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negative numbers

The introduction of the negative numbers is due to the need for subtraction to be performable without restriction. In the domain of positive numbers the subtraction a - b = c can only be carried out if a > b (a is greater than b). If, on the other hand, a < b (a is less than b) we define c = -(b - a), e.g., 5 - 7 = (-2). Here the "- sign" on the left hand side of the equation represents an operation; on the right hand side it forms part of the number itself. In the case of positive numbers the associated sign (+) may be omitted, but not in the case of negative numbers. Representation of the numbers on a straight line clarifies the notion of a negative number.

Long denied legitimacy in mathematics, negative numbers are nowhere to be found in the writings of the Babylonians, Greeks, or other ancient cultures. On the contrary, because Greek mathematics was grounded in geometry, and the concept of a negative distance is meaningless, negative numbers seemed to make no sense. They surface for the first time in in bookkeeping records seventh-century India and in a chapter of a work by the Hindu astronomer Brahmagupta. Their earliest documented use in Europe is in 1545 in the Ars magna of Girolamo Cardano. By the early 17th century, Renaissance mathematicians were explicitly using negative numbers but also meeting with heavy opposition. René Descartes called negative roots "false roots", and Blaise Pascal was convinced that numbers "less than zero" couldn't exist. Gottfried Leibniz admitted that they could lead to some absurd conclusions, but defended then as useful aids in calculation. By the 18th century, negative numbers had become an indispensable part of algebra.

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