Negative number

Unit Circle on a Complex Plane
containing +/-, real/imaginary numbers

In mathematics, a negative number is a real number that is less than zero. Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal. They are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. 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 thing. Negative numbers are usually written with a minus sign at the front.

Quotes

A Mandelbrot Fractal
with the "power" setting at (-1.5, 0)

• If we subtract an additive number from an empty power, [$0x^{n}-ax^{n}$] the same subtractive power remains; if we subtract the subtractive number from an empty power, [$0x^{n}-(-ax^{n})$] the same additive power remains. If we subtract an additive number from a subtractive number, the remainder is their subtractive sum; if we subtract a subtractive number from a greater subtractive number, the result is their subtractive difference; if the number from which one subtracts is smaller than the number subtracted, the result is their additive difference.
  • Al-Samawal al-Maghribi, al-Bahir fi'l-jabr (c. 1149) as quoted by Victor J. Katz, Karen Hunger Parshall, Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century (2014) p. 160.

• Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity.
  • W. W. Rouse Ball, A History of the Study of Mathematics at Cambridge (1889) p. 45.

• Accountants eventually became comfortable with using negative numbers... but for a long time mathematicians remained wary... the negatives were known as absurd numbers—numeri absurdi...Consider this equation:${\frac {-1}{\quad 1}}={\frac {\quad 1}{-1}}$...it states that the ratio of a smaller number, -1, to a larger number, 1, is equal to the ratio of a larger number, 1, to a smaller one, -1. The paradox was much discussed... To make sense of negative numbers, many mathematicians, including Leonhard Euler, came to the bizarre conclusion that they were larger than infinity. ...considering the following sequence:${\frac {10}{3}},{\frac {10}{2}},{\frac {10}{1}},{\frac {10}{\frac {1}{2}}}...\quad [{\frac {10}{\frac {1}{n}}},n\to \infty ]$As the... denominator [${\frac {1}{n}}$] gets lower... the... fraction [${\frac {10}{\frac {1}{n}}}$] approaches infinity. It was assumed that when the denominator reaches zero, the fraction is infinity, and when the denominator [passes zero to become]... negative—the [negative] fraction [${\frac {10}{\frac {-1}{\;n}}}$] must be larger than infinity.
  • Alex Bellos, The Grapes of Math (2014)

• One voice of clarity among the confusion belonged to... John Wallis, who devised a powerful visual interpretation for the negative numbers. In his 1685 work A Treatise of Algebra, he first described the "number line,"... By replacing the idea of quantity with the idea of position, Wallis argued that negative numbers were neither "Unuseful [nor] Absurd,"...it is the most successful explicatory diagram of all time.
  • Alex Bellos, The Grapes of Math (2014)

• Nearly half a century has elapsed since Baron Maseres published his use of the negative sign, or that mark in algebra which denotes the operation of subtraction. The work was acknowledged to have great merit, and the arguments in it were never answered; but still, such is the force of custom, very eminent mathematicians could not break themselves of their old habits, and they continued to use the sign of subtraction without a preceding number, from which that to which the sign was annexed was to be subtracted. Hence a number was said to be less than nothing, other numbers were called impossible, and these ideal beings became the objects of demonstration. The difficulties thus introduced into science gave such a mysterious appearance to algebra, that few would study it; and it is conceived at the present moment to require transcendant abilities and application to make any progress in it.
  ...men are springing up who seem determined to restore algebra to that purity with which it was taught originally by Vieta. ...Mr. [William] Frend published his ""Principles of Algebra," in which he excludes entirely the notion of either a number less than nothing, or an imaginary, or an impossible number; and he asserts, that whenever such an appearance takes place, "the error is either in the person who proposed, or in him who attempted to solve the proposed equation." This work was followed by an appendix by Baron Maseres, who overthrew entirely a supposed demonstration given by Clairaut of the existence of negative numbers, and the possibility of their producing by multiplication a positive number, and shewed that it was in the very outset founded on error.
  Carnot... sees in the same point of view the confusion that has arisen in science by the introduction of a fiction. Instead of rejecting it however entirely, as they have done, he wishes to make a kind of compromise, and declaring it to be absurd, he would leave it in possession of the rights and privileges with which from the time of Descartes it has been indulged. Perhaps he thinks this an easier way of overthrowing the system in his country, for he accumulates instances of the false consequences that arise in reasoning upon the present plan, and concludes that they must have their effect at last in opening people's eyes, and restoring science to its ancient footing.
  Among the instances which he adduces of the absurdity of considering negative numbers as capable of any of the processes of arithmetic, he introduces the following: First from the laws laid down by Euclid['s]... fifth book... since it is supposed that1 : -1 :: -1 : 1and -1 being less than nothing, 1 is evidently greater that -1; therefore in the proportion the first term is greater than the second, but the third is not greater than the fourth. The proportion then is evidently false, -1 x -1 cannot be equal to +1. Again, -3 is less than 2; therefore -3² is less than 2². But -3², according to the prevailing doctrine, is equal to 9, and 9 is greater than 4. Therefore the square of -3 is greater than the square of 2, which is absurd; and, as the author says, "Cette theorie est done complettement fausse." This theory is absolutely false. In the application to geometry the same is evident... In the same manner it will follow, from an algebraical expression for the radius of the circle, and allowing the usual doctrine of negative quantities, that the radius of a circle may be both negative and positive. ...
  The fact is, algebra knows nothing of position; that is peculiar to geometry; and when from the consideration of lines an equation is formed in algebra, the rules of algebra alone can be used in the solution of it; and the geometrician must previously tell what quantities he chooses to be greater or less than others, before the algebraist can give an answer to the question.
  • Boosey, "Carnot's Geometrie de Position," The Annual Review and History of Literature, Vol. 2 (1804) ed., Arthur Aikin, pp. 821-823.

