What is Absolute: Definition and 1000 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. S

    Absolute convergence of series?

    The question is: Show that if \suman from n=1 to ∞ converges absolutely, then \suman2 from n=1 to converges absolutely. I'm not sure which approach to take with this. I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or...
  2. ElijahRockers

    Integrating absolute values over infinity

    Homework Statement Find <x> in terms of X0 if X0 is constant and \Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}} and <x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx where Psi* is the complex conjugate of Psi. Since there is no imaginary component, this is effectively Psi2. so, from...
  3. B

    Is |a|<c possible when |a-b|<c-b?

    Homework Statement If a,b,c and are all positive, and if |a-b| < c-b , then prove or find a counterexample to |a|<c Homework Equations The Attempt at a Solution So far I have been able to show |a-b|<c but don't know what to do next. THanks! BiP
  4. D

    How to find the absolute extrema of a function on a given interval?

    Homework Statement Locate the absolute extrema of the given function on the indicated interval. f(x) = -x^2 + 3x ; interval: [0 , 3 ] Homework Equations The Attempt at a Solution f'(x) => -2x + 3x = 0 => x = 0 f(0) = -(0)^2 + 3(0) = 0 f(3) = -(3)^2 + 3(3) = 0 My answer: Absolute...
  5. F

    Absolute Magnitude of the Full Moon

    Hey guys, I was doing some homework and found something I don't quite understand. When calculating the absolute magnitude of the Full Moon, I came across a square root of its Albedo. (here - http://space.wikia.com/wiki/Absolute_magnitude). No idea why it's squared, but I can't get the +0,25...
  6. M

    Proving |xy-ab| is Less Than Epsilon: Absolute Value Question | Homework Help"

    Homework Statement Given: |x-a|<ε |y-b|<ε. proove: |xy-ab|<ε(|a|+|b|+ε) Homework Equations I need a direction for this proof. The Attempt at a Solution I tried by the info: -ε+a<x<ε+a and -ε+b<y<ε+b to ,multiply these inequalities, but it's not true. and i tried with the...
  7. P

    Inequality with two absolute values

    Homework Statement Find all real values of x that satisfy the following inequality. Homework Equations |x-3| > |x + 1| The Attempt at a Solution Splitting up the inequality into cases I get: 1. |x-3| > x + 1 and 2. |x-3| < -x - 1 1. x-3 > x + 1 or x-3 < -x - 1...
  8. S

    Is the proof that ##0^2>0## false if a =0 possible?

    Which of the following two definitions is correct: 1) ##\forall x\forall y[ |x|=y\Longleftrightarrow( x\geq 0\Longrightarrow x=y)\wedge(x<0\Longrightarrow x=-y)]## 2) ##\forall x\forall y[ |x|=y\Longleftrightarrow( x\geq 0\Longrightarrow x=y)\vee(x<0\Longrightarrow x=-y)]## I think the...
  9. N

    Does anyone know how to calculate absolute uncertainty?

    Does anyone know how to calculate absolute uncertainty?? A student repeatedly measures the time taken by an object to cover a certain distance and obtains the following data: t1 = 2.432 s, t2 = 2.393 s, t3 = 2.402 s, t4 = 2.412 s, t5 = 2.424 s The instrument uncertainty is 0.001 s What...
  10. A

    Can Absolute Value Inequalities Prove This Expression?

    Homework Statement we know that |a| < c and |bl < c prove that : (la+bl + la-bl)/2 < c The Attempt at a Solution all I've gotten to so far is this : la+bl < 2c lal + lbl < 2c we have : la+bl < lal+lbl then la+bl < 2c i need to prove that la-bl < 0.? also by squaring all i...
  11. H

    Absolute within an absolute

    Solve: | |x+1| +2| - | x-2 | = 3 Relevant equations: if |x| = a, then x = a; x = -a My attempt: |x+1| +2 - (x-2) = 3 ; |x+1| + 2 - (x-2) = -3 (by theorem provided by teacher above) |x+1| = x- 1 ; |x+1| = x-7 if |x+1| < 0: -(x+1) = x -1 -x - 1 = x - 1 -2x = 0 x = 0...
  12. R

    Why do absolute values appear in the simplification of square roots?

