields that same number. Also, the multiplicative inverse is the reciprocal of any number (except zero; zero is the only number without a multiplicative inverse), that is, multiplying the reciprocal of any number by the number itself yields the multiplicative identity.

The product of a and b is written as a × b or a • b. When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab. In computer programming languages and software packages in which one can only use characters normally found on a keyboard, it is often written with an asterisk: a * b.

Division (÷ or /)[link]

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient is greater than one, otherwise it is less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × ¹/_b. When written as a product, it obeys all the properties of multiplication.

Decimal arithmetic[link]

Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal)" positional notation; however, any numeral system based on powers of ten, e.g., Greek, Cyrillic, roman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".

Modern methods for four fundamental operations (addition, subtraction. multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indoram" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", “hundredths place"). For example, 507.36 denotes 5 hundreds (10²), plus 0 tens (10¹), plus 7 units (10⁰), plus 3 tenths (10⁻¹) plus 6 hundredths (10⁻²).

Zero as a number comparable to the other basic digits is a concept that is essential to this notation, as is the concept of zero’s use as a placeholder, and as is the definition of multiplication and addition with zero. The use of zero as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci^[3] using the Hindu-Arabic numeral system.

Algorism comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value nine, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one. This adjustment is termed a carry of the value one.

The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds nine, the carry adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from one to eight (9 × 9 = 81). Additional steps define the final result.

Similar techniques exist for subtraction and division.

The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the exponent for the radix (base) of ten increases by one (to the left) or decreases by one (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10ⁿ with integer n. The list of values corresponding to all possible positions for a single digit is written as {..., 10², 10, 1, 10⁻¹, 10⁻², ...}.

Repeated multiplication of any value in this list by ten produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of number theory.

Compound Unit Arithmetic[link]

Compound^[4] Unit Arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches, gallons and pints, pounds shillings and pence and so on. Prior to the use of decimal-based systems of money and units of measure, the use of compound unit arithmetic formed a significant part of commerce and industry.

Basic arithmetic operations[link]

The techniques used for compound unit arithmetic was developed over many centuries and are well documented in many textbooks in many different languages.^{[5][6][7][8][9]} In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions:

• Reduction where a compound quantity is reduced to a single quantity, for example conversion of a distance expressed in yards, feet and inches to one expressed in inches.^[10]
• Expansion, the inverse function to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to 1 lb, 8 oz.
• Normalisation is the conversion of a set of compound units to a standard form - for example rewriting "1 ft 13 in" as "2 ft 1 in".

Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic.

Principles of compound unit arithmetic[link]

There are two basic approaches to compound unit arithmetic:

• Reduction-Expansion Method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach suited for automated calculations. A typical example is the handling of time by Microsoft Excel where all time intervals are processed internally as days and decimal fractions of a day.
• On going Normalisation Method in which each unit is treated separately and the problem is continuously normalised as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations.

An example of the on-going normalisation method as applied to addition is shown below.

The addition operation is carried out from right to left; in this case, pence are processed first, then shillings followed by pounds. The numbers below the "answer line" are intermediate results.

The total in the pence column is 25. Since there are 12 pennies in a shilling, the number 24 is divided by 12 to give 2 with a remainder of 1. The value "1" is then written to the answer row and the value "2" carried forward to the shillings column. This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column. The intermediate total is divided by 20 rather than 12 as there are 20 shillings in a pound. The pound column is then processed, but as pounds are the largest unit that is being considered, no values are carried forward from the pounds column.

It should be noted that for the sake of simplicity, the example chosen did not have farthings.

Operations in practice[link]

A scale calibrated in imperial units with an associated cost display.

During the nineteenth and twentieth century various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings and "Ready Reckoners" - books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet^[11] that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".

The cumbersome nature of compound unit arithmetic has been recognised for many years - in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth").^[12] in which he declared that the universal introduction of decimal coinage, measures, and weights to be merely a question of time while in the modern era, many conversion programs, such as that embedded in the calculator supplied as a standard part of the Microsoft Windows 7 operating system display compound units in a reduced decimal format rather than using an expanded format (ie "2.5 ft" is displayed rather than "2 ft, 6  in").

