“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."”

— Morris Kline

However, negative numbers gained acceptance slowly.
Source: Mathematical Thought from Ancient to Modern Times (1972), p. 185.

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Morris Kline42

American mathematician 1908–1992

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), pp. 252-253.

“The Hindus saw clearly that if the arithmetic operations… were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers. …To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens.
What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 51.

“As for negative numbers… most mathematicians of the sixteenth and seventeenth centuries did not accept them… In the fifteenth century Nicolas Chuquet and, in the sixteenth, Stifel both spoke of negative numbers as absurd numbers. …Descartes accepted them, in part. …he had shown that, given an equation, one can obtain another whose roots are larger than the original one by any given quantity. Thus an equation with negative roots could be transformed into one with positive roots. Since we can turn false roots into real roots, Descartes was willing to accept negative numbers. Pascal regarded the subtraction of 4 from zero as utter nonsense.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 252.

“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 (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 592.

“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 (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 192.

“The intelligence of that creature known as a crowd is the square root of the number of people in it.”

Terry Pratchett book Jingo

Source: Jingo

“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 (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), pp. 49-50.

“In strong families, positive strokes out-number negative broadsides by a wide margin.”

Joyce Brothers (1927–2013) Joyce Brothers

10 Keys to a Strong Family (2002)
Context: In strong families, positive strokes out-number negative broadsides by a wide margin. Members regularly express appreciation: "Thanks for fixing the drainpipe." "You look so nice in that dress." "The dinner was great." Criticism is offered gently. After all, strong families figure, if we can be kind to strangers, why not to one another?

“Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his Arithmetica Infinitorum (1655), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 253.

“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 (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 143.

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