Sign (mathematics)
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In mathematics, the concept of sign originates from the property of every non-zero real number to be positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero. Along its application to real numbers, "change of sign" is used throughout mathematics and physics to denote the additive inverse (multiplication to −1), even for quantities which are not real numbers (so, which are not prescribed to be either positive, negative, or zero). Also, the word "sign" can indicate aspects of mathematical objects that resemble positivity and negativity, such as the sign of a permutation (see below).
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Sign of a number
Every number has multiple attributes (such as value, sign and magnitude). A real number is said to be positive if its value (not its magnitude) is greater than zero, and negative if it is less than zero. The attribute of being positive or negative is called the sign of the number. Zero itself is not considered to have a sign (though this is context dependent, see below). Also, signs are not defined for complex numbers, although the argument generalizes it in some sense.
In common numeral notation (which is used in arithmetic and elsewhere), the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes "positive three", and −3 denotes "negative three". When no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of negative numbers through subtraction, the minus sign is perceived to have a strong association with negative numbers (of the negative sign). Likewise, "+" associates with positivity.
In algebra, a minus sign is usually thought of as representing the operation of additive inverse (sometimes called negation), with the additive inverse of a positive number being negative and the additive inverse of a negative number being positive. In this context, it makes sense to write −(−3) = +3.
Any non-zero number can be changed to a positive one using the absolute value function. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| = 3 and |3| = 3.
Sign of zero
The number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the additive inverse of itself.
Note that this definition is culturally determined. In France and Belgium, 0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be "strictly positive" resp. "strictly negative".
In some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers (see signed zero).
One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits. This notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively; these behaviours are not necessarily the same.
Terminology for signs
Because zero is neither positive nor negative (in most countries), the following phrases are sometimes used to refer to the sign of an unknown number:
- A number is positive if it is greater than zero.
- A number is negative if it is less than zero.
- A number is non-negative if it is greater than or equal to zero.
- A number is non-positive if it is less than or equal to zero.
Thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero. For example, the absolute value of a real number is always non-negative, but is not necessarily positive.
The same terminology is sometimes used for functions that take real or integer values. For example, a function would be called positive if all of its values are positive, or non-negative if all of its values are non-negative.
Sign convention
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In many contexts the choice of sign convention (which range of values is considered positive and which negative) is natural, whereas in others the choice is arbitrary subject only to consistency, the latter necessitating an explicit sign convention.
Sign function
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The sign function or signum function is sometimes used to extract the sign of a number. This function is usually defined as follows:
Thus sgn(x) is 1 when x is positive, and sgn(x) is −1 when x is negative. For nonzero values of x, this function can also be defined by the formula
where |x| is the absolute value of x.
Meanings of sign
Sign of an angle
In many contexts, it is common to associate a sign with the measure of an angle, particularly an oriented angle or an angle of rotation. In such a situation, the sign indicates whether the angle is in the clockwise or counterclockwise direction. Though different conventions can be used, it is common in mathematics to have counterclockwise angles count as positive, and clockwise angles count as negative.
It is also possible to associate a sign to an angle of rotation in three dimensions, assuming the axis of rotation has been oriented. Specifically, a right-handed rotation around an oriented axis typically counts as positive, while a left-handed rotation counts as negative.
Sign of a change
When a quantity x changes over time, the change in the value of x is typically defined by the equation
Using this convention, an increase in x counts as positive change, while a decrease of x counts as negative change. In calculus, this same convention is used in the definition of the derivative. As a result, any increasing function has positive derivative, while a decreasing function has negative derivative.
Sign of a direction
In analytic geometry and physics, it is common to label certain directions as positive or negative. For a basic example, the number line is usually drawn with positive numbers to the right, and negative numbers to the left:
As a result, when discussing linear motion, displacement or velocity to the right is usually thought of as being positive, while similar motion to the left is thought of as being negative.
On the Cartesian plane, the rightward and upward directions are usually thought of as positive, with rightward being the positive x-direction, and upward being the positive y-direction. If a displacement or velocity vector is separated into its vector components, then the horizontal part will be positive for motion to the right and negative for motion to the left, while the vertical part will be positive for motion upward and negative for motion downward.
Signedness in computing
| most-significant bit | |||||||||
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | = | 127 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | = | 126 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | = | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | = | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | = | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | = | −1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | = | −2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | = | −127 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | = | −128 |
| Most computers use two's complement to represent the sign of an integer. | |||||||||
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In computing, an integer value may be either signed or unsigned, depending on whether the computer is keeping track of a sign for the number. By restricting an integer variable to non-negative values only, one more bit can be used for storing the value of a number. Because of the way integer arithmetic is done within computers, the sign of a signed integer variable is usually not stored as a single independent bit, but is instead stored using two's complement or some other signed number representation.
In contrast, real numbers are stored and manipulated as Floating point values. The floating point values are represented using three separate values, mantissa, exponent, and, sign. Given this separate sign bit, it is possible to represent both positive and negative zero. Most programming languages normally treat positive zero and negative zero as equivalent values, albeit, they provide means by which the distinction can be detected.
Other meanings
In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematics and the sciences:
- Words up to sign mean that for a quantity q is known that either q = Q or q = −Q for certain Q. It is often expressed as q = ±Q. For real numbers, it means that only the absolute value |q| of the quantity is known. For complex numbers and vectors, a quantity known up to sign is a stronger condition than a quantity with known magnitude: aside Q and −Q, there are many other possible values of q such that |q| = |Q|.
- The sign of a permutation is defined to be positive if the permutation is even, and negative if the permutation is odd.
- In graph theory, a signed graph is a graph in which each edge has been marked with a positive or negative sign.
- In mathematical analysis, a signed measure is a generalization of the concept of measure in which the measure of a set may have positive or negative values.
- In a signed-digit representation, each digit of a number may have a positive or negative sign.
- The ideas of signed area and signed volume are sometimes used when it is convenient for certain areas or volumes to count as negative. This is particularly true in the theory of determinants.
- In physics, any electric charge comes with a sign, either positive or negative. By convention, a positive charge is a charge with the same sign as that of a proton, and a negative charge is a charge with the same sign as that of an electron.