• Every negative quantity standing by itself is a mere creature of the mind and... those which are met with in calculations are only mere algebraical forms, incapable of representing any thing real and effective. ...That each of these algebraical forms being taken, with a proper consideration of its sign, is nothing else but the difference of two other absolute quantities, of which the greatest in the case of which the reasoning depends, is the least in the case in which the result of the calculation is to be applied.
  To say of a quantity that it becomes negative, is to employ an improper expression, and one leading... into error; and the true meaning of the expression is, that this absolute quantity does not belong to the system on which the reasonings have been established, but to another with which it is related; so that to apply the forms applicable to it for this first system, the sign + must be changed into the sign - in the preceding form.
  ...Nothing is to be changed... we have only to substitute a clear and true idea in the room of one faulty and useless; and this is the object of this work. I think that I have accomplished it by substituting for the notion of positive and negative quantities, which I have opposed, that of quantities which I call direct and inverse; and the geometry of position is that where the notion of positive and negative quantities standing by themselves is supplied by that of quantities direct and inverse.
  ...however these unintelligible forms ought not to be neglected, and... they may be employed like real forms, because they may be made to disappear by simple algebraical transformations; and there will then remain only explicit forms immediately applicable to the object proposed, provided, as it has been before observed, that these forms, conditions proposed, and suppositions on which the reasoning is grounded, are all consistent with each other.
  • Lazare Carnot, Geometrie de Position (1803) as quoted by Boosey, "Carnot's Geometrie de Position," The Annual Review and History of Literature, Vol. 2 (1804) ed., Arthur Aikin, pp. 821-823.

• SNOWMAN...
  That ice
  cream cone'll
  drip?
      •
  Curious
  key hole.
      •
  I want to go into the immense
  blue yonder
  
  and I've built a negative number
  times three.
  • Robert Creeley, "SEVEN: A Suite for Robert Therrein," Windows (1990)

• The imaginary expression ${\sqrt {-a}}$ and the negative expression -b have this resemblence, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are equally imaginary, since 0 - a is as inconceivable as ${\sqrt {-a}}$. What then is the difference of signification? The following problems will elucidate this.
  • Augustus De Morgan, On the Study and Difficulties of Mathematics (1831) p. 53.

• The ideas of number are the clearest and most distinct in the human mind; the acts of the mind upon them are equally simple and clear. There cannot be confusion in them, unless numbers too great for the comprehension of the learner are employed, or some arts are used which are not justifiable. The first error in teaching the principles of algebra is obvious on perusing a few pages only in the first part of Maclaurin's Algebra. Numbers are there divided into two sorts, positive and negative; and an attempt is made to explain the nature of negative numbers, by allusions to book-debts and other arts. Now, when a person cannot explain the principles of a science without reference to metaphor, the probability is, that he has never thought accurately upon the subject. A number may be greater or less than another number; it may be added to, taken from, multiplied into, and divided by another number; but in other respects it is very untractable: though the whole world should be destroyed, one will be one, and three will be three; and no art whatever can change their nature. You may put a mark before one, which it will obey: it submits to be taken away from another number greater than itself, but to attempt to take it away from a number less than itself is ridiculous. Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary. Hence they talk of two roots to every equation of the second order, and the learner is to try which will succeed in a given equation: they talk of solving an equation which requires two impossible roots to make it solvible: they can find out some impossible numbers, which, being multiplied together, produce unity. This is all jargon, at which common sense recoils; but, from its having been once adopted, like many other figments, it finds the most strenuous supporters among those who love to take things upon trust, and hate the labour of a serious thought.
  • William Frend, The Principles of Algebra (1796)

• The next Tract in this Collection is the 10th Chapter of the 2nd Book of Mr. John Kersey's Elements of Algebra,—a most valuable work in two small volumes, folio, that was published in the year 1673. This book is now too little known, and deserves to be strongly recommended to all students of Algebra on account of the great fulness and perspicuity with which it is written; in which important qualities it very far surpasses most of the Treatises of Algebra that have been written since, excepting Dr. Saunderson's Elements of Algebra, in two volumes, quarto, which was published in the year 1740. And in some respects it is even preferable to Saunderson's Algebra; because it gives a fuller account of the Diophantine problems, and because it deals less in the absurd mysteries of negative roots, and the negative quantities in general, and the doctrine of the generation of higher affected equations from lower equations by multiplication, which has been the source of great obscurity arid perplexity to all the students of Algebra ever since it's introduction into that science by the publication of Harriot's Algebra in the year 1631 and Des Cartes's Geometry in the year 1637. These absurd doctrines seem to have delighted Dr. Saunderson, who frequently insults and rails at such of his readers as shall find a difficulty in Understanding them, as being persons of narrow minds and incorrigible dullness and stupidity: but Mr. Kersey, though he did not venture boldly to reject them, as either positively absurd and unintelligible or, at least, leading to obscurity, yet says but little about them, and seems desirous, as much as possible, to keep clear of them. His book therefore deserves to be studied by all persons who desire to become skilfull in Algebra...
  • William Frend, John Kersey, Tracts on the Resolution of Affected Algebräick Equations by Dr. Halley's, Mr. Raphson's, and Sir Isaac Newton's, Methods of Approximation (1800) Preface, pp. lvii-lviii.

