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Negative and non-negative numbers

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A signed number is a number which has either a positive or negative sign (or in the case of zero, neither). A negative number is a number that is less than zero, such as -3. A positive number is a number that is greater than zero, such as 3.

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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).

Non-Negative Numbers A non-negative 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

Addition and Subtraction

Adding a negative number is the same as subtracting the corresponding positive number:

5 + (-3) = 5 - 3 = 2
-2 + (-5) = -2 - 5 = -7

Subtracting a positive number from a smaller positive number yields a negative result:

4 - 6 = -2 (if you have $4 and spend $6 then you have a debt of $2).

Subtracting a positive number from any negative number yields a negative result:

-3 - 6 = -9 (if you have a debt of $3 and spend another $6, you have a debt of $9).

Subtracting a negative is equivalent to adding the corresponding positive:

5 - (-2) = 5 + 2 = 7 (if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).

Also:

(-8) - (-3) = -5 (if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).

Multiplication

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.

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:

(3 + (-3)) · (-4) = 3 · (-4) + (-3) · (-4).
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 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,

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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