Absolute value

The graph of the absolute value function for real numbers

The absolute value of a number may be thought of as its distance from zero.

In mathematics, the absolute value or modulus of a real number $x$ , denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x|} , is the (non-negative) magnitude of $x$ measured without regard to its sign. Namely, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x|=x} if $x$ is a positive number, and $|x|=-x$ if $x$ is negative (in which case Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -x} is positive), and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |0|=0} . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

Terminology and notation

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for the complex absolute value,^[1]^[2] and it was borrowed into English in 1866 as the Latin equivalent modulus.^[1] The term absolute value has been used in this sense from at least 1806 in French^[3] and 1857 in English.^{[lower-alpha 1]} The notation $| x |$ , with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.^[5] Other names for absolute value include numerical value^[1] and magnitude.^[1] The absolute value of $x$ has also been denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {abs} x} in some mathematical publications,^[6] and in spreadsheets, programming languages, and computational software packages, the absolute value of ${\textstyle x}$ is generally represented by abs(x), or a similar expression,^[7] as it has been since the earliest days of high-level programming languages.^[8]

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant.^[9] Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example, a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the Euclidean norm^[10] or sup norm^[11] of a vector in $\mathbb {R} ^{n}$ , although double vertical bars with subscripts (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \|\cdot \|_{2}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \|\cdot \|_{\infty }} , respectively) are a more common and less ambiguous notation.

Definition and properties

Real numbers

For any real number $x$ , the absolute value or modulus of $x$ is denoted by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x|} , with a vertical bar on each side of the quantity, and is defined as^[12] Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}

The absolute value of $x$ is thus always either a positive number or zero, but never negative. When $x$ itself is negative ( $x<0$ ), then its absolute value is necessarily positive (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x|=-x>0} ).^[13]

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, and more generally, the absolute value of the difference of two real numbers (their absolute difference) is the distance between them.^[13] The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference.^[14] See § Distance below.

Since the square root symbol represents the unique positive square root, when applied to a positive number, it follows that^[15] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| = \sqrt{x^2}.} This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.^[16]

The absolute value has the following four fundamental properties (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle b} are real numbers), that are used for generalization of this notion to other domains:^[17]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a\| \ge 0 }	Non-negativity^[17]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a\| = 0 \iff a = 0 }	Positive-definiteness^[17]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|ab\| = \left\|a\right\| \left\|b\right\|}	Multiplicativity^[17]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a+b\| \le \|a\| + \|b\| }	Subadditivity, specifically the triangle inequality^[17]

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a+b|=s(a+b)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=\pm 1} , with its sign chosen to make the result positive. Now, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 \cdot x \le |x|} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1 \cdot x \le |x|} , it follows that, whichever of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm1} is the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} , one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \cdot x\leq |x|} for all real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . Consequently, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a+b|=s \cdot (a+b) = s \cdot a + s \cdot b \leq |a| + |b|} , as desired.

Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl\| \left\|a\right\| \bigr\| = \|a\|}	Idempotence (the absolute value of the absolute value is the absolute value)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\|-a\right\| = \|a\|}	Evenness (reflection symmetry of the graph)^[18]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a - b\| = 0 \iff a = b }	Identity of indiscernibles (equivalent to positive-definiteness)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a - b\| \le \|a - c\| + \|c - b\| }	Triangle inequality (equivalent to subadditivity)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\|\frac{a}{b}\right\| = \frac{\|a\|}{\|b\|}\ } (if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \ne 0} )	Preservation of division – equivalent to multiplicativity^[19]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a-b\| \geq \bigl\| \left\|a\right\| - \left\|b\right\| \bigr\| }	Reverse triangle inequality – equivalent to subadditivity^[19]

Two other useful properties concerning inequalities are:^[19]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a| \le b \iff -b \le a \le b }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a| \ge b \iff a \le -b\ } or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge b }

These relations may be used to solve inequalities involving absolute values. For example:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|x-3\| \le 9 }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff -9 \le x-3 \le 9 }
	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iff -6 \le x \le 12 }

The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers.

