Absolute value

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The absolute value of a real number $x$ , denoted $|x|$ , is its distance from zero on a number line. If $x\ge 0$ , then $|x|=x$ , and if $x<0$ , then $\displaystyle |x|=-x$ . This is equivalent to "dropping the minus sign."

Similarly, the absolute value of a complex number $z=x+iy$ , where $x,y\in\mathbb{R}$ , is $|z|=\sqrt{x^2+y^2}$ .

Example Problems

Simple Absolute Value Problems

$|x|=5$

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: $x=5$ and $x=-5$ . This works because x can be both positive and negative, and will still give the same result. The answer is $x=\{-5,\,5\}$ .

Now, let's say that you have functions outside your absolute value: $4+3|7x|=151$ .

Just like in the other problem, you must isolate the variable. First, sutract 4 from both sides to get $3|7x|=147$ . Then, divide by three to get $|7x|=49$ .

Now, try to solve it by yourself.

Solution: We first get rid of the absolute value by making two equations: $7x=49$ and $7x=-49$ . Divide everything by 7 to get the answer: $x=\{-7,\,7\}$ .

Practice Problems

$-|x|=x-6$

$|7b|=21$

$5+8|4x|=69$

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 $|x-71|=28$ as your equation, and if you solve it, you get 99 as the maximum and 43 as the minimum.

Problems from Competitions

2000 AMC 12 Problem 5

Generalized Absolute Values

The absolute value functions listed above have three very important properties:

$|x|\ge 0$ for all $x$ , and $|x|=0$ if and only if $x=0$ .
$|x\times y|=|x|\times |y|$ .
$|x+y| \le |x|+|y|$ . (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 $\mathbb{Q}$ , the rational numbers. Let $x=\prod_{i=1}^n p_i^{e_i}$ , where the $p_{i}$ 's are distinct prime numbers, and the $e_i$ 's are (positive, negative, or zero) integers. Define $|x|_{p_i}=p_i^{-e_i}$ . This defines an absolute value on $\mathbb{Q}$ . This absolute value satisfies a stronger triangle inequality:

$|x+y|\le\max(|x|,|y|)$ .

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 $\mathbb{R}$ or $\mathbb{C}$ is archimedian.

The theory of absolute values is important in algebraic number theory. Let $K/\mathbb{Q}$ be a finite Galois extension with $[K:\mathbb{Q}]=n$ , and let $\sigma_1,\ldots,\sigma_n$ be the field automorphisms of $K$ over $\mathbb{Q}$ . Then the only absolute values are the archimedian ones given by $|x|_i=|\sigma_i(x)|$ (the ordinary real or complex absolute values) and the nonarchimedian ones given by $|x|_{\mathfrak{p}}$ for some prime ${\mathfrak{p}}$ of $K$ .

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