Number theory[link]

The term arithmetic also refers to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations in integers, as well as modern research that is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Jean-Pierre Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry. Number theory is also referred to as the higher arithmetic, as in the title of Harold Davenport's book on the subject.

Arithmetic in education[link]

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (using the decimal place-value system). This study is sometimes known as algorism.

The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.^[13]

See also[link]

• Lists of mathematics topics
• Mathematics
• Outline of arithmetic

Related topics[link]

Notes[link]

References[link]

External links[link]

Look up arithmetic in Wiktionary, the free dictionary.

Wikibooks has more on the topic of Arithmetic

Wikimedia Commons has media related to: Arithmetic

• What is arithmetic?
• MathWorld article about arithmetic
• Interactive Arithmetic Lessons and Practice
• Talking Math Game for kids
• The New Student's Reference Work/Arithmetic (historical)
• Arithmetic Game
• Math Games for kids and adults
• The Great Calculation According to the Indians, of Maximus Planudes – an early Western work on arithmetic at Convergence

Brooke Fraser
Background information
Birth name	Brooke Gabrielle Fraser
Born	(1983-12-15) 15 December 1983 (age 28) Wellington, New Zealand
Origin	Wellington, New Zealand
Genres	Pop, folk rock
Years active	2002–present
Labels	Sony Music, Wood + Bone
Associated acts	Hillsong
Website	www.brookefraser.com

Brooke Gabrielle Fraser Ligertwood, better known as Brooke Fraser (born 15 December 1983) is a New Zealand born award-winning folk-pop and Christian music^[1][2] artist. Her third studio album Flags has become the most successful album of her career, with the single "Something in the Water" reaching number one on the RIANZ singles chart in 2010.

Early life[link]

Fraser is the eldest of the three children born to former All Black Bernie Fraser,^[3] who was born in Lautoka, Fiji, and his wife Lynda Fraser.^[4] She grew up in Naenae, Lower Hutt and attended Dyer Street School, Naenae Intermediate School and Naenae College.^[5]

Fraser took piano lessons between the ages of seven and seventeen. She started writing songs at age twelve and taught herself the acoustic guitar at fifteen. Despite her skills as a singer she has never taken singing lessons.^[5]

She performed at Parachute, an annual New Zealand music festival in 2000 – including a special guest performance in 2007.^[5]

She began writing for the Soul Purpose magazine at age fifteen, and was later made editor in 2002. She gave up her job as editor shortly after moving to Auckland in late 2002 in order to pursue her music career.^[6]

Musical career[link]

In 2002, while Fraser was in Auckland, Scotty Pearson, the drummer for Elemeno P, organised a meeting with producer Matty J for her. Matty J became her manager, liaising with the major labels who had shown interest and compared their offers. Fraser had about five different labels offer her deals and chose to sign a multi-album deal with Sony Music at the age of 17.^[7]

[edit] What to Do With Daylight

Fraser's first album, What to Do with Daylight, was released in New Zealand in late 2003, debuting at number one and achieved gold status in the same week. The album eventually went seven times platinum,^[8] selling over 105,000 copies in New Zealand alone.^[9] It remained on the album charts for sixty-six weeks.^[10] All five singles from the album reached the top twenty in the New Zealand Singles Chart.^[11] Her album also topped the New Zealand Top 50 Albums of 2004.^[12]

Following the release of What to Do with Daylight, Fraser toured Australia and New Zealand with American artist John Mayer and then toured New Zealand with veteran U.K. rock artist David Bowie.^[13][14] Whilst on tour with John Mayer, she met with his guitarist and keyboardist Michael Chaves who, after recording Mayer's album Heavier Things, Fraser enlisted to play on her album and future concerts.^[15]

[edit] Albertine

In 2005, prior to writing and preparing her follow-up album, Fraser took a trip to Africa to immerse herself in Rwanda before visiting her sponsor children in Tanzania, as she is an advocate for child sponsorship. In this trip, she was so moved that she wrote the song Albertine about a young child (named Albertine), whom she met while in Rwanda. It was later decided to make this song the album's title track.^[16]

Fraser performing in Seattle, 8 December 2010

For the second album, Fraser decided to enlist a new band, primarily constructed from American musicians who'd worked with an array of notable artists, both live and recorded.^[17] In 2006, Fraser and the band went into the studio in Los Angeles to record the album. Later Fraser allowed her fans to listen to the album's first single Deciphering Me via her MySpace page. The single was later released initially to radio and ultimately to CD single, and achieved number four in the New Zealand single charts.