Complex numbers

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The absolute value of a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} from the origin. It is also seen in the picture that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} and its complex conjugate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar z} have the same absolute value.

Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x + iy,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} are real numbers, the absolute value or modulus of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|} and is defined by^[20] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z| = \sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}=\sqrt{x^2 + y^2},} the Pythagorean addition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Re}(z)=x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Im}(z)=y} denote the real and imaginary parts of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} , respectively. When the imaginary part Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is zero, this coincides with the definition of the absolute value of the real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .^[20]

When a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is expressed in its polar form as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = r e^{i \theta},} its absolute value is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z| = r.}

Since the product of any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} and its complex conjugate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar z = x - iy} , with the same absolute value, is always the non-negative real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(x^2 + y^2\right)} , the absolute value of a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is the square root of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \cdot \overline{z},} which is therefore called the absolute square or squared modulus of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} :^[20] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z| = \sqrt{z \cdot \overline{z}}.} This generalizes the alternative definition for reals: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |x| = \sqrt{x\cdot x}} .

The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|^n = |z^n|} is a special case of multiplicativity that is often useful by itself.^[20]

Absolute value function

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The graph of the absolute value function for real numbers

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Composition of absolute value with a cubic function in different orders

The real absolute value function is continuous everywhere. It is differentiable everywhere except for $x = 0$ . It is monotonically decreasing on the interval $(-\infty, 0]$ and monotonically increasing on the interval $[0, +\infty)$ .^[21] Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible.^[22] The real absolute value function is a piecewise linear, convex function.^[18]

For both real and complex numbers, the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself).

Relationship to the sign function

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| = x \sgn(x),}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| \sgn(x) = x,}

and for $x \neq 0$ ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sgn(x) = \frac{|x|}{x} = \frac{x}{|x|}.}

Relationship to the max and min functions

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s,t\in\R} , then the following relationship to the minimum and maximum functions hold:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |t-s|= -2 \min(s,t)+s+t}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |t-s|=2 \max(s,t)-s-t.}

The formulas can be derived by considering each case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s>t} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t>s} separately.

From the last formula one can derive also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |t|= \max(t,-t)} .

Derivative

The real absolute value function has a derivative for every $x \neq 0$ , given by a step function equal to the sign function except at $x = 0$ where the absolute value function is not differentiable:^[23]^[24] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d\left|x\right|}{dx} &= \frac{x}{|x|} = \begin{cases} -1 & x<0 \\ 1 & x>0 \end{cases} \\[7mu] &= \sgn x\quad \text{for } x \ne 0. \end{align}}

The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist.

The subdifferential of $| x |$ at $x = 0$ is the interval $[-1, 1]$ .^[25]

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.^[23]

The second derivative of $| x |$ with respect to $x$ is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function.

Antiderivative

The antiderivative (indefinite integral) of the real absolute value function is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \left|x\right| dx = \frac{x\left|x\right|}{2} + C,}

where $C$ is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.

Derivatives and antiderivatives of compositions

The following three formulae are special cases of the chain rule:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\text{d}^n \over \text{d}x^n} f(|x|)= (\sgn x)^n f^{(n)}(|x|)\quad \text{for } x \ne 0\,,}

if the absolute value is inside a function, and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\text{d}^n \over \text{d}x^n} |f(x)|=\sgn(f(x)) f^{(n)}(x)\quad \text{for } f(x) \ne 0\,,}

if another function is inside the absolute value. Combining both, the result is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\text{d}^n \over \text{d}x^n} |f(|x|)|=(\sgn x)^n \sgn(f(|x|)) f^{(n)}(|x|)\quad \text{for } x \ne 0, f(|x|) \ne 0\,.}

From these formulae and using integration by parts, antiderivatives can also be obtained:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {f(|x|)\text{d}x}= \sgn (x) F(|x|)\quad \text{for } x \ne 0\,,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {|f(x)| \text{d}x} = \int {{|f(x)| \over f(x)} f(x) \text{d}x} = \sgn(f(x)) F(x)\quad \text{for } f(x) \ne 0\,,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {|f(|x|)| \text{d}x} = \int {{|f(|x|)| \over f(|x|)} f(|x|) \text{d}x} = \sgn (x) \sgn(f(|x|)) F(|x|)\quad \text{for } x \ne 0, f(|x|) \ne 0\,,}

supposing the derivative of the sign function is 0.

