In
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 real ...
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 n ...
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. 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
The Fahrenheit scale () is a temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686–1736). It uses the degree Fahrenheit (symbol: °F) as the unit. Several accounts of how he originally defined his ...
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, resulti ...
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
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
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, resulti ...
(as a
superscript
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
). 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
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
.
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 n ...
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 language ...
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 imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...
(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 o ...
(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 trea ...
were describing the use of negative numbers.
Islamic mathematicians
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...
further developed the rules of subtracting and multiplying negative numbers and solved problems with negative
coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
.
Prior to the concept of negative numbers, mathematicians such as
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
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:
![Number-line](https://upload.wikimedia.org/wikipedia/commons/9/93/Number-line.svg)
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 real ...
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, result ...
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
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
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
Goal difference, goal differential or points difference is a form of tiebreaker used to rank sport teams which finish on equal points in a league competition. Either "goal difference" or "points difference" is used, depending on whether matches ar ...
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
Hockey is a term used to denote a family of various types of both summer and winter team sports which originated on either an outdoor field, sheet of ice, or dry floor such as in a gymnasium. While these sports vary in specific rules, numbers o ...
; 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
Net run rate (NRR) is a statistical method used in analysing teamwork and/or performance in cricket. It is the most commonly used method of ranking teams with equal points in limited overs league competitions, similar to goal difference in foo ...
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 striki ...
;
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 hock ...
: 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
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 tea ...
: 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), competiti ...
events, such as
sprint races, the
hurdles
Hurdling is the act of jumping over an obstacle at a high speed or in a sprint. In the early 19th century, hurdlers ran at and jumped over each hurdle (sometimes known as 'burgles'), landing on both feet and checking their forward motion. Today, ...
, 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 gr ...
, the
wind assistance
In track and field, wind assistance is the benefit that an athlete receives during a race or event as registered by a wind gauge. Wind is one of many forms of weather that can affect sport.
Due to a tailwind helping to enhance the speed of the at ...
is measured and recorded, and is positive for a
tailwind
A tailwind is a wind that blows in the direction of travel of an object, while a headwind blows against the direction of travel. A tailwind increases the object's speed and reduces the time required to reach its destination, while a headwind has ...
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 pol ...
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 letter l ...
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 c ...
.
*
Topographical
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary sci ...
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 abou ...
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 standardised g ...
, 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 Bank ...
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 Highest temperature recorded on Earth, hottest place on Earth.
Death Valley's Badwater Basin is the ...
, or the elevation of the
Thames Tideway Tunnel
The Thames Tideway Tunnel is a combined sewer under construction running mostly under Tideway, the tidal section (estuary) of the River Thames across Inner London to capture, store and convey almost all the raw sewage and rainwater that curren ...
).
*
Electrical circuits
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...
. 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
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conven ...
have a positive or negative electrical charge.
*
Impedance of an AM broadcast tower used in multi-tower
directional antenna
A directional antenna or beam antenna is an antenna which radiates or receives greater power in specific directions allowing increased performance and reduced interference from unwanted sources. Directional antennas provide increased performance ...
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 Earnings are the net benefits of a corporation's operation. Earnings is also the amount on which corporate tax is due. For an analysis of specific aspects of corporate operations several more specific terms are used as EBIT (earnings before interes ...
).
* 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 o ...
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 u ...
are a negative charge to the card.
* The annual percentage growth in a country's
GDP
Gross domestic product (GDP) is a monetary measure of the market value of all the final goods and services produced and sold (not resold) in a specific time period by countries. Due to its complex and subjective nature this measure is often ...
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 reductio ...
may be negative (
deflation
In economics, deflation is a decrease in the general price level of goods and services. Deflation occurs when the inflation rate falls below 0% (a negative inflation rate). Inflation reduces the value of currency over time, but sudden 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 the ...
, 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 market ...
or the
Dow Jones Dow Jones is a combination of the names of business partners Charles Dow and Edward Jones.
Dow Jones & Company
Dow, Jones and Charles Bergstresser founded Dow Jones & Company in 1882. That company eventually became a subsidiary of News Corp, and ...
.
* 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
![Elevator Negative Floor Numbers in Ireland (16785350923)](https://upload.wikimedia.org/wikipedia/commons/6/6e/Elevator_Negative_Floor_Numbers_in_Ireland_%2816785350923%29.jpg)
* 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 sound ...
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 ...
, 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
A game show is a genre of broadcast viewing entertainment (radio, television, internet, stage or other) where contestants compete for a reward. These programs can either be participatory or demonstrative and are typically directed by a host, sh ...
:
** Participants on ''
QI'' often finish with a negative points score.
** Teams on ''
University Challenge
''University Challenge'' is a British television quiz programme which first aired in 1962. ''University Challenge'' aired for 913 episodes on ITV from 21 September 1962 to 31 December 1987, presented by quizmaster Bamber Gascoigne. The BBC ...