• Our general arithmetic, so far surpassing in extent the geometry of the ancients, is entirely the creation of modern times. Starting originally from the notion of absolute integers, it has gradually enlarged its domain. To integers have been added fractions, to rational quantities the irrational, to positive the negative and to the real the imaginary. This advance, however, has always been made at first with timorous and hesitating step. The early algebraists called the negative roots of equations false roots, and these are indeed so when the problem to which they relate has been stated in such a form that the character of the quantity sought allows of no opposite. But just as in general arithmetic no one would hesitate to admit fractions, although there are so many countable things where a fraction has no meaning, so we ought not to deny to negative numbers the rights accorded to positive simply because innumerable things allow no opposite. The reality of negative numbers is sufficiently justified since in innumerable other cases they find an adequate substratum. This has long been admitted, but the imaginary quantities—formerly and occasionally now, though improperly, called impossible—as opposed to real quantities, are still rather tolerated than fully naturalized, and appear more like an empty play upon symbols to which a thinkable substratum is denied unhesitatingly by those who would not depreciate the rich contribution which this play upon symbols has made to the treasure of the relations of real quantities.
  The author has for many years considered this highly important part of mathematics from a different point of view, where just as objective an existence may be assigned to imaginary as to negative quantities, but hitherto he has lacked opportunity to publish these views. In the present paper the outlines are given briefly; they consist of the following:
  Positive and negative numbers can only find an application when the thing counted has an opposite which when conceived of as united with it has the effect of destroying it. Accurately speaking, this supposition can only be made where the things enumerated are not substances (objects thinkable in themselves) but relations between any two objects. It is postulated that these objects are arranged after a definite fashion in a series e.g., A, B, C, D,... and that the relation of A to B can be regarded as equal to that of B to C, etc. The notion of opposition involves nothing further than the interchange of the terms of the relation so that if the relation of (or transition from) A to B is considered as +1 the relation of B to A must be represented by -1. So far then as such a series is unlimited on both sides, every real integer represents the relation of a term arbitrarily taken as origin to a definite term of the series.
  If, however, the objects are of such a kind that they cannot be arranged in one series, even though unlimited, but only in series of series, or, what amounts to the same thing, they form a manifoldness of two dimensions; if there is the same connection between the relations of one series to another, or the transitions from one to another, as in the case of the transition from one term of a series to another term of the same series, we shall evidently need for the measurement of the transition from one term of the system to another, besides the previous units +1 and -1, two others opposite in character +i and -i.
  Obviously we must also postulate that the unit i shall always mark the transition from a given term of the one series to a definite term of the immediately adjacent series. In this way the system can be arranged in a two-fold manner in series of series.
  The mathematician leaves entirely out of consideration the nature of the objects and the content of their relations. He has simply to do with the enumeration and comparison of the relations. So far as he has assigned sameness of nature to the relations designated by +1 and -1, considered in themselves, he is warranted in extending such sameness to all four elements, +1, -1, +i, -i.
  ...We see further that if we wish to take +1 for the relation previously expressed by +i, we must necessarily take +i for the relation previously expressed by -1. In the language of mathematicians this means that +i is a mean proportional between +1 and -1, or corresponds to the symbol ${\sqrt {-1}}$. We say purposely not the mean proportional because -i has just as good a right to that designation. Here then the demonstrability of an intuitive signification of ${\sqrt {-1}}$ has been fully justified and nothing more is necessary to bring this quantity into the domain of objects of arithmetic.
  • Carl Friedrich Gauss, in Göttingische gelehrte Anzeigen (April 23, 1831) an account of his paper Theoria residuorum biquadraticorum, Commentatio secunda, as quoted in Science, Vol. 6 (Jul-Dec, 1897) ed., John Michels, pp. 304-305.

• Old shoes, old roads—
  the questions keep being new ones.
  Like two negative numbers multiplied by rain
  into oranges and olives.
  • Jane Hirshfield, "Like Two Negative Numbers Multiplied by Rain," Poetry (2012)

• Stevin is... the first mathematician who understands the subtraction of a "number" is the addition of a "negative number". In every one of these theses Stevin is merely assimilating the concept of "number" to operations on "numbers" already long established.
  • Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968)

• While the mathematicians were still looking askance at the Greek gift of the irrational number, the Hindus of India were preparing another brain-teaser, the negative number, which they introduced about A.D. 700. The Hindus saw that when the usual, positive numbers were used to represent assets, it was helpful to have other number represent debts.
  • Morris Kline, Mathematics and the Physical World (1959) pp. 49-50.

• When an equation...clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. ...Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.
  • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 143.

• The Hindus introduced negative numbers... The first known use is Brahmagupta about 628; he also states the rules for the four operations with negative numbers. Bhāskara points out that the square root of a positive number is twofold, positive and negative. He brings up the matter of the square root of a negative number but says that there is no square root because a negative number is not a square. No definitions, axioms, or theorems are given.
  The Hindus did not unreservedly accept negative numbers. Even Bhāskara, while giving 50 and -5 as two solutions of a problem, says, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions." However, negative numbers gained acceptance slowly.
  • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 185.

