
Negative Numbers These include negative integers, negative rational numbers, negative real numbers, negative hyperreal numbers, and negative surreal numbers. Zero itself does not count as a negative number.
Negative integers can be regarded as an extension of the natural numbers, such that the equation x  y = z has a meaningful solution for all values of x and y. The other sets of numbers are then derived as progressively more elaborate extensions and generalizations from the integers.
Negative numbers are useful to describe values on a scale that goes below zero, such as temperature, and also in bookkeeping where they can be used to represent debts. In bookkeeping, debts are often represented by red numbers, or a number in parentheses.
Positive Numbers In the context of complex numbers positive implies real, but for clarity one may say "positive real number". Zero is not a positive number, though in computing zero is sometimes treated as though it were a positive number (due to the way that numbers are typically represented).
NonNegative Numbers A nonnegative number is either zero or a positive number. A number is nonnegative if and only if it is greater than or equal to zero, i.e. positive or zero. Thus the nonnegative integers are all the integers from zero on upwards, and the nonnegative reals are all the real numbers from zero on upwards.
A real matrix A is called nonnegative if every entry of A is nonnegative.
A real matrix A is called totally nonnegative by matrix theorists or totally positive by computer scientists if the determinant of every square submatrix of A is nonnegative.
Arithmetic involving signed numbers
Subtracting a positive number from a smaller positive number yields a negative result:
Subtracting a positive number from any negative number yields a negative result:
Subtracting a negative is equivalent to adding the corresponding positive:
Also:
Mulitplication of two negative numbers yields a positive result: (3) · (4) = 12. This situation cannot be understood as repeated addition, and the analogy to debts doesn't help either. The ultimate reason for this rule is that we want the distributive law to work:
Computing On a computer, the sign of a number (whether it is positive or negative) is usually expressed using the leftmost bit. If the bit is 1, the whole number is negative, otherwise the number is not negative (zero or positive). Such an integer or variable is called signed. There are many different ways to represent numbers; see Integral data type for more information on how integers are typically represented on computers. The most common system for representing negative integers in a fixed set of bits is termed "two's complement", in which negative numbers are represented by complementing the absolute value and then adding one to the value as if it were unsigned. For example, if an integer is expressed by 8 bits,
digits binary actual value 0 00000000 0 1 00000001 1 .... 126 01111110 126 127 01111111 127 128 10000000 128 129 10000001 127 130 10000010 126 .... 254 11111110 2 255 11111111 1
Usually, the computer's central processing unit (CPU) can use both signed and unsigned variables. In typical computer architectures there is no way to determine if a given digit is signed or unsigned at runtime because 255 and 1, for instance, appear the same in memory, and both addition, subtraction and multiplication are identical between signed and unsigned values, assuming overflow is ignored, by simply cutting off higher bits than can be stored. The datatype of the value dictates which operation should be applied.
There is a duplicate material at Computer numbering formats.
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