Is 0 A Negative Number

Real number that is strictly less than zero

This thermometer is indicating a negative Fahrenheit temperature (−4 °F).

In mathematics, a negative number represents an reverse.^[1] In the real number system, a negative number is a number that is less than cipher. Negative numbers are oft used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of every bit a negative asset. If a quantity, such as the charge on an electron, may have either of ii contrary senses, then one may choose to distinguish between those senses—perhaps arbitrarily—every bit positive and negative. Negative numbers are used to depict values on a scale that goes beneath zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetics for negative numbers ensure that the common-sense idea of an contrary is reflected in arithmetic. For case, −(−3) = 3 because the opposite of an opposite is the original value.

Negative numbers are usually written with a minus sign in front end. For example, −three represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". To assist tell the deviation betwixt a subtraction operation and a negative number, occasionally the negative sign is placed slightly higher than the minus sign (as a superscript). Conversely, a number that is greater than zilch is called positive; zero is usually (but not always) thought of as neither positive nor negative.^[two] The positivity of a number may be emphasized past placing a plus sign before it, eastward.g. +3. In full general, the negativity or positivity of a number is referred to equally its sign.

Every real number other than zilch is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with nothing) are referred to as integers. (Some definitions of the natural numbers exclude zero.)

In bookkeeping, amounts owed are often represented by cherry numbers, or a number in parentheses, as an alternative annotation to represent negative numbers.

Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its nowadays grade dates from the menses of the Chinese Han Dynasty (202 BC – AD 220), but may well comprise much older material.^[3] Liu Hui (c. tertiary century) established rules for adding and subtracting negative numbers.^[iv] By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further adult the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients.^[5] Prior to the concept of negative numbers, mathematicians such every bit Diophantus considered negative solutions to problems "fake" and equations requiring negative solutions were described as absurd.^[6] Western mathematicians like Leibniz (1646–1716) held that negative numbers were invalid, but notwithstanding used them in calculations.^[7] ^[viii]

Introduction [edit]

The number line [edit]

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 further 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, fifty-fifty though (positive) $8$ is greater than (positive) $v$ , written

$8 > 5$

negative $8$ is considered to be less than negative $v$ :

$-eight < -5.$

(Because, for case, if yous have £−viii, a debt of £viii, you would take less after adding, say £x, to it than if you accept £−v.) It follows that whatsoever negative number is less than any positive number, and then

$-8 < 5$ and $-5 < viii.$

Signed numbers [edit]

In the context of negative numbers, a number that is greater than zero is referred to as positive. Thus every real number other than zero is either positive or negative, while zero itself is not considered to take a sign. Positive numbers are sometimes written with a plus sign in front, e.g. $+iii$ 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 naught, while nonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number.

As the event of subtraction [edit]

Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative 3 is the result of subtracting three from nil:

$0 - 3 = -three.$

In general, the subtraction of a larger number from a smaller yields a negative issue, with the magnitude of the result beingness the difference between the two numbers. For case,

$5 - viii = -3$

since $8 - 5 = three$ .

Everyday uses of negative numbers [edit]

Sport [edit]

Negative golf game scores relative to par.

Goal difference in association football and hockey; points difference in rugby football; net run rate in cricket; golf scores relative to par.
Plus-minus differential in ice hockey: the difference in total goals scored for the team (+) and against the team (−) when a particular thespian is on the ice is the thespian'southward +/− rating. Players tin can accept a negative (+/−) rating.
Run differential in baseball game: 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 in Formula ane may be given as the difference compared to a previous lap (or sector) (such equally the previous tape, or the lap simply completed past a driver in front), and will be positive if slower and negative if faster.^[11]
In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorded,^[12] and is positive for a tailwind and negative for a headwind.^[thirteen]

Scientific discipline [edit]

Temperatures which are colder than 0 °C or 0 °F.^[14] ^[xv]
Latitudes south of the equator and longitudes west of the prime meridian.
Topographical features of the world's surface are given a elevation above ocean level, which can exist negative (e.g. the surface elevation of the Expressionless Sea or Death Valley, or the elevation of the Thames Tideway Tunnel).
Electric circuits. When a battery is continued in opposite polarity, the voltage applied is said to exist the opposite of its rated voltage. For example, a vi-volt battery continued in opposite applies a voltage of −six volts.
Ions accept a positive or negative electrical charge.
Impedance of an AM broadcast belfry used in multi-belfry directional antenna arrays, which can be positive or negative.