• In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers.
  • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 192.

• By 1700 all of the familiar members of the [number] system... were known. However, opposition to the newer types of numbers was expressed throughout the century. Typical are the objections of... Baron Francis Masères... in 1759 his Dissertation on the Use of the Negative Sign in Algebra... shows how to avoid negative numbers... and especially negative roots, by carefully segregating the types of quadratic equations so that those with negative roots are considered separately; and... the negative roots are to be rejected.
  • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 592.

• Abundant quantities are those which are greater than nothing (majores nihilo,) and carry the idea of increase along with them. These have either no symbol prefixed, or this one +, which is the copulative (copula) of increase. Thus, if you are not in debt, and your wealth be estimated at 100 crowns, these may either be noted 100 crowns, or + 100 crowns; and are read a hundred crowns of increase; always signifying wealth and gain. ...Defective quantities are those which are less than nothing (minores nihilo,) and carry the idea of diminution along with them. These are always preceded by this symbol which is the copulative of diminution. Thus, in the estimation of his wealth whose debts exceed his goods by 100 crowns, justly his funds are thus prenoted,—100 crowns, and are read, a hundred crowns of decrease; signifying always loss and defect. I have already shown that defective quantities have their origin in subtracting the greater from the less.
  • John Napier, Booke of Arithmeticke and Algebra (ca. 1615), as quoted by Mark Napier, "History of the Invention of Logarithms, &c.," Memoirs of John Napier of Merchiston (1834) p. 468.

• Sir Isaac Newton, in his Algebra, says. "Quantitates vel affirmativæ sunt, seu majores nihilo; vel negativæ seu nihilo minores. Sic in rebus humanis possessiones dici possunt bona affirmativa, debita vero bona negativa;" [Quantities are either positive, or greater than zero; or negative, zero or less. Thus, in the affairs of men, possessions of the goods are of the affirmative, but the debit of goods, negative.] the very example which Napier gives. Dr. Horsley, Newton's commentator, observes at this passage; "Albertus Girardus, ni fallor, omnium primus, (quem summum interea mathematicum agnosco,) durâ quâdam verborum figurâ, Diophanto et Vietæ prorsus ignotâ, quam vellem Cartesius et nostrates minus avide arripuissent, nihilo minores, dixit." [Albert Girard, unless I am mistaken, first amongst all (supreme amongst the mathematically knowledgable) with hard words and no figure, totally unknown to Diophantus and Viète, and wished for by Descartes, eagerly seized the minus, less than nothing, he said.] This shows how neglected Napier's great work is by the learned. Horsley, of course, could not know, that in Napier's unpublished manuscript there was a chapter upon this distinction, but he might have read in the Canon Mirificus, c. i. p. 5, "Logarithmos sinuum, qui semper majores nihilo sunt, abundantes vocamus, et hoc signo +, aut nullo prænotamus; logarithmos autem minores nihilo defectivos vocamus, prænotantes; eis hoc signum -." [The logarithms of sines, which are always greater than zero, we call abundant, and this is the sign of +, or nothing is noted down; But less than zero negative logarithms are noted; this is their sign -.] This was published fifteen years before the work of Girard, to which Horsley alludes. Dr. Hutton, in his History of Algebra, has fallen into the same mistake; "Girard was the first who gave the whimsical name of quantities less than nothing to the negative ones." Here is another indication that Hutton analyzed Napier's works, and presumed to attack his character, without reading the original proofs as he ought to have done. Even Leslie and Playfair had not read the Canon Mirficus. The former says, "Girard was possessed of fancy as well as invention; and his fondness for philological speculation led him to frame new terms, and to adopt certain modes of expression which are not always strictly logical; though he stated well the contrast of the signs plus and minus, in reference to mere geometrical position, he first introduced the very inaccurate phrases of greater and less than nothing. Playfair says, "Girard is the author of the figurative expression, which gives the negative quantities the name of quantities less than nothing; a phrase that has been severely censured by those who forget that there are correct ideas which correct language can hardly be made to express. It is, indeed, unphilosophical fastidiousness to call the phrase "very inaccurate." Napier fortified it by a better nomenclature, in the terms abundant and defective, than those now in use,—positive and negative, which are said to convey erroneous impressions. Again, his exemplification of the idea is that which is invariably adopted now, though not from him. Surely Euler was never rummaging in the Merchiston charter-chest? Yet his illustration is identically Napier's; "In algebra, simple quantities are numbers considered with regard to the signs which precede or affect them. Farther, we call those positive quantities, before which the + sign is found; and those are called negative quantities which are affected by the sign -. The manner in which we generally calculate a person's property is an apt illustration of what has just been said; for we denote what a man really possesses, by positive numbers, using or understanding the sign +; whereas his debts are represented by negative numbers, or by using the sign - : Thus, when it is said of any one that he has 100 crowns, but owes 50, this means that his real possession amounts to 100 - 50; or, which is the same thing, + 100 - 50, i.e. 50. Since negative numbers may be considered as debts, because positive numbers represent real possessions, we may say that negative numbers are less than nothing; thus, when a man has nothing of his own, and owes 50 crowns, it is certain that he has 50 crowns less than nothing, for if any one were to make him a present of 50 crowns to pay his debts, he would still be only at the point of nothing, though really richer than before. In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing." Maclaurin, too, defends the phrase, but illustrates the idea more poetically; "the depression of a star below the horizon may be equal to the elevation of a star above it; but those positions are opposite, and the distance of the stars is greater than if one of them was at the horizon so as to have no elevation above it, or depression below it; it is on account of this contrariety that a negative quantity is said to be less than nothing; because it is opposite to the positive, and diminishes it when joined to it, whereas the addition of 0 has no effect; but a negative is to be considered no less as a real quantity than the positive. The opinion of Leslie, who calls the phrase inaccurate, and of Hutton, who calls it whimsical, must go down before the opinions of Napier, Newton, Maclaurin, Euler, and Playfair.
  • Mark Napier, "History of the Invention of Logarithms, &c.," Memoirs of John Napier of Merchiston (1834)

