What is Absolute value: Definition and 367 Discussions

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

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  1. Math Amateur

    MHB Limits and Continuity - Absolute Value Technicality ....

    I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ... I need some help with an inequality involving absolute values in Example 4.1.2 (a) ... Example 4.1.2 (a) ... reads as follows:In the above text we read ...
  2. M

    MHB Absolute Value Statements....2

    Rewrite each statement using absolute values. 1. The number y is less than one unit from the number t. y < | t - 1 | 2. The sum of the distances of a and b from the origin is greater than or equal to the distance of a + b from the origin. | a + b - 0 | > or = | a + b - 0 | Is this correct...
  3. M

    MHB Is this the correct way to rewrite absolute value statements?

    Rewrite each statement using absolute values. 1. The distance between x and 4 is at least 8. | x - 4 | > or = 8 Can this also be expressed as | 4 - x | > or = 8? If so, why? 2. The distance between x^3 and -1 is at most 0.001. | x^3 -(-1) | < or = 0.001 Can this also be expressed as | - 1...
  4. M

    MHB How to Rewrite Absolute Value Expressions Without Absolute Values?

    The | x | = x when x > or = 0. The | x | = - x when x < 0. Rewrite the following expression in a form that does not contain absolute value. | x + 3 | + 4 | x + 3 |, where x < -3 -(x + 3) + 4 -(x + 3) -x - 3 + 4 - x - 3 -2x - 6 + 4 -2x - 2 Correct?
  5. M

    MHB How Do Absolute Values Express At Least and At Most Conditions?

    Rewrite each statement using absolute values. 1. The distance between x and 4 is at least 8. Work: | x - 4 | > or = 8 Correct? Why must we write greater than or equal to for AT LEAST statements? 2. The distance between x^3 and -1 is at most 0.001. Work: | x^3 - (-1) | < or = 0.001...
  6. Bunny-chan

    Epsilon distance between two terms

    Homework Statement How close is x to x_0 (x_0 \neq 0) so that 2. Homework Equations The Attempt at a Solution I tried to use absolute value properties:- \epsilon \lt \frac{\sqrt{x_0^2+1}}{x_0^3} - \frac{\sqrt{x^2+1}}{x^3} \lt \epsilonBy adding in the three sides, we...
  7. peroAlex

    Absolute Value of Magnetization

    Hello, I'm a student of electrical engineering. This task appeared in one of the past exams. I've been using the procedure I believe should yield the correct result, however, it turns out I was wrong. Could somebody please check out where the mistake lays in my calculations? Homework Statement...
  8. M

    MHB Is it Possible for an Absolute Value Equation to Equal a Negative Number?

    Precalculus by David Cohen 3rd Edition Chapter 1, Section 1.2. Question 68, page 11. Before typing the textbook question, I must say that I have not been able to find a satisfactory answer to absolute value equations that equal a negative number. Question: Explain why there are no real...
  9. M

    Integral of absolute value of a Fourier transform

    Homework Statement Hi guys, I am going to calculate the following integral: $$\int_0^{f_c+f_m} |Y(f)|^2\, df$$ where:$$Y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega +...
  10. H

    I Contradiction in an absolute value property?

    An absolute value property is $$\lvert a \rvert \geq b \iff a\leq-b \quad \text{ or } \quad a\geq b,$$ for ##b>0##. Is this true for the case ##a=0##? I mean if ##a=0, \lvert a \rvert =0## so ##0 \geq b##. But ##b## is supposed to be ##b>0##, so we have a contradiction. How can this property...
  11. Mr Davis 97

    I Why Does the Equation ##\sqrt{a^2} = a## Lead to Confusion in Mathematics?

