Difference between revisions of "Absolute value"

Latest revision as of 10:37, 5 January 2009

The absolute value of a real number $x$ , denoted $|x|$ , is the unsigned portion of $x$ . Geometrically, $|x|$ is the distance between $x$ and zero on the real number line.

The absolute value function exists among other contexts as well, including complex numbers.

Real numbers

When $x$ is real, $|x|$ is defined as $|x| = \begin{cases} x & \text{for } x \ge 0,\\ -x & \text{for } x \le 0.\end{cases}$ For all real numbers $x$ and $y$ , we have the following properties:

(Alternative definition) $|x| = \sqrt{x^2}$
(Non-negativity) $|x| \ge 0$
(Positive-definiteness) $|x| = 0 \iff x=0$
(Multiplicativeness) $|xy| = |x| |y|$
(Triangle Inequality) $|x+y| \le |x|+|y|$
(Symmetry) $|x| = |-x|$

Note that

$|x| \le y \iff -y \le x \le y$

and

$|x| \ge y \iff x \ge y \text{ or } x \le -y.$

Complex numbers

For complex numbers $z$ , the absolute value is defined as $|z| = \sqrt{x^2+y^2}$ , where $x$ and $y$ are the real and imaginary parts of $z$ , respectively. It is equivalent to the distance between $z$ and the origin, and is usually called the complex modulus.

Note that $|z| = |\overline{z}| = \sqrt{z\overline{z}}$ , where $\overline{z}$ is the complex conjugate of $z$ .

Examples

If $|x|=k$ , for some real number $k$ , then $x=k$ or $x=-k$ .
If $|ax| = k$ , for some real numbers $a$ , $k$ , then $ax = k$ or $ax = -k$ , and therefore $x = \frac{k}{a}$ or $x = -\frac{k}{a}$ .

Problems

Find all real values of $x$ if $-|x| = x-6$ .
Find all real values of $x$ if $5 + 8 \cdot |4x| = 69$ .
(AMC 12 2000) If $|x - 2| = p$ , where $x < 2$ , then find $x - p$ .

@@ Line 1: / Line 1: @@
-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."
+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]].
-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>.
+The absolute value function exists among other contexts as well, including [[complex numbers]].
+==Real numbers==
-== Introductory Concepts ==
+When <math>x</math> is real, <math>|x|</math> is defined as <cmath> |x| = \begin{cases} x & \text{for } x \ge 0,\\ -x & \text{for } x \le 0.\end{cases} </cmath> For all real numbers <math>x</math> and <math>y</math>, we have the following properties:
-=== Example Problems & How To Solve for Middle School Math Problems===
-Example Problem
-________________
-* [[2000_AMC_12/Problem_5 | 2000 AMC 12 Problem 5]]
-Amc 12 2000, Problem 5:
+* (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>
-. If |x − 2| = p, where x < 2, then x − p
+Note that
-(A) −2
-(B) 2
-(C) 2 − 2p
-(D) 2p − 2
-(E) |2p − 2|
+<cmath>|x| \le y \iff -y \le x \le y </cmath>
-How to solve simple Absolute Value Problems:
+and
-Example:
+<cmath> |x| \ge y \iff x \ge y \text{ or } x \le -y.</cmath>
-|x|=7
-Solution: For this problem, you have to know that absolute value is always a positive number or 0. For example, if you put -7 in, you get 7 out. Another way to look at it, is it takes takes the value inside the absolute value symbols and instead puts the positive difference between itself and 0. If you have just an absolute value function on a graph, it will always appear in quadrant 1 and 2, hence those quadrants have a positive y value or 0. This is how you solve an absolute value problem: You make 2 equations out of this when you take away the absolute value signs. The 2 equations would be x=7 and x=-7. If you havn't learned this already you probably don't know why this works. This is because the other number (x as to -x - opposite) that works, is the number that would equal 0 if you add it to the original number, or in other words, it's opposite.
+==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]].
-Now, let's say that you have functions outside your absolute value.
+Note that <math>|z| = |\overline{z}| = \sqrt{z\overline{z}}</math>, where <math>\overline{z}</math> is the [[complex conjugate]] of <math>z</math>.
-+3|7x|=151
+==Examples==
-Wait one second big guy.........you can't JUST use the information above!
+# 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>.
-Now, just a precaution. Treat numbers outside absolute value functions as it was in ().
+==Problems==
-So, you would first -4 from both sides in your effort to isolate the variable (If you don't know why, you should probably attempt to read an article in Algebra as to why you do this.........I'm not sure if this that is in there).
+# 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>.
-Your new equation is this:
+==See Also==
-|7x|=147
+* [[Magnitude]]
+* [[Norm]]
-Keep on isolating the variable.
+* [[Valuation]]
-Divide both sides by 3 and get:
-|7x|=49
-Now, using the above information, I think we can solve this problem. Try to finish this by yourself, then look at the solution.
-----------------------
-'''Solution:
-We first get rid of the absolute value by making two equations. Thus, we make:
-x=49 and 7x=-49
--  -      -   -
-  7      7   7
-x=7 or x=-7'''
---------------------
-Now, try some of these problems (if you really are having trouble, post something on this forum.
--|x|=x-6
-|b|=21
-+8|4x|=69
--------------------
-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
-|x-71|=28 as your equation, and if you solved it like you did above, you get 99 as the maximum and 43 as the minimum.
-Obviously, that might be too easy for absolute value, but when the numbers get tricky, this format could definately help.
---[[User:Jhredsox|Jhredsox]] 10:25, 4 November 2006 (EST)
-==Generalized absolute values==
-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\times y|=|x|\times |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''.
-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>.
-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 <math>\mathbb{R}</math> or <math>\mathbb{C}</math> is archimedian.
-The theory of absolute values is important in [[algebraic number theory]]. Let <math>K/\mathbb{Q}</math> be a [[finite]] [[Galois extension]] with <math>[K:\mathbb{Q}]=n</math>, and let <math>\sigma_1,\ldots,\sigma_n</math> be the [[field automorphisms]] of <math>K</math> over <math>\mathbb{Q}</math>. Then the only absolute values are the archimedian ones given by <math>|x|_i=|\sigma_i(x)|</math> (the ordinary real or complex absolute values) and the nonarchimedian ones given by <math>|x|_{\mathfrak{p}}</math> for some prime <math>{\mathfrak{p}}</math> of <math>K</math>.
-==See also==
-*[[Algebraic number theory]]
-*[[Completion]]
-*[[p-adic number]]
-*[[Valuation]]

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