    Homework Statement Simplify. a) \sqrt{x^6} b) 8 \sqrt{x^7y^{10}} - 10 \sqrt{x^7y^{10}} For b, it's y^10. I can't make it look right for some reason. Mod note: Fixed the exponent. Homework Equations The Attempt at a Solution I can simplify all of them but I don't know when or where I need to...
  13. U

    How to find the derivative with an absolute value?

    I need to find the derivative of y=| x | / squareroot of 2-x2. We never learned how to find the derivative with an absolute value, so I have absolutely no idea how to do this problem and I can't find an example in the textbook.
  14. B

    Integrating to an absolute value?

    Homework Statement I need some help understanding an integral step in the example below. I get how the integrand was set up, but I don't get how comes to two expressions with the absolute value of z-L/2. (Problem description if that is needed: L is the length of a cylinder with radius R, and P...
  15. D

    MHB Prove Schwarz's & Triangle Ineqs for Inf Seqs: Abs Conv

    Prove the Schwarz's and the triangle inequalities for infinite sequences: If $$ \sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad \sum_{n = -\infty}^{\infty}|b_n|^2 < \infty $$ then $\sum\limits_{n = -\infty}^{\infty} a_nb_n$ converges absolutely. To show this, wouldn't I need to...
  16. T

    Understanding Absolute Value in Thermodynamics

    Hi everyone,I have a problem.I get a short thermodynamics course.Until now,I have thought,that I know,what absolute value means. And here comes my problem.In lecture is writen,that if we want to make a synthesis in cell valid,we must to have:|ΔH|>|TΔS|.So what this expression means?Does it...
  17. S

    Can absolute zero ever be measured?

    HeyI was wondering - is it physically impossible to make something absolute zero and then measure it to check if it was? I often read that we bring things down to "close to absolute zero" but it never is the full way. Is it because if human or electronic device, which ever it was that tried to...
  18. J

    Gamma Function Limits: Proving \Gamma(x) \to 0 as x \to -\infty

    Homework Statement The absolute value of the gamma function \Gamma (x) that is defined on the negative real axis tends to zero as x \to - \infty . Right? But how do I prove it? Homework Equations The Attempt at a Solution I've tried to use Gauss's Formula...
  19. D

    Solving Absolute Value Inequalities: A Deeper Understanding

    I was hoping someone could give a little more insight, or perhaps enlighten me to a better way of approaching solving these seemingly simple Algebra 2 inequalities. I did some google searching but I was not able to find the answers I seek. The problem came up when a friend of mine had an...
  20. N

    Multiple choice question on the absolute uncertainty?

    An experiment is performed to determine a physical quantity and 10 trials are made. The absolute uncertainty in the quantity can be given by a. the mean of the data b. the standard error of the data c. None of the above I'm leaning more towards b because absolute uncertainty has to deal with...
  21. B

    The absolute value function

    The absolute value function f(x)=|x| has a global minimum at x=0. How could we prove this rigorously? In other words, how could we prove that there is no point c \ \epsilon \ ℝ such that f(c)<f(0) (Obviously, the function is not differentiable at x=0 so we cannot apply Fermat's...
  22. P

    Specific Heat of Sodium vs Absolute Temperature

    what is the relation between Specific Heat and Absolute Temperature of any material ? Specifically, Sodium. Cp vs T ( not the change in temperature) Any mathematical formula ? Thanks
  23. B

    Triangle inequality for a normalized absolute distance

    Hi, can you please give me some hints to show that \frac{|a-b|}{1+|a|+|b|} \leq \frac{|a-c|}{1+|a|+|c|}+\frac{|c-b|}{1+|c|+|b|}, \forall a, b, c \in \mathbb{R}. I tried to get this from |a-b| \leq |a-c|+|c-b|, \forall a, b, c \in \mathbb{R}, but I couldn't succeed. Thank you.
  24. T

    An inequality with absolute values

    Homework Statement Solve Ix+3I>2 *I is used for absolute value notation The Attempt at a Solution Considering both a) Ix+3I > 0 then Ix+3I= x+3 b) Ix+3I < 0 then Ix+3I= -(x+3) when solved this would yield to; a) x>-3 and x>-1 b) x<-5 and x<-3 from my general reasoning i...
  25. C

    MHB Finding Absolute Values: Can someone help me with these practice questions?