On 4 December 2006 Albertine was released in New Zealand, achieving double platinum status less than a month after its release and has remained, to date, in the top 20 every week since. The album was released in Australia and internationally on 31 March 2007. In Australia, it charted at number twenty-nine in its first week on 9 April and has thus far achieved Gold sales status.

On 6 April 2007, Fraser performed "Deciphering Me" for the Good Friday Appeal, an annual televised fundraising event to raise money for the Royal Children's Hospital in Melbourne.

In 2008, Fraser appeared on the Dave Barnes song "Believe" from his album Me + You + the World, performing backing vocals.

Albertine was also Fraser's U.S. debut, released 27 May 2008 and entered the Billboard 200 at number ninety on 19 July 2008. Her album propelled in success with online sales after being chosen as Editor's Choice on iTunes. On 4 July 2008 Fraser supported Canadian artist/U2 collaborator Daniel Lanois at the Montreal Jazz Festival. In August she toured the southern U.S.^[18] In September, she re-toured several major U.S. cities and completed the tour at the Shepherd's Bush Empire in London, UK.

[edit] Flags

Flags was recorded at East West Studios in Hollywood in the summer of 2010 ^[19] and released by Wood & Bone Records on 12 October 2010. Fraser celebrated with sold-out tours in the U.S., Australia and New Zealand. Flags reached the #1 spot on the New Zealand album chart,^[20] #3 in Australia,^[21] and entered at #59 on the Billboard 200 in the US,^[22] and climbed as high as #3 on the U.S iTunes album chart. "Flags" has been certified Gold in Australia and 3x Platinum in New Zealand where it was the 6th highest selling album of 2010.^[23] The first single, “Something in the Water,” has achieved extensive radio success across the globe, notably receiving BBC Radio 2 A-list rotation upon release.^[24] The single has gone on to achieve double platinum sales in NZ where it spent 21 weeks in the 2010 singles chart.^[25] The single has reached #2 in the Dutch airplay chart whilst residing in the top 5 singles chart for 6 consecutive weeks.^[26] The single has charted in Poland and Belgium at #15 and #23 respectively,^[27] as well as being certified Gold in both Germany and Italy.^[28] Flags was released in July 2011 in Germany, Austria and Switzerland and debuted in the following positions: #6 in Germany, #33 in Austria & #24 in Switzerland. One month on from release, "Flags" remained in the German top 20 at #13 and "Something in the Water" on the singles charts at #8. "Flags" was released in October 2011 in Hungary, reaching #15 on the national album chart by 2nd week of release.^[29] Flags is due to be released in Sweden, Norway, Denmark and Finland on 7 November 2011.^[27] Flags has received accolades from peers and press alike. Sara Bareilles announced to her 2 million followers on Twitter "If you don’t know her, you should. @brookefraser is a freakin gem. New album out now. Yay." and Women’s Wear Daily has called Brooke "The Kiwi Norah Jones."^[30][31]

In its review of Flags, Glide Magazine said: "Brooke Fraser’s third release, Flags, is a wonder. From the stunning lyrical imagery throughout to the impressive guest vocalists who join her (Cary Brothers, Jon Foreman and Aqualung among them), from Fraser’s ethereal and breathy performances to the wide-ranging soundscapes, this record is drenched in beauty and stands as one of the more remarkable achievements of 2010."^[32]

In support of "Flags" release in Switzerland, Brooke will be appearing in November 2011 at the “ENERGY STARS FOR FREE” event in Zurich alongside Lenny Kravitz and Snow Patrol.^[33]

Personal life[link]

