|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Use dmy dates|date=June 2012}}
| |
| [[Image:US Navy 070317-N-3642E-379 During the warmest part of the day, a thermometer outside of the Applied Physics Laboratory Ice Station's (APLIS) mess tent still does not break out of the sub-freezing temperatures.jpg|thumb|right|This thermometer is indicating a negative temperature (−4 [[Fahrenheit|°F]]).]]
| |
| A '''negative number''' is a [[real number]] that is [[inequality (mathematics)|less than]] [[0 (number)|zero]]. Such numbers are often used to represent the amount of a loss or absence. For example, a [[debt]] that is owed may be thought of as a negative asset, or a decrease in some quantity may be thought of as a negative increase. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and [[Fahrenheit]] scales for temperature.
| |
|
| |
|
| Negative numbers are usually written with a [[Plus and minus signs|minus sign]] in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". To help tell the difference between a [[subtraction]] operation and a negative number, occasionally the negative sign is placed slightly higher than the [[minus sign]] (as a [[superscript]]). Conversely, a number that is greater than zero is called ''positive''; zero is usually thought of as neither positive nor [[negative zero|negative]].<ref>The convention that zero is neither positive nor negative is not universal. For example, in the French convention, zero is considered to be ''both'' positive and negative. The French words [[:fr:Nombre positif|positif]] and [[:fr:Nombre négatif|négatif]] mean the same as English "positive or zero" and "negative or zero" respectively.</ref> The positivity of a number may be emphasized by placing a plus sign before it, e.g. {{math|+3}}. In general, the negativity or positivity of a number is referred to as its [[sign (mathematics)|sign]].
| |
|
| |
|
| In mathematics, every real number other than zero is either positive or negative. The positive whole numbers are referred to as [[natural number]]s, while the positive and negative whole numbers (together with zero) are referred to as [[integer]]s.
| | The ultimate bliss silence, light sigh a way: "I am wrong, although he's a madman, is a powerful madman. Things seem to have got a problem..."<br><br>"You leave him alone, would help us." Win [http://www.mtcaravans.co.nz/ghdnz.html Cheap GHD NZ] or that sentence .<br><br>The sword of the sing again, still can kill do not qin home, still can kill? [http://En.wiktionary.org/wiki/Ultimate+bliss Ultimate bliss] is win, will not allow him to do so. know to ultimate bliss hope to win the severe sing, everyone to quit, JingShuiBuFanHeShui later.<br>Of course, the idea too much wishful thinking. Ultimate bliss and qin home, week home, how can so give up? In ten thousand his family, also don't allow them to pass sang.<br><br>[http://Mondediplo.com/spip.php?page=recherche&recherche=Ultimate+bliss Ultimate bliss] sighed again, " how do you not understand? So dangerous lunatic, of course, is killed to exempt from the edge. Otherwise, he which day again, we are not crazy more dangerous"win in anger way: "song was not mad, but you go and different,"<br>A raised my hand [http://www.mtcaravans.co.nz/ghdnz.html Cheap GHD NZ] against the ultimate bliss, lived the shot of winning acupuncture points, in the mattias lampe shouted up com. "Can solve naval gun sing?<br>Mattias lampe some difficult, just she saw just sing to the unpredictable sword. For the song, the heart without any grasp. But also can say, "it's hard to."<br><br>Naval gun battle is group of weapons, but to sing such strong individual fix for the threat is not very big. Want to hit song, must first lock him. Sing to the breath can transform unfathomable, too b really ray gun power is still strong, but not too may like yuan baby real use sophisticated as practice. The change of the straight line, almost helpless not sing.<br>Of course, warships equipped with various complete skulls, in ten kilometers around the battleship inside, there is no any blind Angle. If it can sing desperate hard rushed in and out he is not easily shot.<br><br>Day ROM warships, if it is far away from the eternal field, too b really ray gun can power to develop acme, can even compare god of top. A cannon kill yuan baby, is not difficult. However, the power of support by the nickel-ferric array method, ghd hair straightener and no country can consume up.