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. M

    Proof of an expression involving absolute value

    Homework Statement Prove: |ab|=|a||b|. (Hint: Consider the cases separately for various signs of a and b.) Homework Equations I think it asks me to prove it only from the definition of absolute value (if it is possible). The Attempt at a Solution |ab|= \frac{(ab), if (ab)\geq0}{-(ab), if...
  2. A

    Derivative of cubic absolute value function

    Homework Statement f(x) = |x+2|^3 -1 The Attempt at a Solution - So, I know the formula is: d|x| / dx = x / |x| - My guess would be: (x+2)^3 / |x + 2|^3 ? I'm trying to find Increase and Decrease intervals for the graph..
  3. K

    Double integral absolute value.

    Homework Statement I just need to know if what i did is correct. The question is as follows: http://imgur.com/1RL7e Homework Equations The Attempt at a Solution What I did is as follows: I split this integral into two parts and solved. [integral from 0 to 1...
  4. P

    How to integrate modulus function ( absolute value) within the specified limits

    Homework Statement integrate modulus of xcos∏x within the limits -1 to 1/2 P.S I'm not allowed to use acaluculator so graphing and finding out the answer not possible Homework Equations it's using the property of definite integral ∫f(x) dx within the limit from a to b = ∫f(x) dx...
  5. T

    Derivatives of ln including Absolute Value

    Homework Statement Determine the domain and find the derivative f(x) = ln|(x+2)/(x3 - 1)| Homework Equations The Attempt at a Solution We can factor x3 - 1 = (x-1)(x2) + x + 1) From this we know that x ≠ 1 or -2 because 1 would be undefined at -2 would cause the function to...
  6. F

    What is the Absolute Value of b(t)?

    I need to find the absolute value of b(t): b(t)=a(t)[1+e^{-j2\pi tT}] Here is the answer in the textbook: \therefore \left | b(t) \right |=\left | a(t) \right |\sqrt{2(1+cos(2\pi tT)}2|cos(\pi tT)| However I got a different answer when working it out, and can't understand how they got...
  7. S

    Did I explain this right? Basic absolute value functions

    Homework Statement I was helping my sister with her high school geometry homework, and she had to graph the absolute value function y=lx+6l using two linear equations. The way i did it was different than the way her book showed. Basically, I told her that when removing the absolute value you...
  8. C

    Understanding Energy of a Signal: Integral & Absolute Value Sign

    Hi All, I am confused about what is meant in the mathematical definition for the energy of a signal. Why is it the integral of the magnitude of the signal squared? ∫(|f(t)|^2) How do I deal with the absolute value sign? I do not have much experience with absolute value signs in...
  9. S

    Does the Limit Exist for Absolute Value as x Approaches -8?

    Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim x→−8 [ 8 − |x| OVER 8 + x ] The attempt at a solution First → abs(x) = -x ; x<0 / x ; x≥0 Therefore, our equation is going to look like 8 − -x OVER 8 + x If approaching -8 from the left, I...
  10. M

    Intergrating a derivative of absolute value in exponential

    hey~ folks! i'm in trouble with this integral problem. the range of the integral is from -∞ to ∞ and it has x as independent value. and the integrand is a derivative of absolute value in exponential; ∫^{∞}_{-∞} (d/dx e^{-a|x|})dx any idea for the problem??
  11. khurram usman

    Understanding the Relationship between Square Roots and Absolute Values

    absolute value help ... i am a bit confused about two things... first: is sqrt(x^2) is the same thing as |x|? if yes then consider a negative no. 'z' ... i had a question like this: sqrt(z^2)...its answer was -z...so i suppose this means that this square root/ square and absolute value thing...
  12. N

    Why Does the Negative Sign Become Positive in Absolute Value Equations?

    Absolute value Quest, urgent Homework Statement a) |x-2| + |6-3x| = 12x Homework Equations The Attempt at a Solution I have a question about y this negetive becomes a positive.. :: Steps :: |x-2| + -3|x-2| |x-2| +3 |x-2| = 12x > this step if i factored out...
  13. A

    Why is the term 'norm' used instead of 'absolute value' in vector spaces?

    I saw some books and say that norm is the absolute value in vector. If it also means absolute value, why don't we use absolute value |\vec{v}| instead we use ||\vec{v}||?
  14. P

    Absolute value equations with extraneous solutions

    I'm a little confused about solving absolute value equations and why sometimes solutions don't seem to make sense. Take a look at case (ii) on this website: http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_AbsoluteValueEquations.xml I understand the process of solving the...
  15. N

    Solving Absolute Value Equations: Understanding Restrictions and Changing Values

    Urgent! Absolute value help Homework Statement Heres an example :: |5x-3| = |x+1| we were given 3 cases for this type of problem in class they went something like this:: c.1 ) Use the 0 value for the right side and change all values in respect to (introduce - to all values) (x <...
  16. N

    Absolute Value Equations: Solving |2x-8| + |12-3x| = 0

    Homework Statement |2x-8| + |12 - 3x | = 0 Homework Equations what do i do when it = 0 The Attempt at a Solution I tried solving it like all the others, but i know there is a unique thing to do when it = 0, i jus don't know what. So , here's what i did. 2|x-4| +...
  17. Jonnyb42

    Integral of 1/x = ln(x) Problem has missing absolute value?