After the success of her first album, Fraser moved to Sydney, where she has lived since 2004. On 17 March 2008 she married Scott Ligertwood, in Sydney, changing her name to Brooke Ligertwood.^[34] "Brooke Fraser" has remained her stage name for her activity as a Sony BMG recording/touring artist. She continues to write songs for Hillsong Church, now as "Brooke Ligertwood". Among her notable congregational songs are "Desert Song" and "Hosanna". After touring her second album Albertine for almost 4 years, Fraser returned home to Sydney. She took almost a year off from music-making. After attending the Coachella Valley Music and Arts Festival in the US and seeing Fleet Foxes ^[35] set she was inspired to work on her latest album Flags.^[36]

Fraser also runs a blog from her website where she comments on everything from touring, to fashion and food.^[37]

Philanthropy[link]

Fraser has enjoyed success in her career for many years, and she has always used her position to highlight many worthy causes and charity projects, and raising large amounts of money for them in the process. In late 2010 whilst on tour in the U.S. promoting her third album Flags, Fraser, in conjunction with charity: water ran a birthday campaign asking her fans to donate $27 in honor of her 27th birthday. The goal was to raise $50,000 to build clean water wells in Ethiopia.^[38] Joining the likes of Justin Bieber,^[39] Will Smith^[40] and Adam Lambert^[41] Fraser's birthday campaign was a success raising over $54,000. She has vowed to continue her work with charity: water on future projects.

Fraser has worked with World Vision as an Artist Associate since 2001. She has visited Cambodia and Tanzania with World Vision, the Philippines with Opportunity International and independently traveled to Rwanda in June 2005, in June 2006 as part of charity event "Hope Rwanda", and in May 2007 when she filmed the music video for the song "Albertine" off her second studio album of the same name. In 2006 she, along with Petra Bagust and Tau from Spacifix, appeared in an advertisement for the World Vision 40 Hour Famine; an event which raises funds for children in third world countries. She also sponsors eleven children through World Vision^[42] and makes child ways of contributing to the work of World Vision (i.e. fundraising t-shirts etc.).

Discography[link]

Studio albums

• What to Do with Daylight (2003)
• Albertine (2006)
• Flags (2010)

Awards and nominations[link]

APRA Awards

New Zealand Music Awards

GMA Dove Awards

References[link]

External links[link]

Wikimedia Commons has media related to: Brooke Fraser

• Official Brooke Fraser website
• Brooke Fraser at Allmusic
• Brooke Fraser on Myspace
• New Zealand Herald article

Albert Allen Bartlett (born 1923 in Shanghai)^[1] is an emeritus Professor of Physics at the University of Colorado at Boulder, USA. As of July 2001^[update] Professor Bartlett had lectured over 1,600 times since September, 1969 on Arithmetic, Population, and Energy.^[2][3] Bartlett regards the word combination "sustainable growth" as an oxymoron, since even modest annual percentage population increases will inevitably equate to huge exponential growth over sustained periods of time. He therefore regards human overpopulation as "The Greatest Challenge" facing humanity.

Career[link]

Bartlett joined the faculty of the University of Colorado in Boulder in September 1950. He has a B.A. degree in physics from Colgate University (1944), and M.A. (1948) and Ph.D. (1951) degrees in physics from Harvard University. In 1978 he was national president of the American Association of Physics Teachers. He is a Fellow of the American Physical Society and of the American Association for the Advancement of Science. In 1969 and 1970 he served two terms as the elected Chair of the four-campus Faculty Council of the University of Colorado.

Views on population growth[link]

Thomas Malthus, pioneer of the study of demographics.

Human population curve showing the power of exponential growth

Chart showing change in oil prices since the 19th century. The top curve is inflation-adjusted

World population from 1800 to 2100, based on UN 2004 projections (red, orange, green) and US Census Bureau historical estimates (black).