Power rule for expressions with absolute values

Using chain and product rules, the power rule for expressions of the type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n |x|^m} can be written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\text{d} \over \text{d}x} x^n |x|^m = (n+m)x^{n-1}|x|^m\,.}

This holds true even for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=0} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\text{d} \over \text{d}x} |x|^m = mx^{-1}|x|^m\,.}

Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.

The standard Euclidean distance between two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = (a_1, a_2, \dots , a_n) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = (b_1, b_2, \dots , b_n) } in Euclidean $n$ -space is defined as:^[14] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\textstyle\sum_{i=1}^n(a_i-b_i)^2}. }

This can be seen as a generalisation, since for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1} real, i.e. in a 1-space, according to the alternative definition of the absolute value,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_1 - b_1| = \sqrt{(a_1 - b_1)^2} = \sqrt{\textstyle\sum_{i=1}^1(a_i-b_i)^2},}

and for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = a_1 + i a_2 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = b_1 + i b_2 } complex numbers, i.e. in a 2-space,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a - b\| }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \|(a_1 + i a_2) - (b_1 + i b_2)\|}
	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \|(a_1 - b_1) + i(a_2 - b_2)\|}
	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2} = \sqrt{\textstyle\sum_{i=1}^2(a_i-b_i)^2}.}

The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.

The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows:

A real valued function $d$ on a set $X \times X$ is called a metric (or a distance function) on $X$ , if it satisfies the following four axioms:^{[lower-alpha 2]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(a, b) \ge 0 }	Non-negativity
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(a, b) = 0 \iff a = b }	Identity of indiscernibles
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(a, b) = d(b, a) }	Symmetry
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(a, b) \le d(a, c) + d(c, b) }	Triangle inequality

Generalizations

Ordered rings

The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if $a$ is an element of an ordered ring R, then the absolute value of $a$ , denoted by $| a |$ , is defined to be: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a| = \left\{ \begin{array}{rl} a, & \text{if } a \geq 0 \\ -a, & \text{if } a < 0. \end{array}\right. } where $- a$ is the additive inverse of $a$ , 0 is the additive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring.^[26]

Fields

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function $v$ on a field $F$ is called an absolute value (also a modulus, magnitude, value, or valuation)^[27]^{[lower-alpha 3]} if it satisfies the following four axioms:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(a) \ge 0 }	Non-negativity
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(a) = 0 \iff a = \mathbf{0} }	Positive-definiteness
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(ab) = v(a) v(b) }	Multiplicativity
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(a+b) \le v(a) + v(b) }	Subadditivity or the triangle inequality

Where 0 denotes the additive identity of $F$ . It follows from positive-definiteness and multiplicativity that $v (1) = 1$ , where 1 denotes the multiplicative identity of $F$ . The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If $v$ is an absolute value on $F$ , then the function $d$ on $F \times F$ , defined by $d (a, b) = v (a - b)$ , is a metric and the following are equivalent:

$d$ satisfies the ultrametric inequality Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x, y) \leq \max(d(x,z),d(y,z))} for all $x$ , $y$ , $z$ in $F$ .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \left\{ v\left( \sum_{k=1}^n \mathbf{1}\right) : n \in \N \right\} } is bounded in R.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\left({\textstyle \sum_{k=1}^n } \mathbf{1}\right) \le 1\ } for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \N} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(a) \le 1 \Rightarrow v(1+a) \le 1\ } for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in F} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(a + b) \le \max \{v(a), v(b)\}\ } for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b \in F} .

