Sign (number)

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A sign or sign (from Latin signum sign) is a sign that is placed in front of a real number to identify it as positive or negative . A negative number is always provided with a minus sign , while a positive number can optionally be preceded by a plus sign . The number zero is mostly seen as unsigned , but a signed zero is sometimes used in computer representation .

Strictly speaking, the sign, which is always unary , must be distinguished from the mathematical operator for addition (binary plus) or subtraction (binary minus) and the inversion operator for addition ( unary minus ). The latter comes closest to the sign of a number constant. However, there are programming languages that know a separate special character to identify negative number constants, for example APL .

For the sign-related variables such as rotation angles and directions , there are often different signs conventions .

Plus and minus signs

In arithmetic , the sign of a number (more precisely: a real number constant) is indicated by a preceding plus or minus sign . The same characters are used here as for the addition and subtraction of two numbers. The sign is attached directly to the first digit without any space . For example denote

{\ displaystyle +3}

and

{\ displaystyle -3}

each the positive and negative number (constant) three . If no sign is given, the number constant is regarded as non-negative. In algebra , the minus sign is also used as a unary minus for "sign reversal ", whereby the respective opposite number (a number constant or a variable) is obtained. For example

{\ displaystyle - (- x) = x}

for everyone in an additive group. The latter need not be arranged for this. And if it is an ordered group, nothing is said about the sign of the variable . ${\ displaystyle x}$ ${\ displaystyle x}$

The amount function reverses the sign of a negative number, while a positive number remains unchanged. For example are

{\ displaystyle | {-3} | = + 3}

and and .

{\ displaystyle | {+3} | = + 3}

{\ displaystyle | {-x} | = | {+ x} |}

A plus minus sign (or the minus plus sign ) is placed in front of a number if the statement is to apply to both versions (plus and minus). If it occurs more than once before a number (variable or constant) in an equation, this means that either the upper character or the lower character is to be chosen everywhere. For example ${\ displaystyle \ pm 3}$ ${\ displaystyle \ mp 3}$

{\ displaystyle \ cos (x \ pm y) = \ cos y \; \ cos x \ mp \ sin x \; \ sin y}

,

that summarizes two equations in one note is either (first + then -) or (first - then +).

Sign of zero

The number zero is neither positive nor negative and therefore has no sign. The opposite of the number zero is the zero itself. Thus denote

{\ displaystyle +0}

and

{\ displaystyle -0}

the same number zero. However, with machine numbers, the positive and negative zeros are sometimes viewed as two different numbers. Examples are the one's complement of integers or the IEEE 754 standard for floating point numbers . In some applications, the notation is also used when a negative number has been rounded to zero. In calculus , the notation is ${\ displaystyle -0}$

{\ displaystyle \ lim _ {x \ to 0+}}

or

{\ displaystyle \ lim _ {x \ to 0 {-}}}

used in the formation of a right-hand or left-hand limit value .

Sign function

The sign function

{\ displaystyle \ operatorname {sgn}}

With the help of the sign function or signum function, the sign of a ( real ) number variable can be determined. The sign function is usually by ${\ displaystyle x}$

{\ displaystyle \ operatorname {sgn} (x) = {\ begin {cases} -1 & {\ text {if}} x <0, \\ ~~ \, 0 & {\ text {if}} x = 0, \ \ ~~ \, 1 & {\ text {falls}} x> 0. \ end {cases}}}

Are defined. Hence , if the number is positive and if it is negative. Is , then the sign function, by also using the absolute value function ${\ displaystyle \ operatorname {sgn} (x) = 1}$ ${\ displaystyle x}$ ${\ displaystyle \ operatorname {sgn} (x) = - 1}$ ${\ displaystyle x \ neq 0}$

{\ displaystyle \ operatorname {sgn} (x) = {\ frac {x} {| x |}}}

To be defined.

Sign conventions

For many directed quantities, a sign is assigned, i.e. which values are viewed as positive and which as negative, in a natural way. In some cases, however, the choice of the sign is arbitrary and is only chosen uniformly for reasons of consistency. In these cases one speaks of a sign convention.

Signs of angles

Counterclockwise angles are considered positive and clockwise angles are considered negative.

In contrast to an undirected angle, a directed angle has an orientation that is indicated by a sign in front of the size of the angle. In particular, in the case of an angle of rotation, the sign indicates whether the rotation is clockwise or counterclockwise . Although different conventions are used for this, it is common in mathematics to view counterclockwise rotations as positive and clockwise rotations as negative.

It is also possible to assign a sign to a rotation in three dimensions, provided the axis of rotation has an orientation. According to the right-hand rule , a counterclockwise rotation around an oriented axis is considered positive in a right-hand system and negative in a left-hand system.

Signs of changes

If a variable changes over time, the change in size is typically defined as ${\ displaystyle x}$ ${\ displaystyle \ Delta x}$

{\ displaystyle \ Delta x = x _ {\ text {end}} - x _ {\ text {start}}}

.

With this convention, an increase corresponds to a positive change, while a decrease corresponds to a negative change. The same convention is used in calculus to define the derivative . As a result, a monotonically increasing differentiable function has a positive derivative, while a monotonically decreasing function has a negative derivative. ${\ displaystyle x}$ ${\ displaystyle x}$

Signs of directions

In analytical geometry and physics , certain directions are often marked as positive or negative. As a basic example, the number line is usually drawn with positive numbers on the right and negative numbers on the left:

Therefore, in the context of uniform motion, displacement or velocity vectors pointing to the right are usually considered positive while a vector pointing to the left is considered negative.

In a Cartesian coordinate system , the right and up directions are usually viewed as positive, with the right direction corresponding to the positive x-axis and the upward direction corresponding to the positive y-axis. If a displacement or velocity vector is broken down into its components, the vertical component will be positive for an upward movement and negative for a downward movement. In geodetic coordinate systems , however, the x and y axes are swapped. In the three-dimensional coordinate system, a distinction is made between “left-handed” and “right-handed” definitions, which correspond to different sign conventions for the direction of rotation .

Sign of physical quantities

The electrical charge is also given a sign, the charge of electrons being defined as negative. This corresponds to the convention of the “technical direction of current” as the direction of movement (often only imaginary) of positive charge carriers .

literature

Heinz-Dieter Ebbinghaus et al .: Numbers . Springer, Berlin 1992, ISBN 3-540-55654-0 .

Remarks

↑ This makes parenthesis of the negative number constants superfluous.

[1] This makes parenthesis of the negative number constants superfluous.