Finance [edit]

Fiscal statements can include negative balances, indicated either by a minus sign or past enclosing the balance in parentheses.^[16] Examples include bank account overdrafts and business losses (negative earnings).
Refunds to a credit card or debit card are a negative charge to the menu.^[17] ^[18]
The almanac percent growth in a country'southward GDP might exist negative, which is one indicator of being in a recession.^[19]
Occasionally, a rate of aggrandizement may be negative (deflation), indicating a fall in average prices.^[twenty]
The daily alter in a share price or stock market alphabetize, such as the FTSE 100 or the Dow Jones.
A negative number in financing is synonymous with "debt" and "arrears" which are also known equally "being in the crimson".
Involvement rates can be negative,^[21] ^[22] ^[23] when the lender is charged to deposit their money.

Other [edit]

Negative storey numbers in an elevator.

The numbering of storeys in a building beneath the ground floor.
When playing an sound file on a portable media player, such as an iPod, the screen brandish may testify the time remaining as a negative number, which increases upward to null time remaining at the same rate as the time already played increases from zero.
Television game shows:
- Participants on QI often stop with a negative points score.
- Teams on University Challenge accept a negative score if their starting time answers are incorrect and interrupt the question.
- Jeopardy! has a negative money score – contestants play for an amount of money and whatsoever wrong answer that costs them more than what they have now tin issue in a negative score.
- In The Price Is Right'due south pricing game Buy or Sell, if an amount of money is lost that is more than the corporeality currently in the banking concern, it incurs a negative score.
The modify in support for a political political party between elections, known every bit swing.
A politician'south blessing rating.^[24]
In video games, a negative number indicates loss of life, harm, a score punishment, or consumption of a resource, depending on the genre of the simulation.
Employees with flexible working hours may accept a negative balance on their timesheet if they take worked fewer full hours than contracted to that indicate. Employees may be able to take more than their annual vacation assart in a year, and carry frontwards a negative balance to the next year.
Transposing notes on an electronic keyboard are shown on the brandish with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one semitone down.

Arithmetic involving negative numbers [edit]

The minus sign "−" signifies the operator for both the binary (ii-operand) operation of subtraction (as in $y - z$ ) and the unary (ane-operand) operation of negation (equally in $- x$ , or twice in $-(- 10)$ ). A special example of unary negation occurs when it operates on a positive number, in which example the outcome is a negative number (equally in $-5$ ).

The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes but one interpretation or the other possible for each "−". However, it can lead to defoliation and exist difficult for a person to empathize an expression when operator symbols announced side by side to one another. A solution can be to parenthesize the unary "−" along with its operand.

For instance, the expression $7 + -5$ may be clearer if written $7 + (-5)$ (even though they mean exactly the same thing formally). The subtraction expression $7 - v$ is a different expression that doesn't stand for the aforementioned operations, but it evaluates to the same event.

Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers every bit in^[25]

$- 2 + - 5$ gives $- seven$ .

Addition [edit]

A visual representation of the improver of positive and negative numbers. Larger balls represent numbers with greater magnitude.

Addition of two negative numbers is very like to addition of 2 positive numbers. For example,

$(-3) + (-5) = -eight$ .

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 every bit positive quantities being subtracted. For example:

$8 + (-3) = viii - 3 = 5$ and $(-2) + 7 = 7 - 2 = v$ .

In the offset case, a credit of $8$ is combined with a debt of $three$ , which yields a full credit of $5$ . If the negative number has greater magnitude, and then the result is negative:

$(-eight) + 3 = three - 8 = -v$ and $2 + (-7) = two - vii = -5$ .

Here the credit is less than the debt, and then the net outcome is a debt.