    Can someone check these ractice questions for me thanks.i need to understand what is being asked and how to go about this. 1.) For what values is it true that x is less than equal to |x| . 2.)For what values is it true that x=|x|?3.)|z|/-|z|,z is not equal to 0 4.)|t|/|t|, t is not equal to...
  26. C

    MHB Solving absolute value problem

    These i have tried to complete .someone please chech and correct them for me lease and thanks. can u give me steps when dealing with each one.like what i should do to attem these a)|-3-2| =|-5| =5 b)|-5|-|2| =-(-5)-2 c)|7|+|-4| =7+(-4)-7-4=3 d)|-11+1| |-10| =10 e)|6|-|-3| =6-(-3) 6-3=3...
  27. P

    More than one identity element for absolute value?

    I was thinking about identities, and seem to have arrived at a contradiction. I'm sure I'm missing something. A(n) (two-sided) identity for a binary operation must be unique. I will reproduce the familiar proof: Proof: Suppose a is an arbitrary element of a set S, e and e' are both...
  28. Artlav

    Relative and absolute potential?

    Hello. I've been thinking about potential on charged capacitors, and got confused. A charge is AFAIK absolute - there is a balance of electrons and protons in matter, which can be biased one way or another. However, voltage is defined as difference of potential, not an absolute value. So...
  29. F

    Does Resistance Truly Reach Zero at Absolute Zero?

    I was wondering, since resistance decreases along with temperature for most metallic conductors (usually power function), in theory, if this conductor were to be at zero kelvin, would the resistance of this material also be precisely zero, or would it just be some extremely small value? If it...
  30. T

    Absolute value |x-3|^2 - 4|x-3|=12

    Solve |x-3|^2 - 4|x-3|=12 The solution to this equation is -3, 9. But I'm not sure on the working. There is a hint to let u=|x-3| So I worked it out the following way. u^2 -4u = 12 u^2 -4u -12 = 0 (u + 2)(u - 6)=0 u /= -2 and u /= 6 |x-3|=-2 (no solution) |x-3|= 6, x = 9 Now to work...
  31. N

    Comparing Absolute Deviation to Mean Absolute Deviation

    Hi Guys, I am trying to measure variability in a part for my sensor which l do not know the true value. So l decided that a good way to measure variability in this case would be to measure precision of my data points as l change this part on the sensor. So l was wondering, can l compare the...
  32. M

    How is the universe exactly 13.7 billion years old, in absence of absolute time?

    Distribution of mass across the universe is not even. Therefore, passage of time should vary according to gravity. Which means at places time will pass at a higher pace or a lower pace than in respect of other places. Then, how is the entire Universe exactly 13.7 billion years old?
  33. A

    Metric space and absolute value of difference.

    I'm beginning self-study of real analysis based on 'Introductory Real Analysis' by Kolmogorov and Fomin. This is from section 5.2: 'Continuous mappings and homeomorphisms. Isometric Spaces', on page 45, Problem 1. This is my first post to these forums, but I'll try to get the latex right...
  34. S

    Is the Sun's luminosity in specific filters greater than its overall luminosity?

    I'm trying to calculate the luminosity of the sun in the infrared (J, H and K filters). According to http://mips.as.arizona.edu/~cnaw/sun.html, the absolute magnitudes of the Sun in each of these filters are J: 3.64 H: 3.32 K: 3.28. The absolute bolometric magnitude of the Sun is about...
  35. N

    What is absolute in this universe?