    If we define ##|a| = \sqrt{a^2}##, then why can't we do something like ##\sqrt{a^2} = (\sqrt{a})^2 = a##? Or equivalently ##\sqrt{a^2} = (a^2)^{1/2} = a^{2/2} = a##? Isn't this a contradiction? Also, how would this relate to showing that ##\sqrt{|a|} = |\sqrt{a}|## is true or false?
  12. mccoy1

    I The absolute value of the frequency in quantum mechanics

    I was reading Bransden's Quantum Mechanics 2nd edition, chapter 2 page 61. There,it says "It should be noted that since E =hv (v for nu), the absolute value of the frequency has no physical significance in Quantum mechanics..." Why is that? Isn't this a contradiction?
  13. M

    B Why is the use of absolute value in vector norms a matter of preference?

    I would like to ask you why the author does not use absolute value of y instead of y? Source: Mathematical Methods in the Physical Sciences by Mary L. Boas Thank you.
  14. Genilson

    I Limit with integral and absolute value

    Hello good evening to all, I was studying here and got stuck with this. I solved the integral and got [x+sin(x) -1] and that´s the farthest that I got. I would appreciate the help.
  15. liluiass

    Prove: |x+y|<|xy+1| for |x|,|y|<1

    Homework Statement X and Y 2 real numbers / |x| <1 and |y|<1 Prove that |x+y|<|xy+1|Homework EquationsThe Attempt at a Solution |x+y|<2 I couldn't prove that |xy+1| >2 And couldn't find a way to solve the problem Please help
  16. N

    Double integral with absolute value

    Homework Statement I am trying to evaluate double integral ∫∫D (|y - x2|)½ D: -1<x<1, 0<y<2 Homework Equations None The Attempt at a Solution I know that in order to integrate with the absolute value I have to split the integral into two parts: y>x^2−−−>√y−x2 y>x^2−−−>√y−x2 I just can't...
  17. Mr Davis 97

    B Solving absolute value equation

    I have the following equation: ##\left | r-5 \right | = \left | r+2 \right |##. What is a general, analytical way that I can solve equations like these? I always get stumped when trying to solve them...
  18. DavidReishi

    I Square of absolute value of amplitude for a single photon

    I understand that this determines a probability, but of what exactly for a single photon? The probability that the photon will be detected on a surface where the photon is pumped, e.g. where on the surface the laser is aimed?
  19. C

    MHB Proving an absolute value inequality

    If $\left| a \right| \le b$, then $-b\le a\le b$. Let $a,b \in\Bbb{R}$ The definition of the absolute value is $ \left| x \right|= x, x\ge 0$ and $\left| x \right|=-x, x< 0$, where x is some real number. Case I:$a\ge 0$, $\left| a \right|=a>b$ Case II: a<0, $\left| a \right|=-a<b$the solution...
  20. G

    Understanding Why We Need Absolute Value for x

    Homework Statement why we need to make x as absolute value ? as we can see, the original is x , why we we need to make x as absolute value ? is the working wrong ? Homework EquationsThe Attempt at a Solution
  21. Mr Davis 97

    Absolute value in a differential equation

    Homework Statement ##\displaystyle (x+3)\frac{dy}{dx} = y - 2##, where x is not 3 and y is not 2. Homework EquationsThe Attempt at a Solution ##\displaystyle (x+3)\frac{dy}{dx} = y - 2## ##\displaystyle \frac{dy}{y-2} = \frac{dx}{x+3}## ##\displaystyle \int \frac{dy}{y-2} = \int...
  22. Odious Suspect

    When to introduce absolute value in hyperbola expression?

    We begin with this definition of a hyperbola. \left(\overline{F_1 P}-\overline{F_2 P}=2 a\right)\land a>0 Perform a few basic algebraic manipulations. \sqrt{(c+x)^2+y^2}-\sqrt{(x-c)^2+y^2}=2 a \sqrt{(c+x)^2+y^2}=2 a+\sqrt{(x-c)^2+y^2} (c+x)^2+y^2=4 a^2+4 a \sqrt{(x-c)^2+y^2}+(x-c)^2+y^2...
  23. O