<br>Ultimate bliss hesitant bottom, don't know if I should continue to call a person. Four yuan baby all can't kill sang, that her strength can only rely on the number of hard....... Sing to want to walk, but who have no holding. Continue to increase the relief troops, meaning do not seem to be.<br><br>Even the qin home, also won't have too many yuan baby the strong. A because financial limit, or will scruple the emperors suspicion.<br>The rare hesitate to ultimate bliss, the battleship battle outside has done.<br>Autumn comes at a time when, more CanRan stars. A billionaire stars gathered to the Milky Way is long, which spans zhongtian, the sky in two.<br>Looking at the stars sing millions, have feeling at heart, the heroism. To the three yuan baby's humanitarian: "three, look at the time, let me send a few the trip"<br><br>ChuYuan is five hundred and fifty years old, he lunar refinement life cut soul arcane, make moves ZhuanSha people out, is the most secretive biting. His age has of course won't easily move by the words to sing, so ZuoSe, also just remind the other two men, not in the bullpen.<br>Just LiYunFei dead of shady, ChuYuan to sing really terrified on top of the world. Hear ChuYuan wrath of drink, and all the other two men with shots at the same time.<br>The other two yuan baby the strong; the name of the one was FengYu, a call CuiZe.<br><br>FengYu good at positive and negative five line destroy god knife. To say this door method tactic, also called on the JueXue is supreme, and nine glory to the travelling the ninja team coordination. Only in FengYu hand, also but is a careful, not to mention the god meaning inheritance, is travelling is incomplete law.<br>But by this is careful, FengYu just progress to the yuan baby realm. Unfortunately, has his talent and the ability to yuan baby state, has been the limit.<br>CuiZe is the evil spirit is security, a frost spell a superb, and shape a more quickly if electricity, in three people, also is the fastest. |
| | |
| In [[bookkeeping]], amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.
| |
| | |
| Negative numbers appeared for the first time in history in the ''[[Nine Chapters on the Mathematical Art]]'', which in its present form dates from the period of the Chinese [[Han Dynasty]] (202 BC – AD 220), but may well contain much older material.<ref name=struik33>Struik, page 32–33. "''In these matrices we find negative numbers, which appear here for the first time in history.''"</ref> [[India]]n mathematicians developed consistent and correct rules on the use of negative numbers. Prior to the concept of negative numbers, negative solutions to problems were considered "false" and equations requiring negative solutions were described as absurd.<ref>[[Diophantus]], ''[[Arithmetica]]''.</ref>
| |
| | |
| ==Introduction==
| |
| | |
| ===As the result of subtraction===
| |
| Negative numbers can be thought of as resulting from the [[subtraction]] of a larger number from a smaller. For example, negative three is the result of subtracting three from zero:
| |
| : {{math| 0 − 3 {{=}} −3.}}
| |
| In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example,
| |
| : {{math| 5 − 8 {{=}} −3}}
| |
| since {{math|8 − 5 {{=}} 3}}.
| |
| | |
| ===The number line===
| |
| {{Main|Number line}}
| |
| The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a '''number line''':
| |
| [[File:Number-line.svg|center|The number line]]
| |
| Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left.
| |
| | |
| Note that a negative number with greater magnitude is considered less. For example, even though (positive) {{math|8}} is greater than (positive) {{math|5}}, written
| |
| : {{math|8 > 5}}
| |
| negative {{math|8}} is considered to be less than negative {{math|5}}:
| |
| : {{math|−8 < −5.}}
| |
| (Because, for example, if you have £-8 you have less than if you have £-5.) Therefore, any negative number is less than any positive number, so
| |
| : {{math|−8 < 5}} and {{math|−5 < 8.}}
| |
| | |
| ===Signed numbers===
| |
| {{main|Sign (mathematics)}}
| |
| In the context of negative numbers, a number that is greater than zero is referred to as '''positive'''. Thus every [[real number]] other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a [[plus sign]] in front, e.g. {{math|+3}} denotes a positive three.
| |
| | |
| Because zero is neither positive nor negative, the term '''nonnegative''' is sometimes used to refer to a number that is either positive or zero, while '''nonpositive''' is used to refer to a number that is either negative or zero. Zero is a neutral number.