Subtraction [edit]

As discussed above, it is possible for the subtraction of two not-negative numbers to yield a negative answer:

$5 - 8 = -iii$

In general, subtraction of a positive number yields the aforementioned upshot every bit the addition of a negative number of equal magnitude. Thus

$5 - eight = 5 + (-eight) = -3$

and

$(-three) - 5 = (-3) + (-5) = -8$

On the other hand, subtracting a negative number yields the same result as the add-on a positive number of equal magnitude. (The thought is that losing a debt is the same affair as gaining a credit.) Thus

$3 - (-five) = three + 5 = eight$

and

$(-5) - (-8) = (-5) + viii = three$ .

Multiplication [edit]

When multiplying numbers, the magnitude of the production is always but the product of the two magnitudes. The sign of the product is adamant past the post-obit rules:

The product of one positive number and one negative number is negative.
The production of 2 negative numbers is positive.

Thus

$(-two) \times three = -6$

and

$(-ii) \times (-3) = half dozen$ .

The reason behind the first example is simple: calculation three $-2$ 's together yields $-half dozen$ :

$(-2) \times 3 = (-two) + (-ii) + (-2) = -6$ .

The reasoning behind the second instance is more complicated. The thought once again is that losing a debt is the same matter as gaining a credit. In this case, losing two debts of three each is the aforementioned as gaining a credit of 6:

$(-ii$ debts $) \times (-3$ each $) = +six$ credit.

The convention that a product of 2 negative numbers is positive is also necessary for multiplication to follow the distributive law. In this case, we know that

$(-2) \times (-3) + 2 \times (-three) = (-2 + 2) \times (-3) = 0 \times (-3) = 0$ .

Since $2 \times (-three) = -6$ , the product $(-two) \times (-3)$ must equal $half dozen$ .

These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign of a every bit follows:

if a is positive, and 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 production of two negative numbers is a positive number tin be observed in the analysis of complex numbers.

Partitioning [edit]

The sign rules for division are the same equally for multiplication. For example,

$8 \div (-2) = -4$ ,

$(-viii) \div 2 = -4$ ,

and

$(-8) \div (-2) = 4$ .

If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.

Negation [edit]

The negative version of a positive number is referred to equally its negation. For instance, $-iii$ is the negation of the positive number $3$ . The sum of a number and its negation is equal to zero:

$iii + (-3) = 0$ .

That is, the negation of a positive number is the additive inverse of the number.

Using algebra, we may write this principle as an algebraic identity:

$x + (- 10) = 0$ .

This identity holds for whatsoever positive number $x$ . It tin can be fabricated to hold for all real numbers by extending the definition of negation to include nothing and negative numbers. Specifically:

The negation of 0 is 0, and
The negation of a negative number is the corresponding positive number.

For case, the negation of $-iii$ is $+3$ . In general,

$-(- x) = 10$ .

The absolute value of a number is the not-negative number with the same magnitude. For case, the accented value of $-three$ and the absolute value of $3$ are both equal to $3$ , and the absolute value of $0$ is $0$ .

Formal structure of negative integers [edit]

In a like manner to rational numbers, we tin can extend the natural numbers N to the integers Z past defining integers equally an ordered pair of natural numbers (a, b). We tin extend improver and multiplication to these pairs with the post-obit rules:

$(a, b) + (c, d) = (a + c, b + d)$

$(a, b) \times (c, d) = (a \times c + b \times d, a \times d + b \times c)$

We define an equivalence relation ~ upon these pairs with the following dominion:

(a, b) ~ (c, d) if and only if a + d = b + c.

This equivalence relation is compatible with the improver and multiplication defined above, and we may ascertain Z to be the caliber set N²/~, i.e. nosotros place two pairs (a, b) and (c, d) if they are equivalent in the to a higher place sense. Note that Z, equipped with these operations of improver and multiplication, is a ring, and is in fact, the prototypical example of a ring.

We can likewise define a total order on Z past writing

$(a, b) \leq (c, d) if and merely if a + d \leq b + c$ .

This will lead to an additive zero of the class (a, a), an condiment inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a + 1, a), and a definition of subtraction

$(a, b) - (c, d) = (a + d, b + c)$ .

This construction is a special case of the Grothendieck construction.

Uniqueness [edit]

The additive inverse of a number is unique, as is shown by the post-obit proof. As mentioned above, an condiment inverse of a number is defined as a value which when added to the number yields cypher.

Permit x be a number and let y be its condiment inverse. Suppose y′ is some other additive inverse of x. Past definition,

10+y'=0,\quad {\text{and}}\quad ten+y=0.

