For questions about or involving the absolute value function.

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3
votes
4answers
64 views

Show that $f(x)=f(y)$ then $|x|=|y|$, where $f(x )=\frac{1+|x|}{x}$

Let $f: \mathbb{R}^{*}\to \mathbb{R}$ function definied by $f(x )=\dfrac{1+|x|}{x}$ Show that $f(x)=f(y)$ then $|x|=|y|$ Indeed, $$f(x)=f(y)\\ \iff \\\dfrac{1+|x|}{x}=\dfrac{1+|y|}{y} \\ \iff \\ ...
-1
votes
0answers
35 views

How to integrate $\int \left|f(x) +g(x) \right|^2dx$? [on hold]

I am dealing with a quantum mechanics exercise at which I need to find the probabilty of $\left| \psi \right |^2$. $\psi$ is composed of 2 real value functions, say $f$ and $g$. So generally, how to ...
1
vote
0answers
35 views

Uniform convergence fourier series |sin(x)|

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|. After thinking about it ...
1
vote
2answers
46 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: $$|x + a| = b$$ and we know the solutions: $$x = 11 \text{ and } x = 5$$ We need to find $a$ and $b$. From $$11 + a = b \\ 5 + a = -b$$ we ...
1
vote
1answer
27 views

How do I represent $f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$ in an integrated form?

Given the following function, how do I define it without the integral symbol? $$f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$$ I don't understand how I determine when $x + t$ is positive and ...
-1
votes
2answers
34 views

Normal distribution, probability and modulus question [closed]

Say $X$ is a random variable which is normally distributed with mean $0$ and variance $1$. How do I find $k$ such that $$\mathbb{P}(|X-k| < |X+k|) = 0.7$$
-2
votes
1answer
40 views

To check differentiability of function [closed]

A function $f: [0,3]\rightarrow \Bbb{R}$ is defined by $$f(x) = |x| + |x-1| + |x-2| + |x-3|\quad \forall x \in [0,3]$$ The number of points in $[0,3]$ where $f$ is not differentiable is
1
vote
0answers
34 views

Find the product all real numbers in an equation

What is an easy and fast way to solve the problem without going through all these possibilities: a) $n^2-9n+20>0, 16-n^2>0$, b) $n^2-9n+20>0, 16-n^2<0$, c) $n^2-9n+20<0, 16-n^2>0$, ...
1
vote
1answer
37 views

Is there an easy way to solve this absolute values problem?

This is a simple problem. What I want to know is whether there is an easy and fast way to solve the problem. I solved this problem by considering four situations: a) $x>1$, b) $0<x<1$, c) ...
0
votes
3answers
35 views

How to solve this problem on absolute value function?

If $a,b\in \mathbb R$ and be distinct numbers satisfying $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then the minimum value of $|a-b|$ is ? ($|...|$ represents absolute value) I tried solving the ...
0
votes
3answers
88 views

Why is $(\sqrt{x^2})$ equal to $|x|$ [duplicate]

I don't understand why you have to write the absolute value sign when solving for the square root of $x$ squared. Shouldn't the answer automatically be positive? Why is the absolute value sign ...
19
votes
2answers
2k views

Significance of $\displaystyle\sqrt[n]{a^n} $?

There is a formula given in my module: $$ \sqrt[n]{a^n} = a \text{ if $n$ is odd } $$ $$ \sqrt[n]{a^n} = |a| \text{ if $n$ is even } $$ I don't really understand the differences between them, ...
3
votes
2answers
25k views

Square root of a number squared is equal to the absolute value of that number [duplicate]

Possible Duplicate: Significance of $\displaystyle\sqrt[n]{a^n} $? The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just ...
0
votes
2answers
516 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
1
vote
2answers
40 views

If $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$

I am trying to find a simple way to show the fact that if $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$ for any number $d$. Is there a way to do this besides breaking it up into the cases 1) ...
1
vote
1answer
11 views

Mixed Integer linear programming - absolute value of a variable not involved n the objective function

I'm looking to find the absolute value of the expression s-t. I have begun by introducing the following constraints: Where A is the absolute value. Unfortunately, A is not involved in the objective ...
0
votes
1answer
426 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
3
votes
1answer
100 views

Close approximation for absolute value function

I made a very acurate approximation function for $\sqrt{n^{2}+1}$ It is $\sqrt{n^{2}+1}\approx\frac{2n(n^{2}+1)}{2n^{2}+1}+\frac{2n^{2}+1}{n(4(2n^{2}+1)^{2}+1)}$ From this I can make a very close ...
0
votes
0answers
25 views

Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} ...
3
votes
4answers
128 views

The Definition of the Absolute Value

The Absolute Value can be defined in many ways, but these are the two most common : 1. As a Piecewise Function $$ |x|= \begin{cases} -x&\text{if } x < 0\\ x&\text{if } x\geq 0 \end{cases} ...
2
votes
1answer
25 views

why is $\left|xy\ log(\left|x\right|+\left|y\right|)\right|\leq\left|(\left|x\right|+\left|y\right|)log(\left|x\right|+\left|y\right|)\right|$?

I should note that this was used by my book in order to show that the limit of $xy\ log(\left|x\right|+\left|y\right|)$ at $(0,0)$ is $0$. After several attempts in vain, I plotted the function ...
0
votes
0answers
56 views

Compare difference between mean and actual

My problem is: I have two sets of numbers as follows: $X = {x_1, x_2, ..., x_n}; Y = {y_1, y_2, ..., y_m}$. Where $r$ is the actual value. $x^*$ is the mean of set X, $y^*$ is the mean of set Y, ...
10
votes
2answers
462 views

Could we invent a new number with $|p|=-1$?