Professor Bartlett often explains how sustainable growth is a contradiction. His view is based on the fact that a modest percentage growth can equate to huge escalations over relatively short periods of time.^[4]

Bartlett argues that, over time, compound growth can yield enormous increases. For example, an investor earning a constant annual 7% return on their investment would find their capital doubling within 10 years. But the same exponential power, so advantageous to patient investors, may be potentially calamitous when applied to human population. A population of 10,000 individuals, if it were to grow at a constant rate of 7% per annum, would reach a population size of 10 million after 100 years.^[5]

Bartlett regards overpopulation as "The Greatest Challenge" facing humanity, and promotes sustainable living. He opposes the cornucopian school of thought (as advocated by people such as Julian Lincoln Simon), and refers to it as "The New Flat Earth Society"^[6]

J. B. Calvert (1999) has proposed that Bartlett's law ^[7] will result in the exhaustion of petrochemical resources due to the exponential growth of the world population (in line with the Malthusian Growth Model).

Bartlett has made two notable statements relating to sustainability:

"The greatest shortcoming of the human race is our inability to understand the exponential function."

and his Great Challenge:

"Can you think of any problem in any area of human endeavor on any scale, from microscopic to global, whose long-term solution is in any demonstrable way aided, assisted, or advanced by further increases in population, locally, nationally, or globally?"

Books[link]

• The Essential Exponential For the Future of Our Planet a collection of essays by Professor Bartlett (2004). Center for Science, Mathematics and Computer Education, University of Nebraska-Lincoln. ISBN 0-9758973-0-6 ^[8]

See also[link]

• Exponential growth
• Thomas Robert Malthus - the originator of the Malthusian catastrophe argument
• Peak oil
• M. King Hubbert, author of the Hubbert Curve.

Notes[link]

References[link]

• Professor talks at an exponential rate Energy Bulletin article by Todd Neff. Retrieved July 2011

External links[link]

• Professor Bartlett's website contains background, articles, book "The Essential Exponential", and links to his talk, "Arithmetic, Population, and Energy". Retrieved July 2011
• Professor Bartlett's page at Colorado University Retrieved July 2011
• Biography of Dr Bartlett Retrieved July 2011
• "Arithmetic, Population, and Energy" by Professor Albert Bartlett Free audio and video 57 minute lecture. Retrieved July 2011
• "Arithmetic, Population, and Energy" by Professor Albert Bartlett Text of lecture (extracts). Retrieved July 2011
• Analysis of Bartlett's "Arithmetic, Population, and Energy" presentation - Exponentialist website. Retrieved July 2011
• Is There a Population Problem? Ecofuture website. Retrieved July 2011
• The Massive Movement to Marginalise the Modern Malthusian Message article by Professor Bartlett. Retrieved July 2011
• Thoughts on Long-term Energy Supplies - Scientists and the Silent Lie article on energy and population in Physics Today (2004). Retrieved July 2011
• Clean Coal Technology Speech by Andrew McNamara MP on clean coal and report by Professor Bartlett
• Arithmetic, Population & Energy, Part I @YouTube.com (Part 1 of 8), (parts 2-4 are linked). Retrieved July 2011
• Arithmetic, Population & Energy, Part II @YouTube.com (Part 5 of 8), (parts 6-8 are linked). Retrieved July 2011
• Arithmetic, Population & Energy @Google (entire video). Retrieved July 2011
• Lecture by Professor Bartlett about the exponential function @Vimeo. Retrieved July 2011
• BLIND SPOT, Documentary, Population & Energy. Retrieved July 2011

v t e Peak oil Core issues Peak oil Mitigation of peak oil Predicting the timing of peak oil Hubbert peak theory Olduvai theory Results/responses Hirsch report Rimini protocol Price of petroleum 2000s energy crisis Energy crisis Export Land Model Food vs. fuel Oil reserves Pickens Plan Simple living Swing producer Transition Towns People Albert Bartlett Colin J. Campbell David Goodstein John Michael Greer Richard Heinberg M. King Hubbert James Kunstler Jeremy Leggett Dale Allen Pfeiffer Richard Rainwater Matthew Simmons Richard C. Duncan Kenneth S. Deffeyes Books The End of Oil The Long Emergency Out of Gas The Party's Over Power Down Beyond Oil Films A Crude Awakening Collapse The End of Suburbia Oil Factor PetroApocalypse Now? How Cuba Survived Peak Oil What a Way to Go Gashole Organizations ASPO The Oil Drum Energy Watch Group ODAC OPEC OAPEC IEA Post Carbon Institute Other "peaks" Peak car Peak coal Peak copper Peak phosphorus Peak gas Peak uranium Peak water Peak wheat Peak wood

This article has multiple issues. Please help improve it or discuss these issues on the talk page. It does not cite any references or sources. Please help improve it by adding citations to reliable sources. Tagged since December 2008. It may require cleanup to meet Wikipedia's quality standards. Tagged since December 2008.