And then, x + y′ = x + y. Using the law of cancellation for improver, information technology is seen that y′ = y. Thus y is equal to any other additive changed of x. That is, y is the unique additive inverse of x.

History [edit]

For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number corporeality of a concrete object, for example "minus-three apples", and negative solutions to problems were considered "false".

In Hellenistic Arab republic of egypt, the Greek mathematician Diophantus in the 3rd century Advertizement referred to an equation that was equivalent to $4x+twenty=iv$ (which has a negative solution) in Arithmetica, maxim that the equation was cool.^[26] 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 business relationship of others.^[27]

Negative numbers appear for the commencement fourth dimension in history in the Nine Chapters on the Mathematical Art (九章算術, Jiǔ zhāng suàn-shù), which in its present class dates from the catamenia of the Han Dynasty (202 BC – 220 Advertizement), but may well comprise much older textile.^[3] The mathematician Liu 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 Chinese natural philosophy fabricated it easier for the Chinese to take the idea of negative numbers.^[four] The Chinese were able to solve simultaneous equations involving negative numbers. The Ix Chapters used red counting rods to denote positive coefficients and black rods for negative.^[4] ^[28] This organisation is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, bookkeeping, and commerce, wherein red numbers announce 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. Cerise counting rods are positive, black counting rods are negative.^[4]

The ancient Indian Bakhshali Manuscript carried out calculations with negative numbers, using "+" equally a negative sign.^[29] The engagement of the manuscript is uncertain. LV Gurjar dates it no later on than the quaternary century,^[30] Hoernle dates information technology between the 3rd and 4th centuries, Ayyangar and Pingree dates information technology to the 8th or ninth centuries,^[31] and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century,^[32]

During the 7th century Advertising, negative numbers were used in Republic of india to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written c. AD 630), discussed the use of negative numbers to produce the general form quadratic formula that remains in employ today.^[26] He besides found negative solutions of quadratic equations and gave rules regarding operations involving negative numbers and nix, such equally "A debt cutting off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt." He called positive numbers "fortunes", zero "a nil", and negative numbers "debts".^[33] ^[34]

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 menstruum remained timid.^[v] Al-Khwarizmi in his Al-jabr wa'l-muqabala (from which the word "algebra" derives) did non employ negative numbers or negative coefficients.^[v] Simply inside fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication $(a\pm b)(c\pm d)$ ,^[35] and al-Karaji wrote in his al-Fakhrī that "negative quantities must be counted as terms".^[5] In the tenth 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.^[35]

By the 12th century, al-Karaji's successors were to state the full general rules of signs and use them to solve polynomial divisions.^[5] Equally 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 nosotros subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if nosotros subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the rest is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if nosotros subtract a negative number from an empty power, the remainder is the same positive number.^[v]

In the 12th century in Republic of india, Bhāskara Two 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 example not to be taken, for information technology is inadequate; people do not approve of negative roots."

Fibonacci allowed negative solutions in fiscal issues where they could be interpreted as debits (chapter xiii of Liber Abaci, 1202 AD) and later as losses (in Fibonacci'southward work Flos ).

In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers equally exponents^[36] but referred to them equally "cool numbers".^[37]

Michael Stifel dealt with negative numbers in his 1544 Advertisement Arithmetica Integra, where he besides chosen them numeri absurdi (absurd numbers).

In 1545, Gerolamo Cardano, in his Ars Magna, provided the first satisfactory treatment of negative numbers in Europe.^[26] He did non let negative numbers in his consideration of cubic equations, so he had to care for, for instance, $ten^{3}+ax=b$ separately from $x^{3}=ax+b$ (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 circuitous numbers, simply understandably liked them even less.)

