This thermometer is indicating a negativeFahrenheit temperature (−4 °F).
Inmathematics, anegative number is theopposite of a positivereal number.[1] Equivalently, a negative number is a real number that isless thanzero. Negative numbers are often used to represent themagnitude of a loss or deficiency. Adebt 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—aspositive andnegative. Negative numbers are used to describe values on a scale that goes below zero, such as theCelsius andFahrenheit 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 aminus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced and read as "minus three" or "negative three". Conversely, a number that is greater than zero is calledpositive; zero is usually (but not always) thought of as neither positive nornegative.[2] 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 itssign.
Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to asnatural numbers (i.e., 0, 1, 2, 3, ...), while the positive and negative whole numbers (together with zero) are referred to asintegers. (Some definitions of the natural numbers exclude zero.)
Inbookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.
Negative numbers were used in theNine Chapters on the Mathematical Art, which in its present form dates from the period of the ChineseHan dynasty (202 BC – AD 220), but may well contain much older material.[3]Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers.[4] By the 7th century, Indian mathematicians such asBrahmagupta were describing the use of negative numbers.Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negativecoefficients.[5] Prior to the concept of negative numbers, mathematicians such asDiophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd.[6] Western mathematicians likeLeibniz held that negative numbers were invalid, but still used them in calculations.[7][8]
The relationship between negative numbers, positive numbers, and zero is often expressed in the form of anumber line:
The number line
Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are lesser. 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)8 is greater than (positive)5, written
8 > 5
negative8 is considered to be less than negative5:
In the context of negative numbers, a number that is greater than zero is referred to aspositive. Thus everyreal number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with aplus sign in front, e.g.+3 denotes a positive three.
Because zero is neither positive nor negative, the termnonnegative is sometimes used to refer to a number that is either positive or zero, whilenonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number.
Negative numbers can be thought of as resulting from thesubtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero:
0 − 3 = −3.
In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example,
Plus-minus differential inice hockey: the difference in total goals scored for the team (+) and against the team (−) when a particular player is on the ice is the player's +/− rating. Players can have a negative (+/−) rating.
Run differential inbaseball: 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.[9][10]
Lap (or sector) times inFormula 1 may be given as the difference compared to a previous lap (or sector) (such as the previous record, or the lap just completed by a driver in front), and will be positive if slower and negative if faster.[11]
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-towerdirectional antenna arrays, which can be positive or negative.
Financial statements can include negative balances, indicated either by a minus sign or by enclosing the balance in parentheses.[16] Examples include bank accountoverdrafts and business losses (negativeearnings).
The annual percentage growth in a country'sGDP might be negative, which is one indicator of being in arecession.[17]
Occasionally, a rate ofinflation may be negative (deflation), indicating a fall in average prices.[18]
The numbering ofstories in a building below the ground floor.
When playing anaudio file on aportable media player, such as aniPod, 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.
Participants onQI often finish with a negative points score.
Teams onUniversity Challenge have a negative score if their first answers are incorrect and interrupt the question.
Jeopardy! has a negative money score – contestants play for an amount of money and any incorrect answer that costs them more than what they have now can result in a negative score.
InThe Price Is Right'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 asswing.
Invideo games, a negative number indicates loss of life, damage, a score penalty, or consumption of a resource, depending on the genre of the simulation.
Employees withflexible working hours may have a negative balance on theirtimesheet 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 anelectronic keyboard are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for onesemitone down.
Theminus sign "−" signifies theoperator for both the binary (two-operand)operation ofsubtraction (as iny −z) and the unary (one-operand) operation ofnegation (as in−x, or twice in−(−x)). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in−5).
The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.
For example, the expression7 + −5 may be clearer if written7 + (−5) (even though they mean exactly the same thing formally). Thesubtraction expression7 − 5 is a different expression that doesn't represent the same operations, but it evaluates to the same result.
Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in[23]
A visual representation of the addition of positive and negative numbers. Larger balls represent numbers with greater magnitude.
Addition of two negative numbers is very similar to addition of two positive numbers. For example,
(−3) + (−5) = −8.
The idea is that two debts can be combined into a single debt of greater magnitude.
When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:
8 + (−3) = 8 − 3 = 5 and (−2) + 7 = 7 − 2 = 5.
In the first example, a credit of8 is combined with a debt of3, which yields a total credit of5. If the negative number has greater magnitude, then the result is negative:
(−8) + 3 = 3 − 8 = −5 and 2 + (−7) = 2 − 7 = −5.
Here the credit is less than the debt, so the net result is a debt.
As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:
5 − 8 = −3
In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus
5 − 8 = 5 + (−8) = −3
and
(−3) − 5 = (−3) + (−5) = −8
On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is thatlosing a debt is the same thing asgaining a credit.) Thus
A multiplication by a negative number can be seen as a change of direction of thevector ofmagnitude equal to theabsolute value of the product of the factors.
When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. Thesign 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
(−2) × 3 = −6
and
(−2) × (−3) = 6.