| |
| | |
| ==Everyday uses of negative numbers==
| |
| ===Sport===
| |
| * [[Goal difference]] in [[association football]] and [[hockey]]; points difference in [[rugby football]]; [[net run rate]] in [[cricket]]; [[golf]] scores relative to [[Golf#Scoring_and_handicapping|par]].
| |
| * [[Plus-minus]] differential in [[ice hockey]]: the difference in total goals scored for the team (+) and against the team (-) when a particular player is on the ice is the player's +/- rating. Players can have a negative (+/-) rating.
| |
| * British football clubs are deducted points if they enter [[Administration (British football)|administration]], and thus have a negative points total until they have earned at least that many points that season.
| |
| * Lap (or sector) times in [[Formula 1]] may be given as the difference compared to a previous lap (or sector) (such as the previous record, or the lap just completed by a driver in front), and will be positive if slower and negative if faster.
| |
| * In some [[Athletics (sport)|athletics]] events, such as [[sprint (running)|sprint race]]s, the [[110 metres hurdles|hurdles]], the [[triple jump]] and the [[long jump]], the [[wind assistance]] is measured and recorded,<ref>[http://london2012.bbc.co.uk/athletics/event/men-long-jump/index.html BBC website]</ref> and is positive for a [[tailwind]] and negative for a headwind.<ref>[http://www.elitefeet.com/how-wind-assistance-works-in-track-field Elitefeet]</ref>
| |
| | |
| ===Science===
| |
| * [[Temperature]]s which are colder than 0°C or 0°F.
| |
| * [[Latitude]]s south of the equator and [[longitude]]s west of the [[prime meridian]].
| |
| * [[Topography|Topographical]] features of the earth's surface are given a [[height]] above [[sea level]], which can be negative (e.g. The surface elevation of The [[Dead Sea]]).
| |
| * [[Electrical circuits]]. When a battery is connected in reverse polarity, the voltage applied is said to be the opposite of its rated voltage. For example a 6(V) battery connected in reverse applies a voltage of -6(V).
| |
| * [[Ions]] have a positive or negative electrical charge.
| |
| | |
| ===Finance===
| |
| * [[Bank account]] balances which are [[Overdraft|overdrawn]].
| |
| * Refunds to a [[credit card]] or [[debit card]] are a negative [[Debits and credits|debit]].
| |
| * A company might make a negative annual [[Profit (accounting)|profit]] (i.e. a loss).
| |
| * The annual percentage growth in a country's [[Gross domestic product|GDP]] might be negative, which is one indicator of being in a [[recession]].<ref>[http://www.bbc.co.uk/news/business-21193525 BBC article]</ref>
| |
| * Occasionally, a rate of [[inflation]] may be negative ([[deflation]]), indicating a fall in average prices.<ref>[http://www.independent.co.uk/news/business/news/first-negative-inflation-figure-since-1960-1671736.html Article in The Independent]</ref>
| |
| * The daily change in a [[stock market index]], such as the [[FTSE 100 Index|FTSE 100]] or the [[Dow Jones Industrial Average|Dow Jones]].
| |
| * A negative number in financing is synonymous with "debt" and "deficit" which are also known as "being in the red".
| |
| | |
| ===Other===
| |
| [[Image:Universitychallenge1994.jpg|thumb|An episode of ''University Challenge'' in 1994]]
| |
| | |
| * The numbering of [[storey]]s in a building below the ground floor.
| |
| * When playing an [[audio file]] on a [[portable media player]], such as an [[iPod]], the screen display may show the time remaining as a negative number, which increases up to zero at the same rate as the time already played increases from zero.
| |
| * Television [[game shows]]:
| |
| ** Participants on ''[[QI]]'' often finish with a negative points score.
| |
| ** Teams on [[University Challenge]] have a negative score if their first answers are incorrect and interrupt the question.
| |
| ** ''[[Jeopardy!]]'' has a negative money score - contestants play for an amount of money and any incorrect answer that costs them more than what they have now can result in a negative score.
| |
| ** ''[[The Price Is Right (U.S. game show)|The Price Is Right]]'' pricing game Buy or Sell, if any money is lost and is more than the amount currently in the bank, it also incurs a negative score.