Last Alliance (ラスト・アライアンス, Rasuto Araiansu^?, stylized as LAST ALLIANCE) is a Japanese rock band based in Machida, Tokyo. They have performed opening and ending theme songs for various anime series such as Ouran High School Host Club, RD Sennō Chōsashitsu, and Hajime No Ippo: New Challenger. To date, they have released five albums, a mini-album, and 8 singles.

Members[link]

• Ryusuke Anzai (安斉 龍介, Anzai Ryūsuke^?) - vocals, rhythm guitar
• Takahiko Matsumara (松村 孝彦, Matsumara Takahiko^?) - vocals, bass
• Shingo Sano (佐野 森吾, Sano Shingo^?) - lead guitar, screaming, backing vocals
• Hiroshi Ozawa (小澤 浩史, Ozawa Hiroshi^?) - drums, backing vocals

Discography[link]

Albums[link]

Tears Library[link]

Tears Library (Stylized TEARS LIBRARY) is Last Alliance's first studio album released on July 19, 2003.

Track listing

1. Boys Don't Cry - 3:57
2. Run into the Freedom - 3:39
3. Itsuwari no Orange (偽りのオレンジ^?, "Orange of Lies") - 3:59
4. Last Alliance - 1:51
5. Rebel Fire - 2:22
6. Beautiful - 0:48
7. Planetarium (プラネタリウム, Puranetariamu^?) - 3:10
8. See You Again - 2:54
9. Sky Is Crying - 3:36
10. Hidarimuki (ヒダリムキ^?, "Facing Left") - 4:22
11. Equal Reason - 4:27

Underground Blue[link]

Underground Blue (Stylized UNDERGROUND BLUE) is Last Alliance's second studio album released on November 8, 2004. The track "Greens Sunlight" was featured in the Nippon TV program Ongaku Senshi: Music Fighter.

Track listing

1. South Wind Knows - 3:44
2. Clone - 4:12
3. Solitude (ソリチュード, Sorichūdo^?) - 5:06
4. Filter (フィルター, Firutā^?) - 4:06
5. One Hot Second - 3:45
6. Musō Jidai (夢想時代^?, "Dream Age") - 4:29
7. Urge - 3:34
8. Greens Sunlight - 3:46
9. Jikū Trip (時空 Trip^?, "Space-Time Trip") - 5:38
10. Okizari Ace (置き去りエース^?, "Desertion Ace") - 3:41
11. Lucky Luciano ni Utareru Mae ni (ラッキー・ルチアーノに射たれる前に^?, "After Lucky Luciano Shot Me") - 3:31
12. Truth in My Arms - 3:40
13. Tadayou Yugure ni, Kiete Yuku Hibi (漂う夕暮れに、消えてゆく日々^?, "Drifting in the Evening, Disappearing Every Day") - 6:10
14. Letter - 3:58

Me and Your Borderline[link]

Me and Your Borderline is Last Alliance's third studio album released on December 6, 2006. The track "Spiral World" was used as the ending theme to the Nippon TV program Ongaku Senshi: Music Fighter and the tracks "Akai Hana" and "Jōka" was re-recorded on the limited edition release of Keep on Smashing Blue.

Track listing

1. Break a mirror - 4:35
2. Konoyubitomare - 4:00
3. Shissou (疾走^?, "Sprint") - 3:54
4. Akai Hana (赤い花^?, "Red Flower") - 4:25
5. Amy Syndrome (エイミーシンドローム, Eimi Shindorōmu^?) - 4:22
6. Oto no Nai Sekai (音の無い世界^?, "A World Without Sound") - 4:18
7. Spiral World - 3:40
8. Zenmai (ゼンマイ^?, "Spring") - 4:26
9. Gray End - 5:31
10. Fantasia - 4:21
11. Lie of Eternity, Paint It Blue - 4:32
12. Prometheus (プロメテウス, Purometeusu^?) - 3:48
13. Sharing Sky (シェアリングスカイ, Shearingu Sukai^?) - 3:57
14. Jōka (浄化^?, "Purification") - 5:13

The Sum[link]

The Sum (Stylized the sum) is Last Alliance's fourth studio album released on June 25, 2008.