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.^[28]

Vector spaces

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on a vector space $V$ over a field $F$ , represented as $Template:Norm$ , is called an absolute value, but more usually a norm, if it satisfies the following axioms:

For all $a$ in $F$ , and $v$ , $u$ in $V$ ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \\|\mathbf{v}\\| \ge 0 }	Non-negativity
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \\|\mathbf{v}\\| = 0 \iff \mathbf{v} = 0}	Positive-definiteness
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \\|a \mathbf{v}\\| = \left\|a\right\| \left\\|\mathbf{v}\right\\| }	Absolute homogeneity or positive scalability
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \\|\mathbf{v} + \mathbf{u}\\| \le \\|\mathbf{v}\\| + \\|\mathbf{u}\\| }	Subadditivity or the triangle inequality

The norm of a vector is also called its length or magnitude.

In the case of Euclidean space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} , the function defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|(x_1, x_2, \dots , x_n) \| = \sqrt{\textstyle\sum_{i=1}^{n} x_i^2}}

is a norm called the Euclidean norm. When the real numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} are considered as the one-dimensional vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^1} , the absolute value is a norm, and is the $p$ -norm (see L^p space) for any $p$ . In fact the absolute value is the "only" norm on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^1} , in the sense that, for every norm $Template:Norm$ on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^1} , $Template:Norm = Template:Norm \cdot | x |$ .

The complex absolute value is a special case of the norm in an inner product space, which is identical to the Euclidean norm when the complex plane is identified as the Euclidean plane Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} .

Composition algebras

Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x.

The real numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} , and quaternions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{H}} are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general, the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x).

Notes

↑ The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term absolute value is also used in contrast to relative value.^[4]
↑ These axioms are not minimal; for instance, non-negativity can be derived from the other three: $0 = d (a, a) \leq d (a, b) + d (b, a) = 2 d (a, b)$ .
↑ This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolute value.

Footnotes

↑ ^1.0 ^1.1 ^1.2 ^1.3 Oxford English Dictionary, Draft Revision, June 2008
↑ Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877
↑ Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105 at Google Books
↑ Peirce, James Mill. A Text-book of Analytic Geometry. p. 42 – via Internet Archive.
↑ Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25
↑ Siegel (1942).
↑ Bluttman (2015), p. 135.
↑ Knuth (1962), p. 43, 126.
↑ Sargent (2025), p. 10.
↑ Spivak (1965), p. 1.
↑ Munkres (1991), p. 4.
↑ Mendelson (2008), p. 2.
↑ ^13.0 ^13.1 Smith (2013), p. 8.
↑ ^14.0 ^14.1 Tabak (2014), p. 150.
↑ Varberg, Purcell & Rigdon (2007), p. 13.
↑ Stewart (2001), p. A5.
↑ ^17.0 ^17.1 ^17.2 ^17.3 ^17.4 Shechter (1997), p. 259.
↑ ^18.0 ^18.1 Varberg, Purcell & Rigdon (2007), p. 32.
↑ ^19.0 ^19.1 ^19.2 Varberg, Purcell & Rigdon (2007), p. 11.
↑ ^20.0 ^20.1 ^20.2 ^20.3 González (1992), p. 19.
↑ Varberg, Purcell & Rigdon (2007), p. 84.
↑ Baronti et al. (2016), p. 37.
↑ ^23.0 ^23.1 "Weisstein, Eric W. Absolute Value. From MathWorld – A Wolfram Web Resource".
↑ Bartle (2011), p. 163.
↑ Curnier (1999), p. 31–32.
↑ Mac Lane & Birkhoff (1999), p. 264.
↑ Shechter (1997), p. 260.
↑ Shechter (1997), pp. 260–261.