| |
| * The change in support for a political party between elections, known as [[Swing (politics)|swing]].
| |
| * In [[video games]], a negative number indicates loss of life, damage, a score penalty, or consumption of a resource, depending on the genre of the simulation.
| |
| * Employees with [[flextime|flexible working hours]] may have a negative balance on their [[timesheet]] if they've worked fewer total hours than contracted to that point. Employees may be able to take more than their annual holiday allowance in a year, and carry forward a negative balance to the next year.
| |
| | |
| ==Arithmetic involving negative numbers==
| |
| The [[Plus and minus signs|minus sign]] "−" signifies the [[Operator (mathematics)|operator]] for both the binary (two-[[operand]]) [[Operation (mathematics)|operation]] of [[subtraction]] (as in {{math|y − z}}) and the unary (one-operand) operation of [[negation]] (as in {{math|−x}}, or twice in {{math|−(−x)}}). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in {{math|−5}}).
| |
| | |
| The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.
| |
| | |
| For example, the expression {{math|7 + −5}} may be clearer if written {{math|7 + (−5)}} (even though they mean exactly the same thing formally). The [[subtraction]] expression {{math|7–5}} is a different expression that doesn't represent the same operations, but it evaluates to the same result.
| |
| | |
| Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in<ref>{{cite book|title=Understanding by design|author1=Grant P. Wiggins|author2=Jay McTighe|page=210|year=2005|publisher=ACSD Publications|isbn=1-4166-0035-3}}</ref>
| |
| :{{math|<sup>−</sup>2 + <sup>−</sup>5}} gives {{math|<sup>−</sup>7}}.
| |
| | |
| ===Addition===
| |
| [[File:AdditionRules.svg|right|thumb|A visual representation of the addition of positive and negative numbers. Larger balls represent numbers with greater magnitude.]]
| |
| Addition of two negative numbers is very similar to addition of two positive numbers. For example,
| |
| :{{math|(−3) + (−5) {{=}} −8}}.
| |
| The idea is that two debts can be combined into a single debt of greater magnitude.
| |
| | |
| When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:
| |
| :{{math|8 + (−3) {{=}} 8 − 3 {{=}} 5}} and {{math|(−2) + 7 {{=}} 7 − 2 {{=}} 5}}.
| |
| In the first example, a credit of {{math|8}} is combined with a debt of {{math|3}}, which yields a total credit of {{math|5}}. If the negative number has greater magnitude, then the result is negative:
| |
| :{{math|(−8) + 3 {{=}} 3 − 8 {{=}} −5}} and {{math|2 + (−7) {{=}} 2 − 7 {{=}} −5}}.
| |
| Here the credit is less than the debt, so the net result is a debt.
| |
| | |
| ===Subtraction===
| |
| As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:
| |
| : {{math| 5 − 8 {{=}} −3}}
| |
| In general, subtraction of a positive number is the same thing as addition of a negative. Thus
| |
| : {{math| 5 − 8 {{=}} 5 + (−8) {{=}} −3}}
| |
| and
| |
| : {{math| (−3) − 5 {{=}} (−3) + (−5) {{=}} −8}}
| |
| | |
| On the other hand, subtracting a negative number is the same as ''adding'' a positive. (The idea is that ''losing'' a debt is the same thing as ''gaining'' a credit.) Thus
| |
| : {{math| 3 − (−5) {{=}} 3 + 5 {{=}} 8}}
| |
| and
| |
| : {{math| (−5) − (−8) {{=}} (−5) + 8 {{=}} 3}}.
| |
| | |
| ===Multiplication===
| |
| When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The [[sign (mathematics)|sign]] of the product is determined by the following rules:
| |
| * The product of one positive number and one negative number is negative.
| |
| * The product of two negative numbers is positive.
| |
| Thus
| |
| : {{math| (−2) × 3 {{=}} −6}}
| |
| and
| |
| : {{math| (−2) × (−3) {{=}} 6}}.
| |
| The reason behind the first example is simple: adding three {{math|−2}}'s together yields {{math|−6}}:
| |
| : {{math| (−2) × 3 {{=}} (−2) + (−2) + (−2) {{=}} −6}}.