We know that how a single definition $i^2=-1$ revolutionized our mathematics and solved many many problems. I wonder whether the definition $|p|=-1$ could have the potential of creating a new ...
0
votes
0answers
13 views

Automorphisms of local field

(1)Suppose that $K$ is a local field but not $\mathbb C$. Then show that any automorphism of $K$ is continuous. (We can assume that $K$ is $\mathbb R$ with classical absolute value or $K$ is a finite ...
0
votes
2answers
33 views

An equation with a parameter

Given the equation $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-4a^2=0$ find all possible $a$ such that this equation has only one solution. I wanted to solve it like this: ...
0
votes
0answers
19 views

$\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
1
vote
2answers
56 views

Is my hypothesis correct? [closed]

$$\left| \left|(a^2) - 25\right|-b\right| + b = 0$$ You have to prove that $b<0$ and $b=0$ at the same time I have no problem to prove that $b$ can be $0$ the thing that I need help with is ...
0
votes
1answer
19 views

Real Analysis Absolute values [closed]

Someone please help me with detailed explanation on how to solve this problem. For all $a, b \in \Bbb R$, show that; $$ | a - b | \geq | a | - | b | $$
6
votes
3answers
502 views

Absolute value graph sketching

Where would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then: ...
2
votes
1answer
43 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (i) and (ii) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
0
votes
1answer
35 views

Does equality of the sum of two such series imply equality of each term of that series?

Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and ...
1
vote
1answer
21 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (iii) and (iv) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
25
votes
1answer
283 views

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 ...
1
vote
1answer
23 views

Expressing the minimum function in terms of the absolute value in a symmetric manner (generalized to more variables)

It is well known that: $$\max(a,b) = \frac12(a+b)+\frac12|a-b|$$ and similarly: $$\min(a,b) = \frac12(a+b)-\frac12|a-b|.$$ In fact, they are equivalent since $\max(a,b) = -\min(-a,-b)$. We can try ...
1
vote
1answer
36 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (v) and (vi) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
2
votes
1answer
19 views

non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
7
votes
3answers
86 views

Solving modulus inequality $|x - 1| + |x - 6|\le11$ geometrically

Find all possible values of $x$ for which $x$ for which the inequality $$|x - 1| + |x - 6|\le11$$ is true. I know this can be easily solved by taking $3$ cases for$x$ and then taking the intersection ...
2
votes
0answers
30 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
2
votes
1answer
41 views

Is there a solution to the absolute value of an expression which results in a negative value?

The equation given: $ \mid x - 4 \mid = -3$. My instinct (and example 2 in this article) tells me that there shouldn't be any solution as there would be no value of x which would result in a negative ...
0
votes
1answer
22 views

absolute value (positive and negative part)

I found a notation that $$|y_i| = y_i^{+}+y_i^{-} $$ where y is y n dimensional vector. what does +/- imply? I understand that we can have both negative and positive value into absolute function, ...
0
votes
0answers
27 views

The maximum of the absolute value of a real-valued function

For a continuous real-valued function $f$ on a compact set $X\neq \emptyset$, we have \begin{equation*} \max_{x\in X}|f(x)| = \max \{\max_{x\in X} f(x), \max_{x\in X} -f(x)\}. ...
0
votes
1answer
24 views

integral of two functions absolute

I've got the two function: f(x) = -4x + x³ and g(x) = 5x they meet each other at -3, 0 and 3, where the areas between -3 and 0 ...
1
vote
1answer
73 views

$\lim_{x\to a}(f(x)+\frac{1}{|f(x)|})=0$. Find $\lim_{x\to a}f(x)$ [duplicate]

Let a function $f$ be defined in a hollow neighborhood of $a\in \mathbb{R}$, and suppose : $$\lim_{x\to a}\left(f(x)+\frac{1}{|f(x)|}\right)=0$$ Find $\lim\limits_{x\to a}f(x)$ and prove that this ...
1
vote
2answers
36 views

Help to solve absolute value inequality

The inequality I have is $\frac {\mid x-1 \mid} {(x+2)} <1 $ what I'm not sure is how I am supposed to proceed. I cannot multiply by (x+2) because it is unknown whether it is positive or negative. ...
0
votes
1answer
38 views

Maximum value and the absolute value

Let $f\colon X \rightarrow \mathbb{R}$ be a function such that $\max_{x\in X} f(x) + \min_{x\in X} f(x) = 0$. Does it then follow that $\max_{x\in X} f(x) = \max_{x\in X} |f(x)|$? I'm quite sure it ...
0
votes
4answers
40 views

What is the solution to this inequality?

This is the given inequality I've been trying to solve $$1/6\leq \frac{1}{\mid x \mid} \leq 1/2$$ However the answer I get is $(0,6] \cup [2,6]$ which is not the answer given in my book. Could you ...
0
votes
1answer
28 views

Absolute value inequality explanation

I was solving the inequality $-4 \le \left|\frac {x+4} {2-x} \right| \le 4$ and I first wrote the domain, which is $(-\infty,-4] \cup (-4,2) \cup (2, \infty)$ and I got the solution that $x \le \frac ...
3
votes
3answers
65 views

What are the solutions of $|x+y|=|x|+|y|$?

So I am having a problem in solving this type of equation. The problem I am dealing with is... $$\left|(2x-1) + \frac{3x-1}x\right| = \left|2x-1\right| + \left|\frac{3x-1}x\right|$$ Please help me ...
2
votes
1answer
26 views

Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
12
votes
3answers
139 views

Why $|x|$ is not rational expression?

I'm 9th grade student, and my teacher said that $|x|$ is not rational expression ( expression like $\frac{p(x)}{q(x)}$ s.t $p(x)$ and $q(x)\neq 0$ are polynomial) but he didn't have convincing reason. ...