Difference between revisions of "Absolute value"

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The '''absolute value''' of a [[real number]] <math>x</math>, denoted <math>|x|</math>, is its distance from 0. Therefore, if <math>x\ge 0</math>, then <math>|x|=x</math>, and if <math>x<0</math>, then <math>\displaystyle |x|=-x</math>.
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The '''absolute value''' of a [[real number]] <math>x</math>, denoted <math>|x|</math>, is its distance from 0. Therefore, if <math>x\ge 0</math>, then <math>|x|=x</math>, and if <math>x<0</math>, then <math>\displaystyle |x|=-x</math>. This is equivalent to "dropping the minus sign."
  
 
Similarly, the absolute value of a [[complex number]] <math>z=x+iy</math>, where <math>x,y\in\mathbb{R}</math>, is <math>|z|=\sqrt{x^2+y^2}</math>.
 
Similarly, the absolute value of a [[complex number]] <math>z=x+iy</math>, where <math>x,y\in\mathbb{R}</math>, is <math>|z|=\sqrt{x^2+y^2}</math>.
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The absolute value functions listed above have three very important properties:
 
The absolute value functions listed above have three very important properties:
  
*<math> |x|\ge 0</math> for all ''x'', and <math>|x|=0</math> [[iff]] ''x=0''.
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*<math> |x|\ge 0</math> for all <math>x</math>, and <math>|x|=0</math> if and only if <math>x=0</math>.
 
*<math> |x\times y|=|x|\times |y|</math>.
 
*<math> |x\times y|=|x|\times |y|</math>.
*<math> |x+y| \le |x|+|y|</math>. ([[Triangle inequality]])
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*<math> |x+y| \le |x|+|y|</math>. (The [[triangle inequality]])
  
 
We call ''any'' function satisfying these three properties ''an absolute value'', or a ''norm''.
 
We call ''any'' function satisfying these three properties ''an absolute value'', or a ''norm''.
  
Another example of an absolute value is the ''p''-[[p-adic|adic]] absolute value on <math>\mathbb{Q}</math>, the [[rational number]]s. Let <math>x=\prod_{i=1}^n p_i^{e_i}</math>, where the <math>p_{i}</math>'s are distinct [[prime number]]s, and the <math>e_i</math>'s are ([[positive]], [[negative]], or [[zero]]) [[integer]]s. Define <math>|x|_{p_i}=p_i^{-e_i}</math>. This defines an absolute value on <math>\mathbb{Q}</math>. This absolute value satisfies a stronger triangle inequality:
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Another example of an absolute value is the ''p''-[[p-adic|adic]] absolute value on <math>\mathbb{Q}</math>, the [[rational number]]s. Let <math>x=\prod_{i=1}^n p_i^{e_i}</math>, where the <math>p_{i}</math>'s are distinct [[prime number]]s, and the <math>e_i</math>'s are ([[positive]], [[negative]], or [[zero (constant) | zero]]) [[integer]]s. Define <math>|x|_{p_i}=p_i^{-e_i}</math>. This defines an absolute value on <math>\mathbb{Q}</math>. This absolute value satisfies a stronger triangle inequality:
  
 
*<math> |x+y|\le\max(|x|,|y|)</math>.
 
*<math> |x+y|\le\max(|x|,|y|)</math>.

Revision as of 16:26, 15 August 2006

The absolute value of a real number $x$, denoted $|x|$, is its distance from 0. Therefore, if $x\ge 0$, then $|x|=x$, and if $x<0$, then $\displaystyle |x|=-x$. This is equivalent to "dropping the minus sign."

Similarly, the absolute value of a complex number $z=x+iy$, where $x,y\in\mathbb{R}$, is $|z|=\sqrt{x^2+y^2}$.


Introductory Concepts

Example Problems


Generalized absolute values

The absolute value functions listed above have three very important properties:

We call any function satisfying these three properties an absolute value, or a norm.

Another example of an absolute value is the p-adic absolute value on $\mathbb{Q}$, the rational numbers. Let $x=\prod_{i=1}^n p_i^{e_i}$, where the $p_{i}$'s are distinct prime numbers, and the $e_i$'s are (positive, negative, or zero) integers. Define $|x|_{p_i}=p_i^{-e_i}$. This defines an absolute value on $\mathbb{Q}$. This absolute value satisfies a stronger triangle inequality:

  • $|x+y|\le\max(|x|,|y|)$.

An absolute value satisfying this strong triangle inequality is called nonarchimedian. If an absolute value does not satisfy the strong triangle inequality, then it is called archimedian. The ordinary absolute value on $\mathbb{R}$ or $\mathbb{C}$ is archimedian.

The theory of absolute values is important in algebraic number theory. Let $K/\mathbb{Q}$ be a finite Galois extension with $[K:\mathbb{Q}]=n$, and let $\sigma_1,\ldots,\sigma_n$ be the field automorphisms of $K$ over $\mathbb{Q}$. Then the only absolute values are the archimedian ones given by $|x|_i=|\sigma_i(x)|$ (the ordinary real or complex absolute values) and the nonarchimedian ones given by $|x|_{\mathfrak{p}}$ for some prime ${\mathfrak{p}}$ of $K$.

See also