| |
| The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:
| |
| : {{math| (−2}} debts {{math|) × (−3}} each{{math|) {{=}} +6}} credit.
| |
| The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the [[distributive law]]. In this case, we know that
| |
| : {{math| (−2) × (−3) + 2 × (−3) {{=}} (−2 + 2) × (−3) {{=}} 0 × (−3) {{=}} 0}}.
| |
| Since {{math|2 × (−3) {{=}} −6}}, the product {{math|(−2) × (−3)}} must equal {{math|6}}.
| |
| | |
| These rules lead to another (equivalent) rule—the sign of any product ''a'' × ''b'' depends on the sign of ''a'' as follows:
| |
| * if ''a'' is positive, then the sign of ''a'' × ''b'' is the same as the sign of ''b'', and
| |
| * if ''a'' is negative, then the sign of ''a'' × ''b'' is the opposite of the sign of ''b''.
| |
| The justification for why the product of two negative numbers is a positive number can be observed in the analysis of [[complex numbers]].
| |
| ===Division===
| |
| The sign rules for [[Division (mathematics)|division]] are the same as for multiplication. For example,
| |
| :{{math|8 ÷ (−2) {{=}} −4}},
| |
| :{{math|(−8) ÷ 2 {{=}} −4}},
| |
| and
| |
| :{{math|(−8) ÷ (−2) {{=}} 4}}.
| |
| If dividend and divisor have the same sign, the result is always positive.
| |
| | |
| ==Negation==
| |
| {{main|Negation (algebra)}}
| |
| The negative version of a positive number is referred to as its [[negation (algebra)|negation]]. For example, {{math|−3}} is the negation of the positive number {{math|3}}. The [[addition|sum]] of a number and its negation is equal to zero:
| |
| :{{math|3 + −3 {{=}} 0}}.
| |
| That is, the negation of a positive number is the [[additive inverse]] of the number.
| |
| | |
| Using [[algebra]], we may write this principle as an [[algebraic identity]]:
| |
| :{{math|''x'' + −''x'' {{=}} 0}}.
| |
| This identity holds for any positive number {{math|''x''}}. It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. Specifically:
| |
| * The negation of 0 is 0, and
| |
| * The negation of a negative number is the corresponding positive number.
| |
| For example, the negation of {{math|−3}} is {{math|+3}}. In general,
| |
| :{{math|−(−''x'') {{=}} ''x''}}.
| |
| | |
| The [[absolute value]] of a number is the non-negative number with the same magnitude. For example, the absolute value of {{math|−3}} and the absolute value of {{math|3}} are both equal to {{math|3}}, and the absolute value of {{math|0}} is {{math|0}}.
| |
| | |
| ==Formal construction of negative integers==
| |
| {{See also|Integer#Construction}}
| |
| In a similar manner to [[rational number]]s, we can extend the [[natural number]]s '''N''' to the integers '''Z''' by defining integers as an [[ordered pair]] of natural numbers (''a'', ''b''). We can extend addition and multiplication to these pairs with the following rules:
| |
| :(''a'', ''b'') + (''c'', ''d'') = (''a'' + ''c'', ''b'' + ''d'')
| |
| :(''a'', ''b'') × (''c'', ''d'') = (''a'' × ''c'' + ''b'' × ''d'', ''a'' × ''d'' + ''b'' × ''c'')
| |
| | |
| We define an [[equivalence relation]] ~ upon these pairs with the following rule:
| |
| :(''a'', ''b'') ~ (''c'', ''d'') if and only if ''a'' + ''d'' = ''b'' + ''c''.
| |
| This equivalence relation is compatible with the addition and multiplication defined above, and we may define '''Z''' to be the [[quotient set]] '''N'''²/~, i.e. we identify two pairs (''a'', ''b'') and (''c'', ''d'') if they are equivalent in the above sense. Note that '''Z''', equipped with these operations of addition and multiplication, is a [[Ring (mathematics)|ring]], and is in fact, the prototypical example of a ring.
