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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a negative number represents an opposite. In the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
system, a negative number is a number that is
less than In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different ...
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
. Negative numbers are often used to represent the magnitude of a loss or deficiency. A
debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another party, the creditor. Debt is a deferred payment, or series of payments, which differentiates it from an immediate purchase. The ...
that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as ''positive'' and ''negative''. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a
minus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
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 The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
(as a superscript). Conversely, a number that is greater than zero is called ''positive''; zero is usually ( but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign. Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. (Some definitions of the natural numbers exclude zero.) In
bookkeeping Bookkeeping is the recording of financial transactions, and is part of the process of accounting in business and other organizations. It involves preparing source documents for all transactions, operations, and other events of a business. Tr ...
, 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 ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'', which in its present form dates from the period of the Chinese
Han Dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
(202 BC – AD 220), but may well contain much older material.Struik, pages 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
(c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz (1646–1716) held that negative numbers were invalid, but still used them in calculations.


Introduction


The number line

The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a 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) is greater than (positive) , written negative is considered to be less than negative : (Because, for example, if you have £−8, a debt of £8, you would have less after adding, say £10, to it than if you have £−5.) It follows that any negative number is less than any positive number, so


Signed numbers

In the context of negative numbers, a number that is greater than zero is referred to as positive. Thus every
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, res ...
in front, e.g. 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.


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: 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, since .


Everyday uses of negative numbers


Sport

* Goal difference in
association football Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
and hockey; points difference in
rugby football Rugby football is the collective name for the team sports of rugby union and rugby league. Canadian football and, to a lesser extent, American football were once considered forms of rugby football, but are seldom now referred to as such. The ...
; net run rate in
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by st ...
;
golf Golf is a club-and-ball sport in which players use various clubs to hit balls into a series of holes on a course in as few strokes as possible. Golf, unlike most ball games, cannot and does not use a standardized playing area, and coping wi ...
scores relative to par. * Plus-minus differential in
ice hockey Ice hockey (or simply hockey) is a team sport played on ice skates, usually on an ice skating rink with lines and markings specific to the sport. It belongs to a family of sports called hockey. In ice hockey, two opposing teams use ice ...
: 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. *
Run differential In baseball, run differential is a cumulative team statistic that combines offensive and defensive scoring. Run differential is calculated by subtracting runs allowed from runs scored. Run differential is positive when a team scores more runs th ...
in
baseball Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding t ...
: the run differential is negative if the team allows more runs than they scored. * Clubs may be deducted points for breaches of the laws, 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 Formula One (also known as Formula 1 or F1) is the highest class of international racing for open-wheel single-seater formula racing cars sanctioned by the Fédération Internationale de l'Automobile (FIA). The World Drivers' Championship, ...
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 Athletics may refer to: Sports * Sport of athletics, a collection of sporting events that involve competitive running, jumping, throwing, and walking ** Track and field, a sub-category of the above sport * Athletics (physical culture), competi ...
events, such as sprint races, the hurdles, the
triple jump The triple jump, sometimes referred to as the hop, step and jump or the hop, skip and jump, is a track and field event, similar to the long jump. As a group, the two events are referred to as the "horizontal jumps". The competitor runs down th ...
and the
long jump The long jump is a track and field event in which athletes combine speed, strength and agility in an attempt to leap as far as possible from a takeoff point. Along with the triple jump, the two events that measure jumping for distance as a ...
, the wind assistance is measured and recorded, and is positive for a tailwind and negative for a headwind.


Science

*
Temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
s which are colder than 0 °C or 0 °F. *
Latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
s south of the equator and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
s west of the
prime meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great ...
. * Topographical features of the earth's surface are given a
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
above
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardis ...
, which can be negative (e.g. the surface elevation of the
Dead Sea The Dead Sea ( he, יַם הַמֶּלַח, ''Yam hamMelaḥ''; ar, اَلْبَحْرُ الْمَيْتُ, ''Āl-Baḥrū l-Maytū''), also known by other names, is a salt lake bordered by Jordan to the east and Israel and the West Ban ...
or
Death Valley Death Valley is a desert valley in Eastern California, in the northern Mojave Desert, bordering the Great Basin Desert. During summer, it is the hottest place on Earth. Death Valley's Badwater Basin is the point of lowest elevation in Nort ...
, or the elevation of the Thames Tideway Tunnel). * 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-volt battery connected in reverse applies a voltage of −6 volts. * Ions have a positive or negative electrical charge. * Impedance of an AM broadcast tower used in multi-tower directional antenna arrays, which can be positive or negative.