    MHB Limit of Absolute Values and Metric Spaces

    Let $\lim_{{k}\to{\infty}}d\left({x}_{m\left(k\right)},{x}_{m\left(k\right)-1}\right)=\varepsilon$ and $\lim_{{k}\to{\infty}}d\left({x}_{n\left(k\right)},{x}_{m\left(k\right)}\right)=\varepsilon$...Can we say that...
  24. Steve Turchin

    Complex absolute value inequality

    Solve the following inequality. Represent your answer graphically: ## |z-1| + |z-5| < 4 ## Homework Equations ## z = a + bi \\ |x+y| \leq |x| + |y| ## Triangle inequality The Attempt at a Solution ## |z-1| + |z-5| < 4 \\ \\ x = z-1 \ \ , \ \ y = z-5 \\ \\ |z-1+z-5| \leq |z-1| + |z-5| \\...
  25. Mr Davis 97

    Solving Absolute Value Limit: x→2

    Homework Statement Solve: ##\displaystyle \lim_{x\rightarrow 2} \frac{\left | x^2 + 3x + 2 \right |}{x^2 - 4}##. Homework EquationsThe Attempt at a Solution I am trying to solve the following limit: ##\displaystyle \lim_{x\rightarrow 2} \frac{\left | x^2 + 3x + 2 \right |}{x^2 - 4} =...
  26. moondaaay

    How do I solve absolute value inequalities involving polynomials?

    1. Homework Equations Solving Polynomial Inequalities The Attempt at a Solution Then I used the property of absolute value inequality to get rid of it. But I really don't know if I'm doing the right step. Is this correct? So that I could separate them in two cases and find the...
  27. J

    How can the absolute value of x be negative?

    Homework Statement http://imgur.com/RlIdmFh http://imgur.com/3dnLK3m Homework Equations |x| = -x? The Attempt at a Solution I'm trying to make sense of this definition in my book because they are trying to prove the triangle inequality(second link), yet it keeps saying that the absolute value...
  28. Fancypen

    What are the rules to know when to use absolute value?

    Specifically when doing integration problems. I know the indef integral of cosx/sinx+1 is ln(sinx+1) + C, but absvalue is not required here. I think it's because the sine fn must be >= 0 or it's undefined? What about in other cases, is there a general rule to know when to use absvalue? Thanks
  29. B

    Why is the absolute value of 16 not equal to 4?

    If ##\sqrt{x^2} = |x|##, why ##\sqrt{16} ≠ |4|## instead of 4 (please see below image)?
  30. barryj

    How to solve absolute value equation with two absolute values

    How does one solve an equation with two absolute value functions as below My algebra book does not show how to solve with two abs functions. 2|4x-1| = 3|4x+2| I thought this might work.. |4x-1|/|4x+2| = 3/2 then |(4x-1)/(4x+2)| = 3/2 and solve the normal way..
  31. G

    MHB Sketching absolute value graph

    Sketch the region in the plane consisting of all points (x,y) such that |x-y|+|x|-|y|<=2
  32. H

    Separable differential equations

    Homework Statement [/B]Homework Equations The Attempt at a Solution I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...
  33. P

    Definite integral of an absolute value function

    Can we integrate: $$\int_a^b |x| dx$$ using an antiderivative of ##|x|##, namely ##\frac{1}{2} x |x|##, instead of splitting up the integration interval? I know this is not particularly useful for integrals such as: $$\int_{-5}^5 |t^3 - 8| dt$$ However, for absolute value functions with linear...
  34. PsychonautQQ

    Integral of absolute value function

    how do i evaluate say a definite integral from [-3,3] of |x+1|? Any advice? I'm so confused.
  35. shanepitts

    Why is the absolute value only in the numerator here?

  36. K

    MHB Absolute Value of Complex Integral

    Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{b}^{a} f(t)dt \ \bigg| \leq \int_{b}^{a} \bigg|f(t)\bigg| dt,$$ where the first integral is a complex integral, and the second integral is a definite real integral...
  37. anemone

    MHB What is the minimum value of y in the Absolute Value Function?