• Al-Samawal wrote his famous work, al-Bahir fi'l-jabr when he was only 19 years old. The Dazzling... emphasizes the principles of arithmetization of algebra, explaining how unknown arithmetic quantities, or variables, can be treated just like ordinary numbers when considering arithmetic operations. Many scholars consider [it] to be the first treatise to assert that $x^{0}=1$. He was also quite comfortable with using negative numbers and zero in his work, considering such concepts as 0 - a = -a. He also understood how to handle multiplication involving negative numbers...
  • Clifford A. Pickover, The Math Book (2009)

• + whych betokeneth too muche, as this line - plain without a crosse line betokeneth too little.
  • Robert Recorde, Grounds of Artes (1540) as quoted by W. W. Rouse Ball, A Short Account of the History of Mathematics (1888)

• — 2 and 4.
  If negative multiplies with positive
  their creation is incomplete
  a kind of un-being
  negative numbers have empty hearts,
  must reach out to others alike
  else we are bound to fall
  into shadow quadrants that measure
  loss like childhood fears, doubts
  trapped in closets
  • Erik Richardson, "love on the windswept Cartesian plane" (2014)

• The "Appendix to the Principles of Algebra by Francis Maseres, Esq. F.R.S. Cursitor Baron of his Majesty's Court of Exchequer," is written as a supplement to Mr. [William] Frend's treatise on that science, which we introduced to our readers in our Register for the year 1796. ...what will particularly engage the attention of the mathematical world, it contains an unequivocal and perfect approbation of Mr. Frend's doctrine respecting negative numbers. The assent of a person of such eminence in algebraic science to the new opinion, shows, at least, that it has not been adopted without weighty and forcible reasons for its truth; and may, perhaps, encourage other mathematicians to throw off all dread of innovation, all implicit scientific faith, or habit of taking for granted that which has not been previously proved, and to question some other long received dogmas, which certainly wear the appearance of contradiction and absurdity. Is not this the case with the doctrines of infinity and imaginary quantities, as explained by algebraists, and that of the asymptotè in conic sections?
  • G.G.J. and J. Robinson, "Domestic Literature of the Year 1798," The New Annual Register, Or, General Repository of History, Politics, and Literature for the Year 1798 (1799) pp. 240-241.

• The next work in which we... find arguments against integrating the negative numbers into the overall conception of real numbers was, in contrast to Fontenelle's, a textbook explicitly addressing laymen. ...Clairaut wrote two elementary textbooks ...for an elegant public intent on dabbling in leisurely in mathematics without having to shoulder any real effort. The two textbooks on geometry (1741) and algebra (1746) attempt to realize a methodical approach... which... evolves in a seemingly "natural way" from simple inquiries or from useful problems. ...a particular stumbling block in his eyes was multiplication. ...he adapted part of Fontenelle's arguments in favor of separating positive from negative quantities. ...guiding beginners gradually toward an understanding of the necessity of operating with negative numbers, and of the appropriate rules—in particular the rule of signs. ...He ...developed a method of interpreting negative solutions away ...liberating operating with negative quantities from everything "shocking," permitting the reader to recognize the nature of negative problem solutions. One should assume the unknown to be of opposite direction... Clairaut... admitted genuine negative values of unknowns, a fact relieving him from changing the equations... He even extended this conception by permitting negative values as well for the coefficients—in contrast to... quantities preceded by a minus sign, as was then current practice in France. ...He was the first to directly tackle, in a textbook of modern times, the question of how to interpret negative solutions in equations with concrete quantities.
  • Gert Schubring, "Reinterpreting the Negative as Positive," Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17th-19th Century France and Germany (2006) pp. 102-104.

• William Frend (1757-1841) whose daughter Sophia Elizabeth became De Morgan's wife (1837) ...came under De Morgan's definition of a true paradoxer, carrying on a zealous warfare for what he thought right. As a result of his Address to the Inhabitants of Cambridge 1787 and... he was deprived of his tutorship in 1788. A little later he was banished from Cambridge because of his denunciation of the abuses of the Church and his condemnation of the liturgy. His eccentricity is seen in his declining to use negative quantities in the operations of algebra. He finally became an actuary at London and was prominent in radical associations. He was a mathematician of ability, having been second wrangler and having nearly attained the first place, and he was also an excellent scholar in Latin Greek and Hebrew.
  • David Eugene Smith, ed., footnote in Augustus De Morgan's A Budget of Paradoxes (1915) Vol. 1 p. 196.

