Origin of Negative Numbers

Number is a fundamental concept in mathematics that has taken shape in a long historical development. The origin and formulation of this concept occurred simultaneously with the dawn, the birth, and development of mathematics. Man's practical activities, on the one hand, and the internal demands of mathematics, on the other, determined the development of the concept of number. The need to count objects led to the emergence of the concept of Natural number.

All nations that developed forms of writing introduced the concept of Natural number and developed a counting system. The subsequent development of the concept of number proceeded mainly due to the development of mathematics itself. Negative numbers first appear in ancient China. The Chinese were used to calculating with two collections of bars - red for positive numbers and black for negative numbers. However, they did not accept the idea that a negative number could be a solution to an equation.

Indian mathematicians discovered negative numbers when trying to formulate an algorithm for solving quadratic equations. Brahomagupta's contributions are an example of this, since the systematized arithmetic of negative numbers is first found in his work. The rules about quantities were already known from the Greek subtraction theorems, such as (a -b) (c -d) = ac + bd -ad -bc, but Hindus converted them into numerical rules.
about negative and positive numbers.

Diophantus (3rd century) operated easily with negative numbers. They constantly appeared in intermediate calculations on many problems of their "Aritmetika", however there were certain problems for which the solutions were negative integers such as:

4 = 4x +20
3x -18 = 5x ^ 2

In these situations Diophantus merely classified the problem as absurd. In the sixteenth and seventeenth centuries, many European mathematicians did not appreciate negative numbers, and if these numbers appeared in their calculations, they considered them false or impossible. An example of this would be Michael Stifel (1487-1567) who refused to admit negative numbers as the roots of an equation, calling them "numeri absurdi". Cardano used the negative numbers while calling them "numeri ficti". The situation changed from the 18th century when a geometric interpretation of positive and negative numbers was discovered as segments of opposite directions.

Euler, a virtuoso of calculus as found in his scientific articles by the bold manner in which he handled relative numbers and without raising questions as to the legitimacy of his constructions, provided an explanation or justification for the rule's signs. Consider your arguments:

1 - Multiplying a debt by a positive number is not difficult, since 3 debts of a escudos is a debt of 3 escudos, so (b). (- a) = -ab.

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2 - By commutativity, Euler deduced that (-a). (B) = -ab
From these two arguments it follows that the product of a positive quantity by a negative quantity and vice versa is a negative quantity.

3 - It remains to be determined which product of (-a) by (-b). Of course, says Euler, the absolute value is ab. It is therefore necessary to decide between ab or -ab. But since (-a) 'b is -ab, it only remains as the only possibility that (-a). (- b) = + ab.

Of course, this kind of argument demonstrates that any more zealous "spirit", such as Stendhal, cannot be satisfied, since mainly Euler's third argument cannot consistently prove or even justify that - by - = +. Basically, this type of argument denotes that Euler was not yet knowledgeable enough to justify these results acceptably. In the same work of Euler we can see that he understands the negative numbers as just a quantity that can be represented by a letter preceded by the sign - (minus). Euler does not yet understand that negative numbers are quantities less than zero.

Next: Origin of Probabilities