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2 Positive numbers 3 Non-negative numbers 4 Arithmetic involving signed numbers 5 Computing |
Negative numbers
These include negative integers, negative rational numbers, negative real numbers, negative hyperreal numbers, and negative surreal numbers.
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.
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).
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.
For purposes of addition and subtraction, one can think of negative numbers as debts.
Adding a negative number is the same as subtracting the corresponding positive number:
Multiplication of a negative number by a positive number yields a negative result: (-2) · 3 = -6. The reason is that this multiplication can be understood as repeated addition: (-2) · 3 = (-2) + (-2) + (-2) = -6. Alternatively: if you have a debt of $2, and then your debt is tripled, you end up with a debt of $6.
Multiplication 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:
On a computer, the sign of a number (whether it is positive or negative) is usually expressed using the left-most 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,
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.
To represent both positive and negative (signed) integers, the convention is that the most significant bit of the binary representation of the number will be used to indicate the sign of the number, rather than contributing to its magnitude; three formats have been used for representing the magnitude: sign-and-magnitude, one's complement, and two's complement, the latter being by far the most common nowadays.
Positive numbers
Non-negative numbers
Arithmetic involving signed numbers
Addition and subtraction
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: Multiplication
The left hand side of this equation equals 0 · (-4) = 0. The right hand side is a sum of -12 + (-3) · (-4); for the two to be equal, we need (-3) · (-4) = 12.
Computing
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
Complement
Complementing a binary number means changing all the 0s to 1s and all the 1s to 0s; a Boolean NOT on each bit. A byte, holding 8 bits, can represent the values 00000000 (0) to 11111111 (25510), if all bits
are used to represent the magnitude of the number. This is called an unsigned integer.Sign-and-magnitude
Sign-and-magnitude is the simplest and most like human writing forms. The MSB is set to 0 for a positive number or zero, and set to 1 for a negative number. The remaining bits in the number indicate the (positive) magnitude. Hence in a byte with only seven bits (apart from the sign bit), the magnitude can range from 0000000 (0) to 1111111 (127). Thus you can represent numbers from -12710 to +12710. -43 encoded in a byte this way is 10101011.Ones' complement
The ones'-complement representation of a negative number is created by taking the complement of its positive counterpart. For example, negated 00101011 (43) becomes 11010100 (-43) (Notice how this is different from the sign-and-magnitude convention where the same bit pattern would be -84). The PDP-1 and UNIVAC 1100/2200 series use ones'-complement arithmetic. The range of signed numbers using one's complement in a byte is -12710 to +12710.