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 the unsigned portion of <math>x</math>. Geometrically, <math>|x|</math> is the [[distance]] between <math>x</math> and [[zero]] on the real [[number line]]. | |
− | The | + | The absolute value function exists among other contexts as well, including [[complex numbers]]. |
− | + | ==Real numbers== | |
− | + | When <math>x</math> is real, <math>|x|</math> is defined as <cmath> |x| = | |
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− | <math>|x|= | ||
− | + | * (Alternative definition) <math>|x| = \sqrt{x^2}</math> | |
+ | * (Non-negativity) <math>|x| \ge 0</math> | ||
+ | * (Positive-definiteness) <math>|x| = 0 \iff x=0</math> | ||
+ | * (Multiplicativeness) <math>|xy| = |x| |y|</math> | ||
+ | * ([[Triangle Inequality]]) <math>|x+y| \le |x|+|y|</math> | ||
+ | * (Symmetry) <math>|x| = |-x|</math> | ||
− | + | Note that | |
− | + | <cmath>|x| \le y \iff -y \le x \le y </cmath> | |
− | + | and | |
− | + | <cmath> |x| \ge y \iff x \ge y \text{ or } x \le -y.</cmath> | |
− | + | ==Complex numbers== | |
− | + | For [[complex number]]s <math>z</math>, the absolute value is defined as <math>|z| = \sqrt{x^2+y^2}</math>, where <math>x</math> and <math>y</math> are the real and imaginary parts of <math>z</math>, respectively. It is equivalent to the distance between <math>z</math> and the [[origin]], and is usually called the [[complex modulus]]. | |
− | <math> | ||
− | <math>| | + | Note that <math>|z| = |\overline{z}| = \sqrt{z\overline{z}}</math>, where <math>\overline{z}</math> is the [[complex conjugate]] of <math>z</math>. |
− | + | ==Examples== | |
− | === | + | # If <math>|x|=k</math>, for some real number <math>k</math>, then <math>x=k</math> or <math>x=-k</math>. |
− | + | # If <math>|ax| = k</math>, for some real numbers <math>a</math>, <math>k</math>, then <math>ax = k</math> or <math>ax = -k</math>, and therefore <math>x = \frac{k}{a}</math> or <math>x = -\frac{k}{a}</math>. | |
− | + | ==Problems== | |
− | + | # Find all real values of <math>x</math> if <math>-|x| = x-6</math>. | |
+ | # Find all real values of <math>x</math> if <math>5 + 8 \cdot |4x| = 69</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== | |
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− | + | * [[Magnitude]] | |
− | + | * [[Norm]] | |
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− | * [[ | ||
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* [[Valuation]] | * [[Valuation]] |
Revision as of 22:08, 16 March 2008
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.
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 .