“Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.”

p, 125
Stetigkeit und irrationale Zahlen (1872)

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Richard Dedekind13

German mathematician 1831–1916

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“Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).”

Henry Burchard Fine (1858–1928) American academic

Source: The Number-System of Algebra, (1890), p. 86; Reported in Moritz (1914, 282)

“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.

“The integers, the rationals, and the irrationals, taken together, make up the continuum of real numbers. It's called a continuum because the numbers are packed together along the real number line with no empty spaces between them.”

Brian Hayes (scientist) (1900) American scientist, columnist and author

Source: Group Theory in the Bedroom (2008), Chapter 11, Identity Crisis, p. 206 (See also: George Cantor)

“One of the first algebraists to accept negative numbers was… placed negative numbers on a par with positive numbers and gave both roots of a quadratic equation, even when both were negative. Both Girard and Harriot used the minus sign for the operation of subtraction and for negative numbers.”

Morris Kline (1908–1992) American mathematician

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

“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 (1908–1992) American mathematician

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

“The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“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.

“I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

“In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iv
A Treatise on Algebra (1842)

“The seventeenth century witnessed the birth of modern science as we know it today. The science was something new, based on a direct confrontation of nature by experiment and observation. But there was another feature of the new science—a dependence on numbers, on real numbers of actual experience.
…The ancients knew a few numerical laws… But prior to the Scientific Revolution, the goal of science (or the study of nature) was not to seek laws of nature expressed in terms of numbers or number relations. …the new science …not only found laws based on numbers but they were also willing to express these laws in terms of higher powers of numbers—squares and cubes.”

I. Bernard Cohen (1914–2003) American historian of science

The Triumph of Numbers: How Counting Shaped Modern Life (2005)

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Richard Dedekind13

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