| |
| | |
| We can also define a [[total order]] on '''Z''' by writing
| |
| :(''a'', ''b'') ≤ (''c'', ''d'') if and only if ''a'' + ''d'' ≤ ''b'' + ''c''.
| |
| | |
| This will lead to an ''additive zero'' of the form (''a'', ''a''), an ''[[additive inverse]]'' of (''a'', ''b'') of the form (''b'', ''a''), a multiplicative unit of the form (''a'' + 1, ''a''), and a definition of [[subtraction]]
| |
| :(''a'', ''b'') − (''c'', ''d'') = (''a'' + ''d'', ''b'' + ''c'').
| |
| This construction is a special case of the [[Grothendieck group#Explicit construction|Grothendieck construction]].
| |
| | |
| ===Uniqueness===
| |
| The negative of a number is unique, as is shown by the following proof.
| |
| | |
| Let ''x'' be a number and let ''y'' be its negative.
| |
| Suppose ''y′'' is another negative of ''x''. By an [[axiom]] of the real number system
| |
| | |
| :<math>x + y \prime = 0,</math>
| |
| :<math> x + y\,\, = 0.</math>
| |
| | |
| And so, ''x'' + ''y′'' = ''x'' + ''y''. Using the law of cancellation for addition, it is seen that
| |
| ''y′'' = ''y''. Thus ''y'' is equal to any other negative of ''x''. That is, ''y'' is the unique negative of ''x''.
| |
| | |
| == History ==<!-- This section is linked from [[History of negative numbers]] (R to section) -->
| |
| For a long time, negative solutions to problems were considered "false". In [[Hellenistic Egypt]], the [[Greek mathematics|Greek]] mathematician [[Diophantus]] in the third century A.D. referred to an equation that was equivalent to 4''x'' + 20 = 0 (which has a negative solution) in ''[[Arithmetica]]'', saying that the equation was absurd.
| |
| | |
| Negative numbers appear for the first time in history in the ''[[Nine Chapters on the Mathematical Art]]'' (''Jiu zhang suan-shu''), which in its present form dates from the period of the [[Han Dynasty]] (202 BC – AD 220), but may well contain much older material.<ref name="struik33"/> The ''Nine Chapters'' used red [[counting rods]] to denote positive [[coefficient]]s and black rods for negative.<ref>Temple, Robert. (1986). ''The Genius of China: 3,000 Years of Science, Discovery, and Invention''. With a forward by Joseph Needham. New York: [[Simon & Schuster]], Inc. ISBN 0-671-62028-2. Page 141.</ref> This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. The Chinese were also able to solve simultaneous equations involving negative numbers.
| |
| | |
| The ancient Indian ''[[Bakhshali Manuscript]]'', which Pearce Ian claimed was written some time between 200 BC. and AD 300,<ref>
| |
| {{cite web|title=The Bakhshali manuscript|author=Pearce, Ian|publisher=The MacTutor History of Mathematics archive|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html|date=May 2002|accessdate=2007-07-24}}</ref> while George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century,<ref>Teresi, Dick. (2002). ''Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas''. New York: Simon & Schuster. ISBN 0-684-83718-8. Page 65–66.</ref> carried out calculations with negative numbers, using "+" as a negative sign.<ref>Teresi, Dick. (2002). ''Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas''. New York: Simon & Schuster. ISBN 0-684-83718-8. Page 65.</ref>
| |
| | |
| During the 7th century AD, negative numbers were used in India to represent debts. The [[Indian mathematics|Indian mathematician]] [[Brahmagupta]], in ''[[Brahmasphutasiddhanta|Brahma-Sphuta-Siddhanta]]'' (written in [[A.D. 628]]), discussed the use of negative numbers to produce the general form [[quadratic formula]] that remains in use today. He also found negative solutions of [[quadratic equation]]s and gave rules regarding operations involving negative numbers and [[0 (number)|zero]], such as ''"A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt. "'' He called positive numbers "fortunes," zero "a cipher," and negative numbers "debts."<ref>Colva M. Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on 9 March 2006.</ref><ref>''Knowledge Transfer and Perceptions of the Passage of Time'', ICEE-2002 Keynote Address by Colin Adamson-Macedo. "''Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (= fortunes) is used to represent positive numbers, whereas 'rina' (= debts) were negative''".</ref>