Track listing

1. Change by 1 - 3:05
2. Proud of Scar - 2:05
3. Mujūryoku Oneway Shuttle (無重力 Oneway Shuttle^?, "Zero-Gravity Oneway Shuttle") - 4:18
4. Sabaku to Gensō (砂漠と幻想^?, "Deserts and Illusions") - 3:36
5. Katahiza no Yogore (片膝の汚れ^?, "Dirty Knee") - 3:20
6. Perfect Game - 4:30
7. Yureta Byoshin (揺れた秒針^?)- 4:48
8. Lonesome World (ロンサムワールド, Ronsamu Wārudo^?) - 4:15
9. Drag On - 4:33
10. World Is Mine - 3:55
11. Ameagari ni Graffiti (雨上がりにグラフィティ^?, "Graffiti After the Rain") - 4:47
12. Gunjo Kurage (群青海月^?, "Ultramarine Jellyfish") - 4:26

Keep on Smashing Blue[link]

Keep on Smashing Blue (Stylized Keep on smashing blue,) is Last Alliance's fifth studio album released on October 27, 2010. The Limited Edition release contains a DVD of the Kawasaki Relax tour held at Club Quattro on January 28, 2010, a PV of "Wing", and two bonus tracks previously featured in Me and Your Borderline: "Akai Hana (Sweet Anzai Boy)", performed by Anzai, and "Jōka (Angry Matsumura Boy)", performed by Matsumura.

1. Blue Lightning - 4:10
2. Everything Is Evanescent - 3:13
3. Ne(w)rotic World - 3:57
4. Loser - 1:17
5. Third Eyes Syndicate (サードアイズシンジケート, Sādo Aizu Shinjikēdo^?) - 3:45
6. Kotō no Basquiat (孤島のバスキア^?) - 2:13
7. Take Over - 1:32
8. Time Will Tell: Dear Youth - 3:03
9. Wing - 4:33
10. Alliance Airlines - 3:42
11. Ayatsuri (あやつり^?) - 3:41
12. Negaikata Monshirochō ga Sora wo Mau (願イ型紋白蝶ガ宙ヲ舞ウ^?) - 4:28
13. Hekireki - 3:45
14. Looking for the Rainy Sky - 3:03
15. Time Capsule (タイムカプセル, Taimu Kapuseru^?) - 4:16
16. Akai Hana (Sweet Anzai Boy) (赤い花(Aweet Anzai Boy)^?) (Limited Edition only) - 4:41
17. Jōka (Angry Matsumura Boy) (浄化(Angry Matsumura Boy)^?) (Limited Edition only) - 5:13

For Staying Real Blue[link]

For Staying Real Blue (Stylized for staying real BLUE.) is Last Alliance's sixth studio album released on June 15, 2011.

1. In My Hand - 2:23
2. Revolution Is Starting - 2:58
3. Capone's Borsalino (カポネのボルサリーノ, Kapone no Borusarīno^?) - 3:29
4. Throat Wart (スロートワート, Surōto Wāto^?) - 3:59
5. One - 3:52
6. Niji vs Hikōkigumo (虹vs飛行機雲^?, "Rainbow vs. Contrail") - 2:26
7. Kengeki no Hibiki (剣戟の響き^?, "Clash of Sound") - 3:57
8. Deandre - 3:34
9. Boku Shinron (僕信論^?, "My Theory") - 2:54
10. Redesign - 3:54
11. μ (Mu) - 3:24
12. Stardoubt - 3:06
13. Empty Heart (エンプティハート, Enputi Hāto^?) - 3:58
14. Precious Line - 3:03
15. Seijaku no Naka, Sono Ya wa Houbutsusen o Egaku (静寂の中、その矢は放物線を描く^?, "Within the Silence, the Arrow Draws a Parabola") - 4:50
16. Ichi Nigiri no Ao (一握りの青^?, "A Handful of Blue") (Limited Edition Bonus Track) - 3:18
17. Melancholy (Limited Edition Bonus Track) - 3:35

Mini-albums[link]

Kawasaki Relax[link]

Kawasaki Relax (Stylized KAWASAKI RELAX) is Last Alliance's first mini-album released on October 10, 2009. It is the first (and currently only) album that contains full English lyrics.

