Difference between revisions of "Absolute value"
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Eyefragment (talk | contribs) (→Generalized Absolute Values: Added (Non-negative) and (Completely Multiplicative) for consistency with (The Triangle Inequality). Added Ultrametric Inequality as alt. name for Strong Tri. Ineq.) |
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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</math>, and <math>|x|=0</math> if and only if <math>x=0</math>. | + | *<math> |x|\ge 0</math> for all <math>x</math>, and <math>|x|=0</math> if and only if <math>x=0</math>. (Non-negative) |
− | *<math> |x\times y|=|x|\times |y|</math>. | + | *<math> |x\times y|=|x|\times |y|</math>. (Completely Multiplicative) |
*<math> |x+y| \le |x|+|y|</math>. (The [[triangle inequality]]) | *<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-adic]] absolute value of <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: | + | Another example of an absolute value is the [[p-adic]] absolute value of <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, often known as the Ultrametric Inequality: |
*<math> |x+y|\le\max(|x|,|y|)</math>. | *<math> |x+y|\le\max(|x|,|y|)</math>. |
Revision as of 21:48, 29 November 2006
The absolute value of a real number , denoted , is its distance from zero on a number line. If , then , and if , then . This is equivalent to "dropping the minus sign."
Similarly, the absolute value of a complex number , where , is , the distance of from the origin.
Contents
[hide]Example Problems
Simple Absolute Value Problems
Solution: You have to isolate the variable, and then make two equations; one negative, the other positive. The variable is already isolated, so we can make the two equations: and . This works because x can be both positive and negative, and will still give the same result. The answer is .
Now, let's say that you have functions outside your absolute value: .
Just like in the other problem, you must isolate the variable. First, sutract 4 from both sides to get . Then, divide by three to get .
Now, try to solve it by yourself.
Solution: We first get rid of the absolute value by making two equations: and . Divide everything by 7 to get the answer: .
Practice Problems
Word Problems
Absolute Value Functions are also very useful for solving problems.
Lets say you have a problem that goes like this:
In Mrs. Barnett's class, the scores on a certain test varied 28 points from 71. What were the minumum and maximum scores on the test?
You would have as your equation, and if you solve it, you get 99 as the maximum and 43 as the minimum.
Problems from Competitions
Generalized Absolute Values
The absolute value functions listed above have three very important properties:
- for all , and if and only if . (Non-negative)
- . (Completely Multiplicative)
- . (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 of , 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, often known as the Ultrametric 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 .