• That totality is named in an algebraic expression:
  Some formula not relevant any more
  To flower children might express it yet
  Like ${\sqrt {({\frac {x}{y}})^{n}}}=1$
  —Or equals zero, one forgets—
  The y standing for you, dear friend, at least
  Until that hour he reaches for me, then
  Leaves me cold, the great god Pain...
  
  Merrill's calculus is multivariable: he will strain to accompany the x factor of his poetry—himself ("X my mark")—with the y factor, "you,"... it is the equation itself that holds the most interest. As Stephen Yenser has pointed out, it is partly a recasting of the poem's title word, with the "S" and "r" of Syrinx becoming "square root," the "i"orthographically represembling an "I"... But the formula here is... an equation, a set of relations in which the unity of the world... is defined as both one and "I". That unity of (controversial) unities stems from the mutual cooperation, or productive disharmony, of three elements.
  Since I and zero squared are both just themselves, we can set aside the radical sign... One solution... then finds x and y are equal. ...There is another possible solution: if n is an even number, x and y could be opposite rather than identical. If the poet and his... subject, are fundamentally opposed, mirror images (if, that is to say x = -y), we must rely on the third term—"n," nature, the unspoken word of every lyric poem in this tradition—to balance the equation and avoid the nihilism either of -I or 0. The square root of a negative number cannot be anything but imaginary: all imaginary numbers... include ${\sqrt {-1}}$ as a factor, a nonsense term represented by a lower case "i" that here threatens to supplant the well-rooted, erect "I" that is resembled by the number one. Only when nature is the exponent can a lyric poet expound confidently the communion of humanity.
  • James E. von der Heydt, "Merrill's Expansiveness," At the Brink of Infinity: Poetic Humility in Boundless American Space (2008) with an introductory quote from James Merrill's "Syrinx," Harper's Magazine (Sep, 1970)

A Dissertation On the Use of the Negative Sign in Algebra; (1758)

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 at an Appendix, Mr. Machin's Quadrature of a Circle, by Francis Maseres, M.A. Fellow of Clare-Hall, Cambridge.

• The Design of the following Dissertation, is to remove from some of the abstruse Parts of Algebra, the Difficulties that have arisen therein from the too extensive Use of the Negative Sign, in any other Light, than as the Mark of the Subtraction of a lesser Quantity from a greater.

• It is evident that a single Quantity can never be marked with either of these Signs, or considered as either Affirmative or Negative; for, if any single Quantity, as b is marked either with the Sign + or the Sign -, without assigning any other Quantity... to which it is to be added, or from which it is to be subtracted, the Mark will have no Meaning or Signification. Thus, if it be said, the Square of -5, or the Product of -5 into -5, is equal to +25. Such an Assertion must either signify no more than 5 times 5 is equal to 25, without any Regard to the Signs, or it must be meer Nonsense, and unintelligible Jargon.

• Now, what I here propose is, first to apply the Operations of Arithmetic to Compound Quantities, consisting partly of Affirmative, and partly of Negative Quantities, or to the Differences of Simple Quantities, and to demonstrate the Rules usually given for Multiplication, to wit, that - X + gives -, + X - gives -, + X + gives +, and - X - gives +, or that as often as the Signs of the Multiplier and Multiplicand are alike, the Product is to be marked +, and as often as they are unlike, the Product is to be marked -; and secondly, to shew how the roots of Quadratic and Cubic Equations may be assigned by the common Rules given for that Purpose, without having Recourse to any other Idea of a Negative Quantity, than the very Simple One above defined.

A Short Account of the History of Mathematics (1888)

by W. W. Rouse Ball (quotes may be from various editions, 1888-1912).

• Diophantus not only invented a new language in algebra for expressing results that were already known but applied the subject to a number of problems which had previously baffled all investigation. His work however excited no interest among his contemporaries, and with his death his writings and methods fell into an oblivion from which they were not rescued for many centuries. ...His chief work is his Arithmetic. This is really a treatise on algebra; algebraic symbols are used, and the problems are treated analytically. Diophantus tacitly assumes, as is done in nearly all modern algebra, that the steps are reversible. ...Diophantus commences with some definitions... in giving the symbol for minus he states that a subtraction multiplied by a subtraction gives an addition; by this he means that the product of -b and -d in the expression $(a-b)(c-d)$ is +bd, but in applying the rule he always takes care that the numbers a, b, c, d are so chosen that a is greater than b and c is greater than d. ...Probably the rule for solving any quadratic equation was given in that part of the work which is now lost, but where the equation is of the form $ax^{2}+bx+c=0$ he seems to have multiplied by a and the "completed the square" in much the same way as is now done; when the roots are negative or irrational the equation is rejected as "impossible," and even when both roots are positive he never gives more than one, always taking the positive value of the square root.