In 1748 Leonhard Euler, past formally manipulating complex power series while using the square root of $-1,$ obtained Euler'south formula of complex analysis:^[38]

\cos \theta +i\sin \theta =e^{i\theta }

where $i={\sqrt {-1}}.$

In 1797 AD, Carl Friedrich Gauss published a proof of the fundamental theorem of algebra but expressed his doubts at the fourth dimension most "the true metaphysics of the square root of −1".^[39]

However, European mathematicians, for the well-nigh part, resisted the concept of negative numbers until the middle of the 19th century.^[40] In the 18th century information technology was common do to ignore any negative results derived from equations, on the supposition that they were meaningless.^[41] In 1759 AD, the English mathematician Francis Maseres wrote that negative numbers "darken the very whole doctrines of the equations and make nighttime of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers were nonsensical.^[42]

Meet also [edit]

Signed zippo
Additive inverse
History of goose egg
Integers
Positive and negative parts
Rational numbers
Real numbers
Sign function
Sign (mathematics)
Signed number representations

References [edit]

Citations [edit]

^ "Integers are the set of whole numbers and their opposites.", Richard W. Fisher, No-Nonsense Algebra, 2d Edition, Math Essentials, ISBN 978-0999443330
^ The convention that cipher is neither positive nor negative is not universal. For example, in the French convention, cypher is considered to be both positive and negative. The French words positif and négatif mean the same as English "positive or nix" and "negative or zero" respectively.
^ ^a ^b Struik, pages 32–33. "In these matrices we detect negative numbers, which appear hither for the showtime time in history."
^ ^a ^b ^c ^d Hodgkin, Luke (2005). A History of Mathematics: From Mesopotamia to Modernity . Oxford University Press. p. 88. ISBN978-0-19-152383-0. Liu is explicit on this; at the point where the Nine Chapters give a detailed and helpful 'Sign Dominion'
^ ^a ^b ^c ^d ^e ^f Rashed, R. (30 June 1994). The Development of Arabic Mathematics: Between Arithmetics and Algebra. Springer. pp. 36–37. ISBN9780792325659.
^ Diophantus, Arithmetica.
^ Kline, Morris (1972). Mathematical Thought from Aboriginal to Modernistic Times. Oxford Academy Press, New York. p. 252.
^ Martha Smith. "History of Negative Numbers".
^ "Saracens bacon cap alienation: Premiership champions will non contest sanctions". BBC Sport . Retrieved 18 Nov 2019. Mark McCall's side have later on dropped from third to bottom of the Premiership with −22 points
^ "Bolton Wanderers 1−0 Milton Keynes Dons". BBC Sport . Retrieved 30 Nov 2019. Simply in the tertiary minute of stoppage time, the striker turned in Luke White potato'south cross from eight yards to earn a third direct League One win for Hill's side, who started the campaign on −12 points later on going into administration in May.
^ "Glossary". Formula1.com. Retrieved 30 November 2019. Delta time: A term used to describe the time divergence betwixt two dissimilar laps or two different cars. For example, there is usually a negative delta between a commuter'south best practice lap fourth dimension and his best qualifying lap time because he uses a low fuel load and new tyres.
^ "BBC Sport - Olympic Games - London 2012 - Men's Long Jump : Athletics - Results". 5 Baronial 2012. Archived from the original on 5 August 2012. Retrieved 5 December 2018.
^ "How Air current Assistance Works in Track & Field". elitefeet.com. three July 2008. Retrieved eighteen November 2019. Current of air assist is normally expressed in meters per second, either positive or negative. A positive measurement ways that the air current is helping the runners and a negative measurement means that the runners had to piece of work against the wind. And then, for example, winds of −ii.2m/s and +1.9m/s are legal, while a wind of +2.1m/southward is also 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 (−)
^ Forbes, Robert B. (6 January 1975). Contributions to the Geology of the Bering Sea Basin and Adjacent Regions: Selected Papers from the Symposium on the Geology and Geophysics of the Bering Ocean Region, on the Occasion of the Inauguration of the C. T. Elvey Building, University of Alaska, June 26-28, 1970, and from the 2d International Symposium on Chill Geology Held in San Francisco, February 1-4, 1971. Geological Society of America. p. 194. ISBN9780813721514.
^ Wilks, Daniel S. (half dozen Jan 2018). Statistical Methods in the Atmospheric Sciences. Academic Press. p. 17. ISBN9780123850225.
^ Carysforth, Carol; Neild, Mike (2002), Double Award, Heinemann, p. 375, ISBN978-0-435-44746-five