Finance

* Financial statements can include negative balances, indicated either by a minus sign or by enclosing the balance in parentheses. Examples include bank account
overdraft An overdraft occurs when something is withdrawn in excess of what is in a current account. For financial systems, this can be funds in a bank account. For water resources, it can be groundwater in an aquifer. In these situations the account is s ...
s and business losses (negative earnings). * Refunds to a
credit card A credit card is a payment card issued to users (cardholders) to enable the cardholder to pay a merchant for goods and services based on the cardholder's accrued debt (i.e., promise to the card issuer to pay them for the amounts plus the ...
or
debit card A debit card, also known as a check card or bank card is a payment card that can be used in place of cash to make purchases. The term '' plastic card'' includes the above and as an identity document. These are similar to a credit card, but ...
are a negative charge to the card. * The annual percentage growth in a country's GDP might be negative, which is one indicator of being in a
recession In economics, a recession is a business cycle contraction when there is a general decline in economic activity. Recessions generally occur when there is a widespread drop in spending (an adverse demand shock). This may be triggered by various ...
. * Occasionally, a rate of
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduct ...
may be negative ( deflation), indicating a fall in average prices. * The daily change in a share price or
stock market index In finance, a stock index, or stock market index, is an index that measures a stock market, or a subset of the stock market, that helps investors compare current stock price levels with past prices to calculate market performance. Two of th ...
, such as the
FTSE 100 The Financial Times Stock Exchange 100 Index, also called the FTSE 100 Index, FTSE 100, FTSE, or, informally, the "Footsie" , is a share index of the 100 companies listed on the London Stock Exchange with (in principle) the highest marke ...
or the Dow Jones. * A negative number in financing is synonymous with "debt" and "deficit" which are also known as "being in the red". *
Interest rates An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
can be negative, when the lender is charged to deposit their money.


Other

* The numbering of
storey A storey (British English) or story (American English) is any level part of a building with a floor that could be used by people (for living, work, storage, recreation, etc.). Plurals for the word are ''storeys'' (UK) and ''stories'' (US). T ...
s in a building below the ground floor. * When playing an
audio Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to: Sound *Audio signal, an electrical representation of sound *Audio frequency, a frequency in the audio spectrum * Digital audio, representation of sou ...
file on a
portable media player A portable media player (PMP) (also including the related digital audio player (DAP)) is a portable consumer electronics device capable of storing and playing digital media such as audio, images, and video files. The data is typically stored o ...
, such as an
iPod The iPod is a discontinued series of portable media players and multi-purpose mobile devices designed and marketed by Apple Inc. The first version was released on October 23, 2001, about months after the Macintosh version of iTunes wa ...
, the screen display may show the time remaining as a negative number, which increases up to zero time remaining 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! ''Jeopardy!'' is an American game show created by Merv Griffin. The show is a quiz competition that reverses the traditional question-and-answer format of many quiz shows. Rather than being given questions, contestants are instead given genera ...
'' 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. ** In '' The Price Is Rights pricing game Buy or Sell, if an amount of money is lost that is more than the amount currently in the bank, it incurs a negative score. * The change in support for a political party between elections, known as
swing Swing or swinging may refer to: Apparatus * Swing (seat), a hanging seat that swings back and forth * Pendulum, an object that swings * Russian swing, a swing-like circus apparatus * Sex swing, a type of harness for sexual intercourse * Swing ri ...
. * A politician's approval rating. * In
video games Video games, also known as computer games, are electronic games that involves interaction with a user interface or input device such as a joystick, controller, keyboard, or motion sensing device to generate visual feedback. This feedbac ...
, 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 flexible working hours may have a negative balance on their timesheet if they have 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. * Transposing notes on an
electronic keyboard An electronic keyboard, portable keyboard, or digital keyboard is an electronic musical instrument, an electronic derivative of keyboard instruments. Electronic keyboards include synthesizers, digital pianos, stage pianos, electronic organs ...
are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one semitone down.


Arithmetic involving negative numbers

The
minus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
"−" signifies the operator for both the binary (two-
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above exam ...
)
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
of subtraction (as in ) and the unary (one-operand) operation of
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
(as in , or twice in ). 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 ). 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 may be clearer if written (even though they mean exactly the same thing formally). The subtraction expression 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


Addition

Addition of two negative numbers is very similar to addition of two positive numbers. For example, 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: In the first example, a credit of is combined with a debt of , which yields a total credit of . If the negative number has greater magnitude, then the result is negative: 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: In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus and On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that ''losing'' a debt is the same thing as ''gaining'' a credit.) Thus and


Multiplication

When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The 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 and The reason behind the first example is simple: adding three 's together yields : 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: 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 Since , the product must equal . 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 are the same as for multiplication. For example, and If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.


Negation

The negative version of a positive number is referred to as its
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
. For example, is the negation of the positive number . The sum of a number and its negation is equal to zero: That is, the negation of a positive number is the additive inverse of the number. Using
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, we may write this principle as an algebraic identity: This identity holds for any positive number . 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 is . In general, The
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of a number is the non-negative number with the same magnitude. For example, the absolute value of and the absolute value of are both equal to , and the absolute value of is .


Formal construction of negative integers

In a similar manner to
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, we can extend the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s N to the integers Z by defining integers as an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
of natural numbers (''a'', ''b''). We can extend addition and multiplication to these pairs with the following rules: We define an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
~ upon these pairs with the following rule: This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, 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, and is in fact, the prototypical example of a ring. We can also define a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
on Z by writing 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 This construction is a special case of the Grothendieck construction.