Track listing

1. Wedge - 3:27
2. Today - 3:27
3. Days - 2:58
4. 8Heartbeat∞ - 3:08
5. Tomorrow - 3:41

Singles[link]

Last Alliance[link]

Last Alliance (Stylized LAST ALLIANCE) is the first self-titled single by Last Alliance released on June 18, 2003. All three songs were later re-recorded in Tears Library. The single is currently out of print.

Track listing

1. Boys Don't Cry - 3:57
2. Last Alliance - 1:54
3. Equal REason - 4:27

YG Service[link]

YG Service (YGサービス, Wai Jī Sābisu^?) is the second single by Last Alliance released on February 18, 2004. The CD for the single contains a music video for the song "Equal Reason".

Track listing

1. Sketch - 3:44
2. Slow Starter (スロースターター, Surō Sutātā^?) - 4:20
3. Maboroshi Memory - 4:54

IO[link]

IO is the third single by Last Alliance released on July 21, 2004. The songs "Solitude", "Musō Jidai", and "Truth in My Arms" were re-recorded in Underground Blue. The single is currently out of print.

Track listing

1. Solitude (ソリチュード, Sorichūdo^?) - 5:09
2. Musō Jidai (夢想時代^?, "Dream Age") - 4:28
3. Melancholy - 3:35
4. Truth in My Arms - 3:38

Re:frain[link]

Re:frain is the fourth single by Last Alliance released on October 5, 2005. The song "Konoyubitomare" was later re-recorded in Me and Your Borderline.

Track listing

1. Color Desert - 4:53
2. Ryusei Namida (流星涙^?) - 4:41
3. Astrogate-0 - 4:38
4. Konoyubitomare - 4:32

Daze & Hope[link]

Daze & Hope (Stylized DAZE&HOPE) is the fifth single by Last Alliance released on May 17, 2006. The CD for the single contains a promotional video for the song "Greens Sunlight". The track "Shissō" was used as the ending theme for the TV anime Ouran High School Host Club and was later re-recorded in Me and Your Borderline.

Track listing

1. Shissō (疾走^?, "Sprint") - 3:55
2. Fly Again, Hero - 3:28
3. Ittō Ryodan (一刀旅団^?, "Cut in Two with a Stroke of a Sword") - 0:25
4. One Drop of Tear - 3:32

Signal 004[link]

Signal 004 is the sixth single by Last Alliance released on November 28, 2007. The song "Drag On" was rerecorded in the sum. A DVD of the single contains music videos of "Drag On" and "Shissō".

Track listing

1. Drag On - 4:39
2. Aoi Jūsō (蒼い銃創^?, "Blue Gunwounds") - 4:08
3. Mafuyu no Semi (真冬の蝉^?, "Midwinter Cicada") - 4:29

Always in My Heart[link]

Always in My Heart is the seventh single by Last Alliance released on April 16, 2008. The track "Katahiza no Yogore" was used as the ending theme for the TV anime RD Sennō Chōsashitsu and was later re-recorded in the sum.

Track listing

1. Katahiza no Yogore (片膝の汚れ^?, "Dirty Knee") - 3:18
2. I.O.J.F.K (intro) - 1:07
3. I.O.J.F.K - 3:28

New Dawn[link]

New Dawn (Stylized new dawn) is the eighth single by Last Alliance released on March 4, 2009. The track "Hekireki" was used the opening theme for the TV anime Hajime No Ippo: New Challenger and was later re-recorded in Keep on Smashing Blue.

Track listing

1. Hekireki - 4:09
2. My Idea - 3:46
3. Rashin (羅針^?, "Compass")- 4:25

External links[link]

• Official site
• Hiroshi's diary - Hiroshi's blog.