    Hello guys, I've been mulling over this thing for as long as I can remember. Since both time and space are relative in nature. Then what is absolute in this universe? Something must be. Because everything can't be relative, else what will they be relative to. Is it consciousness? If it is, then...
  36. R

    Orthogonality of Absolute Derivative

    Hello! Is any covariant vector orthogonal to absolute derivative of its contravariant counterpart? I read a GR book, and it says the tangent vector of a curve is orthogonal to its absolute derivative, that is ##D\lambda^A/dst_A=0##, where ##t^A## is the unit tangent vector of some curve...
  37. R

    What happens to matter near absolute zero

    I'm trying to figure out why things happen to objects placed in environments near absolute zero. so let's take an ice cube. In a freezer it will remain stable, I'm guessing because the atoms that make up the ice cube will bounce off atoms in the air that are roughly similar in energy to the...
  38. J

    Schools Best way for absolute beginner in college to learn math/physics

    I took Calculus AB in my senior year of high school, but I had absolutely no idea what was going on since day 1. Of course, I failed the class. But now I'm in college and I have access to the internet at home now (I've never had internet at my home before this year) and I found all of these...
  39. M

    Is angular frequency absolute?

    I want to treat this from a purely classical stance, ignoring all quantum and relativistic effects. If a body is undergoing centripetal acceleration, is the magnitude of the centripetal force absolute? If so, is its angular frequency also absolute?
  40. C

    Can chemical reactions take place at absolute zero?

    Title says it all. I've been wondering. The argument is, is there a minimal temperature required for any chemical reaction to reach its activation threshold? Or would the energylessness of the atoms prevent bonds from forming (or dissolving, I guess)? Conrad.
  41. A

    Red/Blue Shift for measuring absolute velocity

    Would it be possible to stand at a point, surrounded by sensors and fire beam of light of at specific frequency at every sensor and determine our absolute velocity without a reference based on the amount of red/blue shift detected in the frequency of each beam? Wouldn't light observed by the...
  42. W

    What if , the law of time is an absolute?

    Law of Physic is not that absolute in the first place such as in place like event horizon. Assuming there is one, time travel does not use law of physics in the first place. It may use another set of law that is very obscure to us in current time, the Law of Time. Assuming there is Law of time...
  43. F

    Absolute position in spacetime?

    I am new to the topic of special relativity. Having read a bit of Feynman and googling for answers I am still a bit lost. While I have an aptitude for logic, it does not extend quickly into physics or maths. So quite likely I am barking up the wrong tree and about to demonstrate my ignorance but...
  44. K

    Absolute Convergent, Conditionally Convergent?

    Homework Statement ##\sum _{n=2}\dfrac {\left( -1\right) ^{n}} {\left( \ln n\right) ^{n}}##The Attempt at a Solution I have applied the Alternating Series test and it shows that it is convergent. However, I need to show that it's either absolute conv. or conditionally conv. Next, I tried the...
  45. K

    Absolute Convergence Theorem and Test for Divergence Connection

    Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity. Homework Equations Theorem for absolute convergence. Test for divergence The Attempt...
  46. K

    Determine Absolute Convergence, Conditionally Convergent, or Divergent

    Homework Statement ##\sum _{n=1}^{\infty }\dfrac {\left( -3\right) ^{n}} {n^{3}}## According to Wolfram Alpha the series diverges by the Limit Comparison Test, but I remember that the limit comparison only works with series greater than zero. How is this possible? Homework Equations...
  47. coktail

    Paradoxes in relativity, and an absolute reality

    This is my first post, so forgive me if I'm posting in the wrong place or violating a rule. I've been doing some reading on relativity, and find myself flummoxed. I understand gravitational time dilation and the principals behind relative velocity time dilation, length contraction, and the...
  48. S

    Multiplying a postivie number by a absolute expression

    Homework Statement If I have a expression like 5lx+2l, can I multiply the 5 with x+2? Homework Equations The Attempt at a Solution I'm guessing that I can only do this if the number in front of the absolute expression is a positive number. Because if -5lx+2l=-10 then multiplying...
  49. F

    Absolute Convergence of a Series

    Homework Statement I'm given this series and asked whether it converges, absolutely converges, or diverges. Ʃ(n=0 to infinity) [2(-1^n)(3^(n+1))]/5^n Homework Equations The Attempt at a Solution The answer states that it is absolutely convergent, and that it converges to 15/4. Everything...
  50. S

    First ODE of an absolute value

    so I understand the basic premise of differentiating a first ODE, or I thought I did. I have the equation y'-y=abs(x-1). I have no idea of how to go about this. Can someone walk me through how to do this? I'm attempting to study for a test and this is one of the practice questions he gave us so...
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