Difference between revisions of "Absolute value"
(rewrite.) |
(→See Also) |
||
Line 39: | Line 39: | ||
# ([[2000 AMC 12 Problems/Problem 5|AMC 12 2000]]) If <math>|x - 2| = p</math>, where <math>x < 2</math>, then find <math>x - p</math>. | # ([[2000 AMC 12 Problems/Problem 5|AMC 12 2000]]) If <math>|x - 2| = p</math>, where <math>x < 2</math>, then find <math>x - p</math>. | ||
− | ==See Also== | + | [[Image:Example.jpg]]==See Also== |
* [[Magnitude]] | * [[Magnitude]] | ||
* [[Norm]] | * [[Norm]] | ||
* [[Valuation]] | * [[Valuation]] | ||
+ | x-p=2 |
Revision as of 08:30, 5 January 2009
The absolute value of a real number , denoted , is the unsigned portion of . Geometrically, is the distance between and zero on the real number line.
The absolute value function exists among other contexts as well, including complex numbers.
Contents
[hide]Real numbers
When is real, is defined as For all real numbers and , we have the following properties:
- (Alternative definition)
- (Non-negativity)
- (Positive-definiteness)
- (Multiplicativeness)
- (Triangle Inequality)
- (Symmetry)
Note that
and
Complex numbers
For complex numbers , the absolute value is defined as , where and are the real and imaginary parts of , respectively. It is equivalent to the distance between and the origin, and is usually called the complex modulus.
Note that , where is the complex conjugate of .
Examples
- If , for some real number , then or .
- If , for some real numbers , , then or , and therefore or .
Problems
- Find all real values of if .
- Find all real values of if .
- (AMC 12 2000) If , where , then find .
File:Example.jpg==See Also==
x-p=2