References

Baronti, Marco; De Mari, Filippo; van der Putten, Robertus; Venturi, Irene (2016). Calculus Problems. Springer. doi:10.1007/978-3-319-15428-2. ISBN 978-3-319-15428-2.
Bartle, Sherbert (2011). Introduction to real analysis (4th ed.). John Wiley & Sons. ISBN 978-0-471-43331-6.
Bluttman, Ken (2015). "Ignoring signs". Excel Formulas and Functions For Dummies. John Wiley & Sons. p. 135. ISBN 9781119076780.
Curnier, A. (1999). Wriggers, Peter; Panatiotopoulos, Panagiotis (eds.). New Developments in Contact Problems. Springer. ISBN 3-211-83154-1.
González, Mario O. (1992). Classical Complex Analysis. CRC Press. p. 19. ISBN 9780824784157.
Knuth, D. E. (1962). "Invited papers: History of writing compilers". Proceedings of the 1962 ACM National Conference. ACM Press. doi:10.1145/800198.806098.
Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra. American Mathematical Society. ISBN 978-0-8218-1646-2.
Mendelson, Elliott (2008). Schaum's Outline of Beginning Calculus. McGraw-Hill Professional. ISBN 978-0-07-148754-2.
Munkres, James (1991). Analysis on Manifolds. Boulder, CO: Westview. ISBN 0201510359.
Nahin, Paul J. (1998). An Imaginary Tale (hardcover ed.). Princeton University Press. ISBN 0-691-02795-1.
O'Connor, J.J.; Robertson, E.F. "Jean Robert Argand". Scotland: School of Mathematics and Statistics, University of St Andrews. Archived from the original on 2 April 2019.
Template:Cite tech report
Shechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.
Siegel, Carl Ludwig (1942). "Note on automorphic functions of several variables". Annals of Mathematics. Second Series. 43 (4): 613–616. doi:10.2307/1968953. JSTOR 1968953. MR 0008095.
Smith, Karl (2013). Precalculus: A Functional Approach to Graphing and Problem Solving. Jones & Bartlett Publishers. p. 8. ISBN 978-0-7637-5177-7.
Spivak, Michael (1965). Calculus on Manifolds. Boulder, CO: Westview. ISBN 0805390219.
Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1.
Tabak, John (2014). Geometry: The Language of Space and Form. Facts on File math library. Infobase Publishing. ISBN 978-0-8160-6876-0.
Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007). Calculus (9th ed.). Pearson Prentice Hall. p. 11. ISBN 978-0131469686.

External links

[5] The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term absolute value is also used in contrast to relative value.^[4]

[27] These axioms are not minimal; for instance, non-negativity can be derived from the other three: $0 = d (a, a) \leq d (a, b) + d (b, a) = 2 d (a, b)$ .

[30] This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolute value.

[oed-1] 1.0 ^1.1 ^1.2 ^1.3 Oxford English Dictionary, Draft Revision, June 2008

[2] Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877

[3] Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105 at Google Books

[4] Peirce, James Mill. A Text-book of Analytic Geometry. p. 42 – via Internet Archive.

[6] Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25

[FOOTNOTESiegel1942-7] Siegel (1942).

[FOOTNOTEBluttman2015[httpsbooksgooglecombooksid3pVxBgAAQBAJpgPA135_135]-8] Bluttman (2015), p. 135.

[FOOTNOTEKnuth196243,_126-9] Knuth (1962), p. 43, 126.

[FOOTNOTESargent202510-10] Sargent (2025), p. 10.

[FOOTNOTESpivak19651-11] Spivak (1965), p. 1.

[FOOTNOTEMunkres19914-12] Munkres (1991), p. 4.

[FOOTNOTEMendelson2008[httpsbooksgooglecombooksidA8hAm38zsCMCpgPA2_2]-13] Mendelson (2008), p. 2.

[FOOTNOTESmith2013[httpsbooksgooglecombooksidZUJbVQN37bICpgPA8_8]-14] 13.0 ^13.1 Smith (2013), p. 8.

[FOOTNOTETabak2014[httpsbooksgooglecombooksidr0HuPiexnYwCpgPA150_150]-15] 14.0 ^14.1 Tabak (2014), p. 150.

