Questions tagged [absolute-value]

For questions about or involving the absolute value function also known as modulus function.

The absolute value function, usually denoted by $|x|$, is a function $\mathbb{R} \to [0, \infty)$ which can be defined in three equivalent ways:

  1. $|x| = \begin{cases}x &\ \text{if $x \ge 0$, and} \\ -x &\ \text{if $x < 0$.} \end{cases}$

  2. $|x| = \sqrt{x^2}$, and

  3. $|x| = \max \{x, -x\}$.

This definition extends to complex numbers as the square root of the norm: $|x+iy|=\sqrt{x^2+y^2}$. In both cases, the function may be interpreted geometrically as the distance of the input number from the origin.


More generally, an absolute value may be defined on an field (or integral domain) $k$ as a function $|\cdot | : k \to \mathbb{R}$ which satisfies the axioms

  1. (nonnegativity) $|x| \ge 0$ for all $x \in k$,

  2. (definiteness) $|x| = 0 \iff x = 0 \in k$,

  3. (multiplicativity) $|x y| = |x||y| $ for all $x,y\in k$ ), and

  4. (triangle inequality) $|x+y| \le |x| + |y|$ for all $x,y\in k$.

For example, if $p$ is a fixed prime number and $x \in \mathbb{Q}$, then there exists a unique $n \in \mathbb{Z}$ such that $x$ may be written as $$ x = p^n \frac{a}{b}, $$ where $\gcd(p, a) = \gcd(p, b) = 1$. The function which maps $x$ to $p^{-n}$ is an absolute value on $\mathbb{Q}$, called the $p$-adic absolute value.

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The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm)

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
hattenn
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Prove that $||x|-|y||\le |x-y|$

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please…
Anonymous
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Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
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Why is the absolute value function not differentiable at $x=0$?

They say that the right and left limits do not approach the same value hence it does not satisfy the definition of derivative. But what does it mean verbally in terms of rate of change?
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Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$

Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$ How would we prove this? Does this follow from Cauchy Schwarz? Intuitively this is how I see it: In the LHS we could have a…
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Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
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Significance of $\sqrt[n]{a^n} $?!

There is a formula given in my module: $$ \sqrt[n]{a^n} = \begin{cases} \, a &\text{ if $n$ is odd } \\ |a| &\text{ if $n$ is even } \end{cases} $$ I don't really understand the differences between them, kindly explain with an example.
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Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns out to be periodic with period 9, i.e., $$ a_n =…
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My teacher describes absolute value confusingly: $|x|=\pm x,\quad \text{if}\enspace x>0 $.

He says (direct quote): "In higher mathematics the absolute value of a number, $|x|$, is equal to positive and negative $x$, if $x$ is a positive number." Then he wrote: $|x|=\pm x,\quad \text{if}\enspace x>0 $. I think he misunderstood the…
user466742
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Converting absolute value program into linear program

I have the generic optimization problem: $$ \max c^T|x|$$ $$ \text{s.t. } Ax \le b $$ $x$ is unrestricted How do I convert it into a linear programming problem? Online I read something about letting $x$ equal the difference of 2 positive numbers…
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Where is the absolute value when computing antiderivatives?

Here is a typical second-semester single-variable calculus question: $$ \int \frac{1}{\sqrt{1-x^2}} \, dx $$ Students are probably taught to just memorize the result of this since the derivative of $\arcsin(x)$ is taught as a rule to memorize.…
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Is there a geometric analog of absolute value?

I'm wondering whether there exists a geometric analog concept of absolute value. In other words, if absolute value can be defined as $$ \text{abs}(x) =\max(x,-x) $$ intuitively the additive distance from $0$ to $x$, is there a geometric…
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Proving square root of a square is the same as absolute value

Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't the same equation. For example, say I put $-1$ into…
Cole Tobin
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How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ then equal holds if we suppose that $a\leq b\leq…
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Quadratic inequality puzzle: Prove$ |cx^2 + bx + a| ≤ 2$ given $|ax^2+bx+c| ≤ 1$

I came across this problem as part of a recreational mathematics challenge on university: Suppose $a, b, c$ are real numbers where for all $ -1 \le x \le 1 $ we have $|ax^2 + bx + c| \le 1$. Prove that for all $-1 \le x \le 1 ,$ $$ |cx^2 + bx +…
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