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Negative number

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 - 1x - 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.
— Negative number hate
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.
— Negative number positive
Albert Girard was the first who understood the use of negative roots in the solution of geometric problems.
— Negative number 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.
— Negative number science
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.
— Negative number time
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.
— Negative number writing
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.
— Negative number time
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.
— Negative number success
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.
— Negative number art
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.
— Negative number time
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.
— Negative number art
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.
— Negative number books

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