| |
| | |
| In the 9th and 10th century AD, [[Islamic mathematicians]] were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.<ref name=Rashed>{{Cite book| publisher = Springer| isbn = 9780792325659| last = Rashed| first = R.| title = The Development of Arabic Mathematics: Between Arithmetic and Algebra| date =1994-06-30|pages=36-37}}</ref> [[Al-Khwarizmi]] in his ''Al-jabr wa'l-muqabala'' (from which we get the word "algebra") did not use negative numbers or negative coefficients, although [[al-Karaji]] wrote in his ''al-Fakhrī'' that "negative quantities must be counted as terms".<ref name=Rashed /> The first and only known medieval Islamic text that uses negative numbers is ''A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen'' by [[Abū al-Wafā' al-Būzjānī]].<ref>{{cite encyclopedia | editor = Thomas Hockey et al | last = Hashemipour | first = Behnaz | title=Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī | encyclopedia = The Biographical Encyclopedia of Astronomers | publisher=Springer | year = 2007 | location = New York | pages = 188–9 | url=http://islamsci.mcgill.ca/RASI/BEA/Buzjani_BEA.htm | isbn=978-0-387-31022-0| ref=harv}} ([http://islamsci.mcgill.ca/RASI/BEA/Buzjani_BEA.pdf PDF version])</ref>
| |
| | |
| However, by the 12th century AD, al-Karaji's successors were to state the general rules of signs, or as [[al-Samaw'al]] writes:
| |
| <blockquote>the product of a negative number — ''al-nāqiṣ'' — by a positive number — ''al-zāʾid'' — is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (''martaba khāliyya''), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.<ref name=Rashed /></blockquote>
| |
| | |
| In the 12th century AD in India, [[Bhāskara II]] gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "''in this case not to be taken, for it is inadequate; people do not approve of negative roots.''"
| |
| | |
| European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although [[Leonardo of Pisa#Important publications|Fibonacci]] allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of ''[[Liber Abaci]]'', AD 1202) and later as losses (in ''[[Leonardo of Pisa|Flos]]'').
| |
| | |
| In the 15th century, [[Nicolas Chuquet]], a Frenchman, used negative numbers as [[Exponentiation|exponents]] and referred to them as “absurd numbers.”{{Citation needed|date=October 2008}}
| |
| | |
| In A.D. 1759, [[Francis Maseres]], an English mathematician, wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers were nonsensical.<ref>{{cite book |last=Maseres |first=Francis |authorlink=Francis Maseres |title=A dissertation on the use of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle |year=1758 |work=Quoting from Maseres' work: If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of −5, or the product of −5 into −5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon}}</ref>
| |
| | |
| In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless.<ref>{{cite book |first=Alberto A. |last=Martinez |title=Negative Math: How Mathematical Rules Can Be Positively Bent |publisher=[[Princeton University Press]] |year=2006 }} a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.</ref>
| |
| | |
| [[Gottfried Wilhelm Leibniz]] was the first mathematician to systematically employ negative numbers as part of a coherent mathematical system, the [[infinitesimal calculus]]. Calculus made negative numbers necessary and their dismissal as "absurd numbers" quickly faded.
| |
| | |
| == See also ==
| |
| <div style="-moz-column-count:2; column-count:2;">
| |
| * [[−0]]
| |
| * [[Additive inverse]]
| |
| * [[History of zero]]
| |
| * [[Integers]]
| |
| * [[Positive and negative parts]]
| |
| * [[Rational numbers]]
| |
| * [[Real numbers]]
| |
| * [[Sign function]]
| |
| * [[Sign (mathematics)]]
| |
| * [[Signed number representations]]
| |
| </div>
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * Bourbaki, Nicolas (1998). ''Elements of the History of Mathematics''. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8.
| |
| * Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications.