Uniqueness

The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero. Let ''x'' be a number and let ''y'' be its additive inverse. Suppose ''y′'' is another additive inverse of ''x''. By definition, x + y' = 0, \quad \text \quad x + y = 0. 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 additive inverse of ''x''. That is, ''y'' is the unique additive inverse of ''x''.


History

For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false". In Hellenistic Egypt, the Greek mathematician Diophantus in the 3rd century AD referred to an equation that was equivalent to 4x + 20 = 4 (which has a negative solution) in '' Arithmetica'', saying that the equation was absurd. For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others. Negative numbers appear for the first time in history in the ''
Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'' (九章算術, ''Jiǔ zhāng suàn-shù''), which in its present form dates from the period of the
Han Dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
( 202 BC220 AD), but may well contain much older material. The mathematician
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
(c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese
natural philosophy Natural philosophy or philosophy of nature (from Latin ''philosophia naturalis'') is the philosophical study of physics, that is, nature and the physical universe. It was dominant before the development of modern science. From the ancien ...
made it easier for the Chinese to accept the idea of negative numbers. The Chinese were able to solve simultaneous equations involving negative numbers. The ''Nine Chapters'' used red
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
to denote positive coefficients and black rods for negative. 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. Liu Hui writes: The ancient Indian '' Bakhshali Manuscript'' carried out calculations with negative numbers, using "+" as a negative sign. The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century, Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries, and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century, During the 7th century AD, negative numbers were used in India to represent debts. The
Indian mathematician Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
, in '' Brahma-Sphuta-Siddhanta'' (written c. AD 630), discussed the use of negative numbers to produce the general form
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
that remains in use today. He also found negative solutions of
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
s and gave rules regarding operations involving negative numbers and
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
, 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". In the 9th century, 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. Al-Khwarizmi in his '' Al-jabr wa'l-muqabala'' (from which the word "algebra" derives) did not use negative numbers or negative coefficients. But within fifty years,
Abu Kamil Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ ( Latinized as Auoquamel, ar, أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as ''Al-ḥāsib al-miṣrī''—lit. "the Egyptian reckoner") (c. 850 – ...
illustrated the rules of signs for expanding the multiplication (a \pm b)(c \pm d), and
al-Karaji ( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works a ...
wrote in his ''al-Fakhrī'' that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in ''
A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( fa, ابوالوفا بوزجانی or بوژگانی) (10 June 940 – 15 July 998) was a Persian mathematician a ...
''. By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes:
the product of a negative number—''al-nāqiṣ'' (loss)—by a positive number—''al-zāʾid'' (gain)—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.
In the 12th century 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." Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of '' Liber Abaci'', 1202 AD) and later as losses (in '' Fibonacci's work ''Flos''''). In the 15th century,
Nicolas Chuquet Nicolas Chuquet (; born ; died ) was a French mathematician. He invented his own notation for algebraic concepts and exponentiation. He may have been the first mathematician to recognize zero and negative numbers as exponents. In 1475, Jehan ...
, a Frenchman, used negative numbers as exponents but referred to them as "absurd numbers".
Michael Stifel Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena Univ ...
dealt with negative numbers in his
1544 __NOTOC__ Events January–June * January 13 – At Västerås, the estates of Sweden swear loyalty to King Gustav Vasa and to his heirs, ending the traditional electoral monarchy in Sweden. Gustav subsequently signs an allianc ...
AD '' Arithmetica Integra'', where he also called them ''numeri absurdi'' (absurd numbers). In 1545,
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
, in his ''Ars Magna'', provided the first satisfactory treatment of negative numbers in Europe. He did not allow negative numbers in his consideration of cubic equations, so he had to treat, for example, x^3 + a x = b separately from x^3 = a x + b (with a, b > 0 in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with complex numbers, but understandably liked them even less.) In 1748
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
, by formally manipulating complex
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
while using the square root of -1, obtained
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
of complex analysis: \cos \theta + i \sin \theta = e ^ where i = \sqrt. In 1797 AD, Carl Friedrich Gauss published a proof of the fundamental theorem of algebra but expressed his doubts at the time about "the true metaphysics of the square root of −1". However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century. In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless. In 1759 AD, the English mathematician Francis Maseres 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.


See also

* Signed zero * Additive inverse * History of zero * Integers *
Positive and negative parts In mathematics, the positive part of a real or extended real-valued function is defined by the formula : f^+(x) = \max(f(x),0) = \begin f(x) & \mbox f(x) > 0 \\ 0 & \mbox \end Intuitively, the graph of f^+ is obtained by taking the graph of f, ...
*
Rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
*
Real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
* Sign function * Sign (mathematics) * Signed number representations


References


Citations


Bibliography

* Bourbaki, Nicolas (1998). ''Elements of the History of Mathematics''. Berlin, Heidelberg, and New York: Springer-Verlag. . * Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications.


External links


Maseres' biographical information

BBC Radio 4 series ''In Our Time'', on "Negative Numbers", 9 March 2006

Endless Examples & Exercises: ''Operations with Signed Integers''


{{DEFAULTSORT:Negative And Non-Negative Numbers Chinese mathematical discoveries Elementary arithmetic Numbers