    Determine the minimum of $y$ where $y=|x-1|+|2x-1|+|3x-1|+\cdots+|nx-1|$.
  38. A

    Teacher told to set absolute value inequality to equal 0

    So I was helping my sister on homework and there was this problem: 2 abs(2x + 4) +1 > or equal to -3 teacher told her to ignore the -3 and just set it equal to zero. Soo should you? This question got me confused. can't you just go about solving, bringing the 1 to the left and then dividing by 2...
  39. P

    Differentiability of the absolute value of a function

    The derivative of ##|f(x)|## with respect to ##x## is ##f'(x)## for ##f(x) > 0## and ##-f'(x)## for ##f(x) < 0##. However, it is undefined wherever the value of the function is zero. I was wondering, though, if the product of this "undefined derivative" and zero is zero.
  40. Mr Davis 97

    Question on logarithmic differentiation and absolute value

    For the problem of differentiating ##y = x^5(3x-1)^3## using logarithmic differentiation, the solution provides the first step as rewriting the functions as ##\left |y \right | = \left | x \right |^5 \cdot \left | 3x-1 \right |^3##. This confuses me. First, how are we, mathematically, able to...
  41. anemone

    MHB Is the Absolute Value Inequality $|4x-5|-|3x+1|+|5-x|+|1+x|=0.99 Solvable?

    Show that the equation $|4x-5|-|3x+1|+|5-x|+|1+x|=0.99$ has no solutions.
  42. gfd43tg

    Absolute value of exponentials being multiplied

    I know this is an elementary question, but it has been some time since I multiplied exponentials, and with imaginary terms combined with absolute values, things get muddled up so easy that I want to clear this up So if I have $$ \Psi (x,t) = c_{1} \psi_{1} e^{- \frac {i E_{1}}{\hbar} t} +...
  43. M

    Designing a Circuit to Return Absolute Value of 3-bit Number

    Homework Statement Make the design of a circuit returning the absolute value of a number of 3 bits . The input and output must be signed , use the complement 2. Show your approach and draw your track using logic gates. I seriously do not understand this at all. What am I supposed to do...
  44. F

    MHB Does e^{2 \ln{|x|}} = |x^2| or x^2?

    is e^{2 \ln{|x|}} = |x^2| or x^2?
  45. N

    For f(x) = abs(x^3 - 9x), does f'(0) exist

    Homework Statement For f(x) = abs(x^3 - 9x), does f'(0) exist? The Attempt at a Solution [/B] The way I tried to solve this question was to find the right hand and left hand derivative at x = 0. Right hand derivative = (lim h--> 0+) f(h) - f(0) / h = (lim h--> 0+) abs(h^3 - 9h) / h...
  46. S

    MHB Absolute Value with two expressions.

    How do I do this? I have tried a few methods and end up getting x values that don't work when placed back into the equation.
  47. jacobi1

    MHB Trigonometric absolute value integral

    Evaluate \lim_{n \to \infty} \int_0^1 | \sin(nx)| \ dx.
  48. O

    Is the Solution to the Absolute Value Inequality x^2<4 then |x|<=2 Correct?

    Question: True or False If x^2<4 then |x|<=2 My solution: I get -2<x<2 when I solve the problem so it should be false. Yet the text says its true? Is this a mistake? If |x| is equal to 2 then it should be a closed interval, not an open interval which seems to be correct to me.
  49. F

    MHB Derivative of function containing absolute value

    I'm working on a ODE with initial conditions y(2)=4 and y'(2)=1/3. I solved it to be y=\frac{c_1}{|x-6|^8} + c_2|x-6|^{\frac{2}{3}}. How do I apply the second initial condition? I'm stuck at taking the derivative.
  50. B

    Is Absolute Value a Useless Concept?

    Is this mathematical concept EVER used in real life or "higher" levels of math? I just find it to be a practically useless thing. It's like some guys sat around and invented this math concept just for the sake of it. Or, am I wrong?
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