A negative number is an element of the set of negative numbers, which (together with zero) appeared in mathematics during the expansion of the set of natural numbers. The main purpose of the expansion was the desire to make a subtraction operation the same grade as an addition. As the rule of the natural numbers claims, you can deduct only the smaller number from the larger, and the commutative law does not include the deduction - for example, the expression 3 + 4-5 is acceptable, and the expression with transposed operands 3-5 + 4 is unacceptable.
All negative numbers are just less than zero. On the real axis, negative numbers are located on the left of zero. For them, as well as for the positive numbers, order relation is determined allowing the comparison of a single integer to another. For each natural number n, there is one and only one negative number, denoted, -n, which adds n to zero. Both numbers are called the opposite of each other. Subtracting a whole number of other integer b is equivalent to the addition of b opposite to a.
Adding to the natural numbers, negative numbers and zero enables subtraction for all pairs of natural numbers. As a result of this expansion the set "integer" was created. With the further expansion of the set of numbers by rational numbers, real numbers, complex numbers, and others, for them the same way we obtain the corresponding negative values. The concepts of positive and negative numbers can be defined in any ordered set. Most often, these terms refer to one of the following numeric systems:
The above properties of the members on this list have a place in the general case. For complex numbers the concept of "positive" and "negative" are not applicable.
Ancient Egypt, Babylon and Ancient Greece did not use negative numbers, and if they get negative roots of equations (by subtraction), they are dismissed as impossible. The exception was Diophantus that in the III century already knew the rule of signs and know how to multiply a negative number. However, he considered them only as an intermediate step that is useful to calculate the final, positive result. For the first time negative numbers were partially legalized in China, and later (approximately VII century) in India, which were treated as debt (loss), or, as Diophantus, recognized as time values. Multiplication and division for negative numbers had not yet been identified. The usefulness and legality of negative numbers were approved gradually. Indian mathematician Brahmagupta (VII century) had already considered them on a par with the positive.
In Europe, the recognition came a thousand years later and even then for a long time, negative numbers were called "false", "imaginary" or "absurd". The first description of them in European literature appeared in the "Book of abaca" Leonardo of Pisa (1202), which interpreted the negative numbers as a necessity. Bombelli and Girard in his writings considered negative numbers are quite acceptable and beneficial, in particular, to refer to a lack of something. Even in the XVII century, Pascal believed that 0-4 = 0, since "nothing can be less than nothing". An echo of those times is the fact that in modern arithmetic operation that is subtraction, the sign of negative numbers is indicated by the same symbol (minus), although it is algebraically completely different concepts.
In the XVII century, with the advent of analytic geometry, negative numbers got a visual geometrical representation on the number line. From this moment begins their full equality. Nevertheless, the theory of negative numbers for a long time was in its infancy. Animatedly discussed, for example, was a strange proportion 1: (- 1) = (-1): 1 - its first term on the left is more than the second, and the right - on the contrary, it turns out that the more equal the lesser ( 'Arnaud paradox "). Wallis believed that negative numbers are less than zero, but at the same time more than infinity. It was not clear also, what sense has the multiplication of negative numbers, and why the multiplication of the negative numbers is positive; on this subject heated discussions were held. Gauss in 1831 considered that it is necessary to clarify that negative numbers in principle have the same rights as positive and that they are not applicable to all things, does not mean anything, because the fractions are also not applicable to all things (for example, do not apply when counting people).
The full and quite rigorous theory of negative numbers were created only in the XIX century by William Hamilton and Hermann Grassmann. Negative numbers are subject to substantially the same algebraic rules as natural but have some peculiarities. If any set of positive numbers bounded below, then any set of negative numbers is bounded from above. When multiplying integers, the rule of signs: the product of numbers with different signs is negative, with the same - positive. Multiplying both sides of the inequality by a negative number the inequality sign is reversed. For example, multiplying the inequality 3 <5 to -2, we get: -6> -10. When dividing the remainder with quotient can have either sign, but the remainder, by convention, always is non-negative (otherwise it is not uniquely determined).
Two's complement (the name for an additional code of negative numbers) is the most common way to represent negative integers in computers. It allows you to change the operation of subtraction into the addition operation, and to do addition and subtraction equally to signed and unsigned numbers to then simplify the architecture of the computer.
Additional code of a negative number can be obtained by inverting the module binary number (first addition) and the addition to the inversion unit (second addition), or by subtracting the number from zero. An additional code (2's complement) binary number is obtained by adding 1 to the least significant bit of its complement to 1. Supplement 2 to the binary number is defined as the value obtained by subtracting the number of the largest power of two.
When writing numbers in the additional code, MSB is a landmark. If its value is 0, the remaining bits is recorded as a positive binary number, which coincides with the direct code. Binary 8-bit signed integer in the additional code can be any integer in the range -128 to +127. If the MSB is zero, the largest integer that can be stored in the remaining 7 bits is 2 ^ 7-1, which is equal to 127. The same principle can be used in a computer representation of decimal numbers: for each figure category X is replaced by the 9-X, and the resulting number is added to 1. For example, when using a four-digit number -0081, it is replaced by 9919 (9919 + 0081 = 0000, the fifth category ejected).