Sign (mathematics)

In mathematics, the word **sign** refers to the property of being positive or negative.  Every real number that is non-zero is either positive or negative, and therefore has a sign.  Zero itself is without a sign, or signless.  In addition to putting signs into real numbers, the word sign is used throughout mathematics to indicate parts of mathematical objects that mean positivity and negativity. Usually, if numbers are seen without a sign, they are seen as a positive number. Otherwise, a minus sign "" is appended before the number to indicate that the number is a negative of another number.

The word sign is also sometimes used to refer to various mathematical signs, such as the plus and minus signs and the multiplication sign.

## Sign of a number

A real number is said to be positive if it 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.

In arithmetic, the sign of a number is often denoted by placing a plus or minus sign before the number.  For example, +3 would denote a positive 3, and −3 would denote a negative 3.  When no plus or minus sign is given, the main way of seeing it is that a number is positive. In algebra equations, a positive number may need to have a positive sign before it. For example, in algebraic subtraction, you would write (+8) + (-6).

The sign of any number that is not zero can be changed to positive 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 as |−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 mean the same number 0.

## Meanings of signs

Because zero is neither positive nor negative, the following 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 definition 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 of an angle

In many texts, it is common to see a sign together with the measure of an angle, particularly a located angle or an angle of rotation.  In such a situation, the sign says 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 put a sign to an angle of rotation in three dimensions, assuming the axis of rotation has been oriented.  Specifically, a right-handed turn around an axis usually counts as positive, while a left-handed rotation counts as negative.

### Sign of a direction

In arithmetic 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:

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.

### Other meanings

In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematics and other sciences:
- In graph theory, a signed graph is a graph in which each edge has been marked with a positive or negative sign.
- In physics, any electric charge comes with a sign, either positive or negative.  By general rules, 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.

## Related pages

- List of mathematical symbols