[FOOTNOTEVarbergPurcellRigdon2007[httpsarchiveorgdetailsmatematika-a-purcell-calculus-9th-edpage13_13]-16] Varberg, Purcell & Rigdon (2007), p. 13.

[FOOTNOTEStewart2001A5-17] Stewart (2001), p. A5.

[FOOTNOTEShechter1997[httpsbooksgooglecombooksideqUv3Bcd56ECpgPA259_259]-18] 17.0 ^17.1 ^17.2 ^17.3 ^17.4 Shechter (1997), p. 259.

[FOOTNOTEVarbergPurcellRigdon2007[httpsarchiveorgdetailsmatematika-a-purcell-calculus-9th-edpage32_32]-19] 18.0 ^18.1 Varberg, Purcell & Rigdon (2007), p. 32.

[FOOTNOTEVarbergPurcellRigdon2007[httpsarchiveorgdetailsmatematika-a-purcell-calculus-9th-edpage11_11]-20] 19.0 ^19.1 ^19.2 Varberg, Purcell & Rigdon (2007), p. 11.

[FOOTNOTEGonzález1992[httpsbooksgooglecombooksidncxL7EFr7GsCpgPA19_19]-21] 20.0 ^20.1 ^20.2 ^20.3 González (1992), p. 19.

[FOOTNOTEVarbergPurcellRigdon2007[httpsarchiveorgdetailsmatematika-a-purcell-calculus-9th-edpage84_84]-22] Varberg, Purcell & Rigdon (2007), p. 84.

[FOOTNOTEBaronti_et_al.2016[httpbooksgooglecombooksiddBFuDQAAQBAJpgPA37_37]-23] Baronti et al. (2016), p. 37.

[MathWorld-24] 23.0 ^23.1 "Weisstein, Eric W. Absolute Value. From MathWorld – A Wolfram Web Resource".

[FOOTNOTEBartle2011163-25] Bartle (2011), p. 163.

[FOOTNOTECurnier1999[httpsbooksgooglecombooksidtiBtC4GmuKcCpgPA31_31&ndash;32]-26] Curnier (1999), p. 31–32.

[FOOTNOTEMac_LaneBirkhoff1999[httpsbooksgooglecombooksidL6FENd8GHIUCpgPA264_264]-28] Mac Lane & Birkhoff (1999), p. 264.

[FOOTNOTEShechter1997[httpsbooksgooglecombooksideqUv3Bcd56ECpgPA260_260]-29] Shechter (1997), p. 260.

[FOOTNOTEShechter1997[httpsbooksgooglecombooksideqUv3Bcd56ECpgPA260_260–261]-31] Shechter (1997), pp. 260–261.

[1]

[2]

[3]

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[5]

[6]

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[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[lower-alpha 2]

[26]

[27]

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[28]

[4]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|a - b\| }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \|(a_1 + i a_2) - (b_1 + i b_2)\|}
	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \|(a_1 - b_1) + i(a_2 - b_2)\|}
	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2} = \sqrt{\textstyle\sum_{i=1}^2(a_i-b_i)^2}.}

Absolute value

Page version status

Contents

Terminology and notation

Definition and properties

Real numbers

Complex numbers

Absolute value function

Relationship to the sign function

Relationship to the max and min functions

Derivative

Antiderivative

Derivatives and antiderivatives of compositions

Power rule for expressions with absolute values

Distance

Generalizations

Ordered rings

Fields

Vector spaces

Composition algebras

See also

Notes

Footnotes

References

External links

Navigation menu

Absolute value

Terminology and notation

Definition and properties

Real numbers

Complex numbers

Absolute value function

Relationship to the sign function

Relationship to the max and min functions

Derivative

Antiderivative

Derivatives and antiderivatives of compositions

Power rule for expressions with absolute values

Distance

Generalizations

Ordered rings

Fields

Vector spaces

Composition algebras

See also

Notes

Footnotes

References

External links

Navigation menu

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