'' 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 Right
''The Price Is Right'' is a television game show franchise created by Bob Stewart, originally produced by Mark Goodson and Bill Todman; currently it is produced and owned by Fremantle. The franchise centers on television game shows, but also inc ...
s 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.
* A politician's
approval rating
An opinion poll, often simply referred to as a survey or a poll (although strictly a poll is an actual election) is a human research survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions ...
.
* 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, game controller, controller, computer keyboard, keyboard, or motion sensing device to gener ...
, 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
Flextime (also spelled flexitime (British English, BE) or flex-time) is a flexible hours Schedule (workplace), schedule that allows workers to alter their workday and decide/adjust their start and finish times. In contrast to traditional Wage labo ...
may have a negative balance on their
timesheet
A timesheet (or time sheet) is a method for recording the amount of a worker's time spent on each job. Traditionally a sheet of paper with the data arranged in tabular format, a timesheet is now often a digital document or spreadsheet. The time c ...
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 an ...
are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
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, resulti ...
"−" 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 examp ...
)
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-Ma ...
of
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
(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
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
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
![AdditionRules](https://upload.wikimedia.org/wikipedia/commons/5/51/AdditionRules.svg)
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
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
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 mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
. 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
.
Division
The sign rules for
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
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
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
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 a ...
, we may write this principle as an
algebraic identity
In mathematics, an identity is an equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of ...
:
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), an ...
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 ration ...
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 n ...
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 relation ...
~ 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
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, 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
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
'' of (''a'', ''b'') of the form (''b'', ''a''), a multiplicative unit of the form (''a'' + 1, ''a''), and a definition of
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
This construction is a special case of the
Grothendieck construction The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory.
Definition
Let F\colon \mathcal \rightarrow \mathbf be a functor from any small category to the category of sma ...
.
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,
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
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
mathematician
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
in the 3rd century AD referred to an equation that was equivalent to
(which has a negative solution) in ''
Arithmetica
''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate e ...
'', 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 imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...
(
202 BC
__NOTOC__
Year 202 BC was a year of the Roman calendar, pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Geminus and Nero (or, less frequently, year 552 ''Ab urbe condita''). The denomination 202 BC for this ye ...
–
220 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 o ...
(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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior throu ...
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 fo ...
to denote positive
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s 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 trea ...
, 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, gr ...
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 (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s and gave rules regarding operations involving negative numbers and
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, 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
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...
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
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
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
,
and
al-Karaji
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian people, Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal sur ...
wrote in his ''al-Fakhrī'' that "negative quantities must be counted as terms".
In the 10th century,
Abū al-Wafā' al-Būzjānī
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 ...
considered debts as negative numbers in ''
''.
By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve
polynomial division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, becaus ...
s.
As
al-Samaw'al
Al-Samawʾal ibn Yaḥyā al-Maghribī ( ar, السموأل بن يحيى المغربي, ; c. 1130 – c. 1180), commonly known as Samau'al al-Maghribi, was a mathematician, astronomer and physician. Born to a Jewish family, he concealed his c ...
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
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...
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
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of ''
Liber Abaci
''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.
''Liber Abaci'' was among the first Western books to describe ...
'',
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 ...
, a Frenchman, used negative numbers as
exponents
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
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 Universit ...
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
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 Universit ...
'', 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 equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
s, so he had to treat, for example,
separately from
(with
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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, 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 in ma ...
, 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 const ...
while using the square root of
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 for an ...
of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
:
where
In
1797 AD
Events
January–March
* January 3 – The Treaty of Tripoli, a peace treaty between the United States and Ottoman Tripolitania, is signed at Algiers (''see also'' 1796).
* January 7 – The parliament of the Cisalpine Republ ...
,
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
published a proof of the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomial ...
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
In Great Britain, this year was known as the ''Annus Mirabilis'', because of British victories in the Seven Years' War.
Events
January–March
* January 6 – George Washington marries Martha Dandridge Custis.
* January 11 &ndas ...
AD, the English mathematician
Francis Maseres
Francis Maseres (15 December 1731 – 19 May 1824) was an English lawyer. He is known as attorney general of the Province of Quebec, judge, mathematician, historian, member of the Royal Society, and cursitor baron of the exchequer.
Biography
F ...
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
Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
*
Additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
*
History of 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 multiplying digits to the left of 0 by the radix, usual ...
*
Integers
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 language o ...
*
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 rationa ...
*
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 real ...
*
Sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
*
Sign (mathematics)
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
*
Signed number representations
In computing, signed number representations are required to encode negative numbers in binary number systems.
In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU regis ...
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 informationBBC Radio 4 series ''In Our Time'', on "Negative Numbers", 9 March 2006Endless Examples & Exercises: ''Operations with Signed Integers''
{{DEFAULTSORT:Negative And Non-Negative Numbers
Chinese mathematical discoveries
Elementary arithmetic
Numbers