• The earliest instances of the use of the signs + and - of which we have any knowledge occur in the Mercantile arithmetic of Johann Widman of Eger... in 1489. They are however not used by him as symbols of operation, but apparently merely as marks signifying excess or deficiency. The corresponding use of the word surplus or overplus is still retained in commerce. It is noticeable that with very few exceptions these signs only occur in practical mercantile questions: hence it has been conjectured that they were originally warehouse marks. Some kinds of goods were sold in a sort of wooden chest called a lagel, which when full was apparently expected to weigh roughly either three or four centners; if one of these cases was a little lighter, say 5 lbs., than four centners Widman describes it as 4c.- 5 lbs.: if it was 5 lbs. heavier than the normal weight it is described as 4c+5 lbs.: and there are some slight reasons slight for thinking that these marks were chalked on to the chests as they came into the warehouses. The symbols are used as if they would be familiar to his readers. ...the vertical line in the symbol for excess printed... is somewhat shorter than the horizontal line. This is also the case with Stifel and most of the early writers who used the symbol: some presses continued to print it in this, its earliest form, up to the end of the seventeenth century. Xylander on the other hand in 1575 has the vertical bar much longer than the horizontal line, and the symbol is something like †.
  ...these signs in Widman are only abbreviations and not symbols of operation; he attached little or no importance to them, and would no doubt have been amazed if he had been told that their introduction was preparing the way for a complete revolution of the processes used in algebra.

• The signs + and - to indicate addition and subtraction occur in Widman's arithmetic published in 1849, but were first brought into general notice, at any rate as symbols of operation, by Stifel in 1544. They occur, however, in a work by G. V. Hoecke, published... in 1514. I believe I am correct in saying that Vieta in 1591 was the first well-known writer who used these signs consistently throughout his work, and that it was not until the beginning of the seventeenth century that they became recognized as well-known symbols. The sign = to denote equality was introduced by Record in 1557.

• Descartes researches in geometry are given in the third section of the Discours. This is divided into three books the first two of these treat of analytical geometry... The third book of the section on geometry is an analysis of the algebra then current, and it largely affected the language of the subject by fixing the custom of employing the letters at the beginning of the alphabet to denote known quantities and those at the end of the alphabet to denote unknown quantities, and by introducing the system of indices now in use. I think Descartes was the first to realize that his letters might represent any quantities, positive or negative, and that it was sufficient to prove a proposition for one general case.

• John Wallis ...was in ...1649 elected to the Savilian professorship of geometry at Oxford a chair which he continued to occupy till his death. ...The most important of his works is his Arithmetica infinitorum published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematized and greatly extended. It at once became the standard book on the subject and Fermat, Barrow, Newton, and Leibnitz all constantly refer to it. Wallis commences it by shewing that $x^{0},x^{-1},x^{-2}...$ stand for $1,{\frac {1}{x}},{\frac {1}{x^{2}}}...$ that $x^{\frac {1}{2}}$ stands for the square root of x, and that $x^{\frac {2}{3}}$ stands for the cube root of $x^{2}$ &c. and that thus the law of indices in algebra in quite general.

• Stifel wrote a small treatise on algebra, but his chief mathematical work is his Arithmetica Integra published at Nuremberg in 1544, with a preface by Melanchthon. The first two books of the Arithmetica integra deal with surds and incommensurables, and are Euclidean in form. The third book is on algebra... This work is chiefly noticeable for having called general attention to the German practice of using the signs + and - to denote addition and subtraction. There are faint traces of their being occasionally employed by Stifel as symbols of operation and not only as abbreviation...

A History of Mathematics (1893)

by Florian Cajori

• Diophantus was one of the last and most fertile mathematicians of the second Alexandrian school. ..He is, as far as we know, the first to state that "a negative number multiplied by a negative number gives a positive number." This is applied to the multiplication of differences, such as (x - 1)(x - 2). It must be remarked, however, that Diophantus had no notion whatever of negative numbers standing by themselves. All he knew were differences, such as (2x - 10), in which 2x could not be smaller than 10 without leading to an absurdity. He appears to be the first who could perform such operations as (x - 1) X (x - 2) without reference to geometry.

• Bhaskara advances far beyond the Greeks and even beyond Brahmagupta when he says that "the square of a positive, as also of a negative number, is positive; that the square root of a positive number is twofold, positive and negative. There is no square root of a negative number, for it is not a square."

• An ambassador from Netherlands once told Henry IV that France did not possess a single geometer capable of solving a problem propounded to geometers by a Belgian mathematician, Adrianus Romanus. It was the solution of the equation of the forty fifth degree:—$45y-3795y^{3}+95634y^{3}-\ldots +945y^{41}-45y^{43}+y^{45}=C$...Vieta, who, having already pursued similar investigations, saw at once that this awe-inspiring problem was simply the equation by which C = 2 sin φ was expressed in terms of y = 2 sin 1⁄45 φ that since 45 = 3·3·5, it was necessary only to divide an angle once into 5 equal parts, and then twice into 3,—a division which could be effected by corresponding equations of the fifth and third degrees. Brilliant was the discovery by Vieta of 23 roots to this equation, instead of only one. The reason why he did not find 45 solutions, is that the remaining ones involve negative sines, which were unintelligible to him.

