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>. | + | 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 | + | *<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>. ([[ | + | *<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: | + | 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 15:26, 15 August 2006
The absolute value of a real number , denoted , is its distance from 0. Therefore, if , then , and if , then . This is equivalent to "dropping the minus sign."
Similarly, the absolute value of a complex number , where , is .
Contents
[hide]Introductory Concepts
Example Problems
Generalized absolute values
The absolute value functions listed above have three very important properties:
- for all , and if and only if .
- .
- . (The triangle inequality)
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 , the rational numbers. Let , where the 's are distinct prime numbers, and the 's are (positive, negative, or zero) integers. Define . This defines an absolute value on . This absolute value satisfies a stronger triangle inequality:
- .
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 or is archimedian.
The theory of absolute values is important in algebraic number theory. Let be a finite Galois extension with , and let be the field automorphisms of over . Then the only absolute values are the archimedian ones given by (the ordinary real or complex absolute values) and the nonarchimedian ones given by for some prime of .