| |
| | |
| ==External links==
| |
| *[http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maseres.html Maseres' biographical information]
| |
| *[http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml BBC Radio 4 series "In Our Time," on ''Negative Numbers'', 9 March 2006]
| |
| *[http://www.free-ed.net/sweethaven/Math/arithmetic/SignedValues01_EE.asp Endless Examples & Exercises: ''Operations with Signed Integers'']
| |
| *[http://mathforum.org/dr.math/faq/faq.negxneg.html Math Forum: Ask Dr. Math FAQ: Negative Times a Negative]
| |
| | |
| {{Number Systems}}
| |
| | |
| {{DEFAULTSORT:Negative And Non-Negative Numbers}}
| |
| [[Category:Elementary arithmetic]]
| |
| [[Category:Integers]]
| |
| [[Category:Numbers]]
| |
The ultimate bliss silence, light sigh a way: "I am wrong, although he's a madman, is a powerful madman. Things seem to have got a problem..."
"You leave him alone, would help us." Win Cheap GHD NZ or that sentence .
The sword of the sing again, still can kill do not qin home, still can kill? Ultimate bliss is win, will not allow him to do so. know to ultimate bliss hope to win the severe sing, everyone to quit, JingShuiBuFanHeShui later.
Of course, the idea too much wishful thinking. Ultimate bliss and qin home, week home, how can so give up? In ten thousand his family, also don't allow them to pass sang.
Ultimate bliss sighed again, " how do you not understand? So dangerous lunatic, of course, is killed to exempt from the edge. Otherwise, he which day again, we are not crazy more dangerous"win in anger way: "song was not mad, but you go and different,"
A raised my hand Cheap GHD NZ against the ultimate bliss, lived the shot of winning acupuncture points, in the mattias lampe shouted up com. "Can solve naval gun sing?
Mattias lampe some difficult, just she saw just sing to the unpredictable sword. For the song, the heart without any grasp. But also can say, "it's hard to."
Naval gun battle is group of weapons, but to sing such strong individual fix for the threat is not very big. Want to hit song, must first lock him. Sing to the breath can transform unfathomable, too b really ray gun power is still strong, but not too may like yuan baby real use sophisticated as practice. The change of the straight line, almost helpless not sing.
Of course, warships equipped with various complete skulls, in ten kilometers around the battleship inside, there is no any blind Angle. If it can sing desperate hard rushed in and out he is not easily shot.
Day ROM warships, if it is far away from the eternal field, too b really ray gun can power to develop acme, can even compare god of top. A cannon kill yuan baby, is not difficult. However, the power of support by the nickel-ferric array method, ghd hair straightener and no country can consume up.
Ultimate bliss hesitant bottom, don't know if I should continue to call a person. Four yuan baby all can't kill sang, that her strength can only rely on the number of hard....... Sing to want to walk, but who have no holding. Continue to increase the relief troops, meaning do not seem to be.
Even the qin home, also won't have too many yuan baby the strong. A because financial limit, or will scruple the emperors suspicion.
The rare hesitate to ultimate bliss, the battleship battle outside has done.
Autumn comes at a time when, more CanRan stars. A billionaire stars gathered to the Milky Way is long, which spans zhongtian, the sky in two.
Looking at the stars sing millions, have feeling at heart, the heroism. To the three yuan baby's humanitarian: "three, look at the time, let me send a few the trip"
ChuYuan is five hundred and fifty years old, he lunar refinement life cut soul arcane, make moves ZhuanSha people out, is the most secretive biting. His age has of course won't easily move by the words to sing, so ZuoSe, also just remind the other two men, not in the bullpen.
Just LiYunFei dead of shady, ChuYuan to sing really terrified on top of the world. Hear ChuYuan wrath of drink, and all the other two men with shots at the same time.
The other two yuan baby the strong; the name of the one was FengYu, a call CuiZe.
FengYu good at positive and negative five line destroy god knife. To say this door method tactic, also called on the JueXue is supreme, and nine glory to the travelling the ninja team coordination. Only in FengYu hand, also but is a careful, not to mention the god meaning inheritance, is travelling is incomplete law.
But by this is careful, FengYu just progress to the yuan baby realm. Unfortunately, has his talent and the ability to yuan baby state, has been the limit.
CuiZe is the evil spirit is security, a frost spell a superb, and shape a more quickly if electricity, in three people, also is the fastest.