^ Gerver, Robert K.; Sgroi, Richard J. (2010), Financial Algebra, Student Edition, Cengage Learning, p. 201, ISBN978-0-538-44967-0
^ What Does a Negative Number on a Credit Card Statement Hateful?, Pocketsense, 27 October 2018.
^ "UK economy shrank at finish of 2012". BBC News. 25 January 2013. Retrieved five December 2018.
^ "Commencement negative inflation figure since 1960". The Independent. 21 April 2009. Archived from the original on 18 June 2022. Retrieved five December 2018.
^ "ECB imposes negative interest charge per unit". BBC News. five June 2014. Retrieved v Dec 2018.
^ Lynn, Matthew. "Think negative involvement rates can't happen hither? Think again". MarketWatch . Retrieved five December 2018.
^ "Swiss involvement rate to plow negative". BBC News. 18 December 2014. Retrieved 5 Dec 2018.
^ Wintour, Patrick (17 June 2014). "Popularity of Miliband and Clegg falls to lowest levels recorded past ICM poll". Retrieved 5 December 2018 – via www.theguardian.com.
^ Grant P. Wiggins; Jay McTighe (2005). Agreement past design . ACSD Publications. p. 210. ISBNane-4166-0035-3.
^ ^a ^b ^c Needham, Joseph; Wang, Ling (1995) [1959]. Scientific discipline and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Globe (reprint ed.). Cambridge: Cambridge Academy Press. p. ninety. ISBN0-521-05801-5.
^ Heath, Thomas Fifty. (1897). The works of Archimedes. Cambridge University Press. pp. cxxiii.
^ Needham, Joseph; Wang, Ling (1995) [1959]. Scientific discipline and Civilisation in People's republic of china: Volume 3; Mathematics and the Sciences of the Heavens and the World (reprint ed.). Cambridge: Cambridge University Printing. pp. 90–91. ISBN0-521-05801-5.
^ Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modernistic Science–from the Babylonians to the Mayas. New York: Simon & Schuster. ISBN 0-684-83718-8. Folio 65.
^ Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 24 July 2007.
^ Hayashi, Takao (2008), "Bakhshālī Manuscript", in Helaine Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, vol. 1, Springer, p. B2, ISBN9781402045592
^ Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon & Schuster. ISBN 0-684-83718-8. Page 65–66.
^ Colva M. Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on ix March 2006.
^ Noesis Transfer and Perceptions of the Passage of Time, ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta'south great work, all the necessary rules for algebra, including the 'dominion of signs', were stipulated, only in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (=fortunes) is used to correspond positive numbers, whereas 'rina' (=debts) were negative".
^ ^a ^b Bin Ismail, Mat Rofa (2008), "Algebra in Islamic Mathematics", in Helaine Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, vol. 1 (2nd ed.), Springer, p. 115, ISBN9781402045592
^ Flegg, Graham; Hay, C.; Moss, B. (1985), Nicolas Chuquet, Renaissance Mathematician: a study with extensive translations of Chuquet'south mathematical manuscript completed in 1484, D. Reidel Publishing Co., p. 354, ISBN9789027718723 .
^ Johnson, Art (1999), Famous Problems and Their Mathematicians, Greenwood Publishing Group, p. 56, ISBN9781563084461 .
^ Euler, Leonard (1748). Introductio in Analysin Infinitorum [Introduction to the Assay of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
^ Gauss, Carl Friedrich (1799) "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse." [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second caste.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)
^ Martinez, Alberto (2014). Negative Math. Princeton University Press. pp. fourscore–109.
^ Martinez, Alberto A. (2006). Negative Math: How Mathematical Rules Tin can Be Positively Aptitude. Princeton University Press. a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.
^ Maseres, Francis (1758). A dissertation on the apply of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle. Quoting from Maseres' piece of work: If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no pregnant or significance, thus if it be said that the square of −5, or the production of −5 into −five, is equal to +25, such an exclamation must either signify no more than 5 times five is equal to 25 without any regard for the signs, or it must exist mere nonsense or unintelligible jargon.

Bibliography [edit]

Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN three-540-64767-8.
Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.

External links [edit]

Maseres' biographical information
BBC Radio 4 series In Our Time, on "Negative Numbers", nine March 2006
Endless Examples & Exercises: Operations with Signed Integers
Math Forum: Inquire Dr. Math FAQ: Negative Times a Negative