• Vieta discarded negative roots of equations. Indeed, we find few algebraists before and during the Renaissance who understood the significance even of negative quantities. Fibonacci seldom uses them. Pacioli states the rule that "minus times minus gives plus," but applies it really only to the development of the product of (a - b) (c - d); purely negative quantities do not appear in his work. The great German "Cossist" (algebraist) Michael Stifel, speaks as early as 1544 of numbers which are "absurd" or "fictitious below zero," and which arise when "real numbers above zero" are subtracted from zero. Cardan, at last, speaks of a "pure minus"; "but these ideas," says Hankel, "remained sparsely, and until the beginning of the seventeenth century, mathematicians dealt exclusively with absolute positive quantities." The first algebraist who occasionally places a purely negative quantity by itself on one side of an equation, is Harriot in England. As regards the recognition of negative roots, Cardan and Bombelli were far in advance of all writers of the Renaissance, including Vieta. Yet even they mentioned these so-called false or fictitious roots only in passing, and without grasping their real significance and importance. On this subject Cardan and Bombelli had advanced to about the same point as had the Hindoo Bhaskara, who saw negative roots, but did not approve of them. The generalisation of the conception of quantity so as to include the negative, was an exceedingly slow and difficult process in the development of algebra.

• Albert Girard... was the first who understood the use of negative roots in the solution of geometric problems.

• The application of algebra to the doctrine of curved lines reacted favourably upon algebra. As an abstract science, Descartes improved it by the systematic use of exponents and by the full interpretation and construction of negative quantities.

• The conception of "number" has been much extended in our time. With the Greeks it included only the ordinary positive whole numbers; Diophantus added rational fractions to the domain of numbers. Later negative numbers and imaginaries came gradually to be recognised. Descartes fully grasped the notion of the negative; Gauss that of the imaginary.

History of Mathematics (1925)

by David Eugene Smith, Vol.2

• No trace of the use of the recognition of negative numbers, as distinct from simple subtrahends, has yet been found in the writings of the ancient Egyptians, Babylonians, Hindus, Chinese, or Greeks. Nevertheless the law of signs was established, with the aid of such operations as (10 - 4) ˙ (8 - 2), and was known long before the negative number was considered by itself.

• The Chinese made use of such numbers as subtrahends at a very early date. They indicated positive coefficients by red computing rods, and negative ones by black... The negative number is mentioned, at least as a subtrahend, in the K'iu-ch'ang Suan-shu (c. 200 B.C.)... but the law of signs is not known to have been definitely stated in any Chinese mathematical treatise before 1299, when Chu Shï-kié gave it in his elementary algebra, the Suan-hio-ki-möng (Introduction to Mathematical Studies).

• In the Arithmetica of Diophantus (c. 275) , where the equation $4x+20=4$ is spoken of as absurd, since it would give $x=-4$. Of the negative number in the abstract Diophantus had apparently no conception. On the other hand, the Greeks knew the geometric equivalent of $(a-b)^{2}$ and of $(a+b)(a-b)$; and hence, without recognizing negative numbers, they knew the results of the operations $(-b)\cdot (-b)$ and $(+b)\cdot (-b)$.

• In India the negative number is first definitely mentioned in the works of Brahmagupta (c. 628). He speaks of "negative and affirmative quantities," using them always as subtrahends but giving the usual rules of signs. The next writer to treat of these rules is Mahāvīra (c. 850), and after a time they are found in all Hindu works on the subject.

• The Arabs contributed nothing new to the theory, but al-Khowârizmî (c. 825) states the usual rules, and the same is true of his successors.

• When Fibonacci wrote his Liber Abaci (1202) he followed the Arab custom of paying no attention to negative numbers, but in his Flos (c. 1225) he interpreted a negative root in a financial problem to mean a loss instead of a gain. Little further was done with the subject by medieval writers, but as we approach the Renaissance period we find the negative number as such receiving more and more recognition. For example, among the problems set by Chuquet (1484) is one which leads to and equation with roots "${\tilde {m}}\cdot 7\cdot {\frac {3}{11}}$" and "$27\cdot {\frac {3}{11}}$," that is, $-7{\frac {3}{11}}$ and $27{\frac {3}{11}}$.

• The first of the 16th century writers to give noteworthy treatment to the negative numbers was Cardan. In his Ars Magna (1545) he recognized negative roots of equations and gave a clear statement of the simple laws of negative numbers.

• Stifel (1544) distinctly mentioned negative numbers as less than zero, and showed some knowledge of their use. By this time the rules of operation with numbers involving negative signs were well understood, even though the precise nature of the negative number was not always clear. Thus Bombelli (1572) gave these rules and applied them intelligently to such cases as (+ 15) + (- 20) = - 5. It was due to the influence of men like Vieta, Harriot, Fermat, Descartes, and Hudde, however, that the negative number came to be fully recognized and understood. The idea of allowing a letter, with no sign prefixed, to represent either a positive or a negative number seems due to Hudde (1659).

• In the 15th century the names "positive" and "affirmative" were used to indicate positive numbers, as also "privative" and "negative" for negative numbers,—a usage followed by [Johannes] Scheubel (1551).
  Cardan spoke of "minus in minus" as being plus, but in general he called positive numbers numeri ueri and negative numbers numeri ficti...
  Stifel (1543) called these numbers "absurd"....
  Tartaglia (1556) spoke of the negative number as "the term called minus" laying down the usual rules.
  Bombelli (1572) used the word "minus" (meno) as we do in such rules as "minus times minus gives plus,"...
  Tycho Brahe... (1598), spoke of the negative number as "privative" and indicated it with a minus sign.
  Napier (c. 1600) used the adjectives abudantes and defectivi to designate positive and negative numbers, [Leonhard Christoph] Sturm (1707) spoke of Sache and Mangel, and various other names and symbols have been suggested.

See also

• Arithmetic
• History of algebra
• History of mathematics
• History of science
• Infinity
• Mathematics