Negative number

"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."

"The "Appendix to the Principles of Algebra by , Esq. F.R.S. Cursitor Baron of his Majestys Court of ," is written as a supplement to Mr. [William] Frends 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. Frends 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?"

"The next work in which we... find arguments against integrating the negative numbers into the overall conception of real numbers was, in contrast to Fontenelles, 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 Fontenelles 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."

"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."

"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."

"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."

"— 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"

"In India the negative number is first definitely mentioned in the works of (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."

"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."

"The Hindus introduced negative numbers... The first known use is 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."

"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."

"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."