The reason behind the first example is simple: adding three−2's together yields−6:
(−2) × 3 = (−2) + (−2) + (−2) = −6.
The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:
(−2 debts) × (−3 each) = +6 credit.
The convention that a product of two negative numbers is positive is also necessary for multiplication to follow thedistributive law. In this case, we know that
The negative version of a positive number is referred to as itsnegation. For example,−3 is the negation of the positive number3. Thesum of a number and its negation is equal to zero:
3 + (−3) = 0.
That is, the negation of a positive number is theadditive inverse of the number.
This identity holds for any positive numberx. 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−3 is+3. In general,
−(−x) = x.
Theabsolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of−3 and the absolute value of3 are both equal to3, and the absolute value of0 is0.
In a similar manner torational numbers, we can extend thenatural numbers to the integers by defining integers as anordered pair of natural numbers (a,b). We can extend addition and multiplication to these pairs with the following rules:
This equivalence relation is compatible with the addition and multiplication defined above, and we may define to be thequotient set, i.e. we identify two pairs (a,b) and (c,d) if they are equivalent in the above sense. Note that, equipped with these operations of addition and multiplication, is aring, and is in fact, the prototypical example of a ring.
This will lead to anadditive zero of the form (a,a), anadditive inverse of (a,b) of the form (b,a), a multiplicative unit of the form (a + 1,a), and a definition ofsubtraction
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.
Letx be a number and lety be its additive inverse. Supposey′ is another additive inverse ofx. By definition,
And so,x +y′ =x +y. Using the law of cancellation for addition, it is seen thaty′ =y. Thusy is equal to any other additive inverse ofx. That is,y is the unique additive inverse ofx.
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".
InHellenistic Egypt, theGreek mathematicianDiophantus in the 3rd century AD referred to an equation that was equivalent to (which has a negative solution) inArithmetica, saying that the equation was absurd.[24] 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.[25]
Negative numbers appear for the first time in history in theNine Chapters on the Mathematical Art (九章算術,Jiǔ zhāng suàn-shù), which in its present form dates from theHan period, but may well contain much older material.[3] The mathematicianLiu Hui (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 Chinesenatural philosophy made it easier for the Chinese to accept the idea of negative numbers.[4] The Chinese were able to solve simultaneous equations involving negative numbers. TheNine Chapters used redcounting rods to denote positivecoefficients and black rods for negative.[4][26] 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:
Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative.[4]
The ancient IndianBakhshali Manuscript carried out calculations with negative numbers, using "+" as a negative sign.[27] The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century,[28] Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries,[29] and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century.[30]
By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solvepolynomial divisions.[5] Asal-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.[5]
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 ofLiber Abaci, 1202) and later as losses (inFlos, 1225).
In the 15th century,Nicolas Chuquet, a Frenchman, used negative numbers asexponents[32] but referred to them as "absurd numbers".[33]
In 1545,Gerolamo Cardano, in hisArs Magna, provided the first satisfactory treatment of negative numbers in Europe.[24] He did not allow negative numbers in his consideration ofcubic equations, 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 withcomplex numbers, but understandably liked them even less.)
^"Integers are the set of whole numbers and their opposites.", Richard W. Fisher, No-Nonsense Algebra, 2nd Edition, Math Essentials,ISBN978-0999443330
^The convention that zero is neither positive nor negative is not universal. For example, in the French convention, zero is considered to beboth positive and negative. The French wordspositif andnégatif mean the same as English "positive or zero" and "negative or zero" respectively.
^abStruik, pages 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
^"Bolton Wanderers 1−0 Milton Keynes Dons".BBC Sport. Retrieved30 November 2019.But in the third minute of stoppage time, the striker turned in Luke Murphy's cross from eight yards to earn a third straight League One win for Hill's side, who started the campaign on −12 points after going into administration in May.
^"Glossary". Formula1.com. Retrieved30 November 2019.Delta time: A term used to describe the time difference between two different laps or two different cars. For example, there is usually a negative delta between a driver's best practice lap time and his best qualifying lap time because he uses a low fuel load and new tyres.
^"How Wind Assistance Works in Track & Field".elitefeet.com. 3 July 2008. Retrieved18 November 2019.Wind assistance is normally expressed in meters per second, either positive or negative. A positive measurement means that the wind is helping the runners and a negative measurement means that the runners had to work against the wind. So, for example, winds of −2.2m/s and +1.9m/s are legal, while a wind of +2.1m/s is too much assistance and considered illegal. The terms "tailwind" and "headwind" are also frequently used. A tailwind pushes the runners forward (+) while a headwind pushes the runners backwards (−)
^Teresi, Dick. (2002).Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon & Schuster.ISBN0-684-83718-8. Page 65.
^Pearce, Ian (May 2002)."The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved24 July 2007.
^Teresi, Dick. (2002).Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon & Schuster.ISBN0-684-83718-8. Page 65–66.
^abBin Ismail, Mat Rofa (2008), "Algebra in Islamic Mathematics", inHelaine Selin (ed.),Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, vol. 1 (2nd ed.), Springer, p. 115,ISBN9781402045592
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