    Integral of 1/x = ln(x)... Problem has missing absolute value!? This is an ODE problem and solution, but what I'm not understanding here is calculus based: Why does the argument to the red-boxed ln() not in absolute value signs?? ( | | ) u = cos(t) so the numerator can be negative. I am just...
  18. W

    Basic limit involving absolute value

    Homework Statement lim as x\rightarrow-1 = \frac{|x^{2}-1|}{x^{2}+x} Homework Equations N/a The Attempt at a Solution Tried to write this as a piecewise function, but I got lost.
  19. K

    Integral of de absolute value of a derivative

    Hi! Homework Statement I'm trying to work out this integral, without success: \int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a) 2. The attempt at a solution I tried solving this by parts, but I'm stuck: \int_{-a}^{a} (1-\left|x\right|/a)...
  20. S

    Minimum Absolute Value of a nxn Matrix Determinant

    The following matrix problem occurred to me. I figured out the answer and would like to pose the problem. It's easy but would be best for an undergrad math major. The question: Consider a square n by n matrix with entries 1, 2, ..., n squared. Find a way to arrange these entries so that the...
  21. 5

    Solving an equation with an absolute value within an absolute value

    Homework Statement Solve the equation | | 3x + 1 | - x | = 2 Homework Equations |x| = x if x \geq 0 |x| = -x if x < 0 The Attempt at a Solution To start, I have never dealt with such an equation before, and am partly 'iffy' with absolute values in general, but, up...
  22. 1

    Is ln|sin x| equivalent to ln sin x?

    Homework Statement y = ln | sin x | Homework Equations - The Attempt at a Solution No attempt at a solution, because it's more of a "which part of this is incorrect" question. y = ln | sin x | Would everyone agree that this is equivalent? y = ln \sqrt{(sin x^2)} Then...
  23. W

    Why Does the Integral of |x^2 - 9| from 0 to 4 Require Splitting at x=3?

    Homework Statement ∫ |x^2 -9| [0-4] Homework Equations The book answer states the same EXCEPT splits into [0-3] and [3-4]. Other problems split the integral perfectly in half for absolute values...why would it differ and are there rules to figure this out? Larson's Calculus has no...
  24. F

    Interesting Absolute Value problem

    Homework Statement Find all x in |5x-2| = 6x-12 The Attempt at a Solution http://www.wolframalpha.com/input/?i=Plot[|5x-2|+%3D+6x-12%2C{x%2C-1.5%2C1.5}] Notice how only x = 10 is a solution? So my question is, why isn't x = -14/11 a solution? I mean sure they don't...
  25. I

    Finding the absolute value (modulus) and solving equation

    [b]1. | (2-i)(3+2i)(-5-6i)(-7+3i) / (-4+i) | Homework Equations [b]3. I am pretty sure I am not computing it right but this is what i come up with. square root of 5* square root of 13* square root of 61* square root of 58 / square root of 17.
  26. N

    Precal, Inequalities involving absolute value

    Homework Statement DIRECTIONS: Express the intercal in terms of an inequality involving absolute value. PROBLEM: (-4, 4) MY STEPS: 1: (-4, 4) 2: -4<x<4 3: |x|< 4 MY ANSWER Is that correct? Is step 3 correct? The only reason that I included that part is becuase it says...
  27. F

    Absolute Value with Quadratic: What Are the Solutions to |x^2 + 6x + 16| < 8?

    Homework Statement Very embarrassing that I had problems with this. I managed to solved it at the end, but I am posting this for two reasons 1) I like someone else to work it out in a different method, I wnt to see what other approaches there are 2) Whether mine is taking too long or not. I...
  28. A

    Integrating Factor and absolute value

    Hi, I have a general question regarding the integrating factor of first-oder linear DEs. All textbooks that I've seen (which aren't too many) simply drop the absolute symbol when the factor has the form exp(ln(abs(x))). This would evaluate to abs(x), yet the books use simply x. Why is that...
  29. I

    Proving Absolute Value Convergence of Sequence to A

    Homework Statement If the absolute value of a sequence, an converges to absolute value of A, does sequence, an necessarily converge to A? Homework Equations convergence: a sequence { an}n=1-->infinity, converges to A є R (A is called the limit of the sequence) iff for all є > 0, there...
  30. G

    Absolute value of complex number

    Homework Statement Let R be a set of real numbers derived from rational numbers. And, let R* be a set consisting of all ordered pairs of the form (x,0) where x is an element of R. For a complex number z = (a,b) = a+ib, I've learned that the absolute value of z is the number...
  31. T

    Absolute value function is the Heaviside step function

    Homework Statement [PLAIN]http://img845.imageshack.us/img845/1706/delta.png The Attempt at a Solution This question has come after a chapter on Dirac's delta-function although the question doesn't say use it. Could I use it? If not how do I do it directly?
  32. ECHOSIDE

    Absolute value appears on both sides of equals sign

    Homework Statement solve for x: 5|7-3x|+2=4|7-3x|-1 Homework Equations The Attempt at a Solution 7-3x>0 -3x>-7 x<3/7 {x|x<3/7} quantity is positive {x|x>3/7} quantity is negative I have never solved an equation with more than one absolute value quantity in it and am not sure how to start...
  33. W

    How Do Absolute Value Inequalities Apply to Different Sign Scenarios?

    Homework Statement I was trying to show that 1) |a+b|≤|a|+|b| 2) |a+b|≥|a|-|b| and find out how they were true when a,b>0, a,b<0, and a>0,b<0 Homework Equations 1) |a+b|≤|a|+|b| 2) |a+b|≥|a|-|b| The Attempt at a Solution For |a+b|≤|a|+|b| a,b>0 I got that |a+b|=a+b...
  34. V

    Where is it appropriate to place absolute value bars while solving this diffeq

    Homework Statement just a simple problem as an example dy/dx = y/x dy/y = dx/x Homework Equations The Attempt at a Solution dy/y = dx/x ln|y| = ln|x| + c then when I take exponents do I still have to include the absolute value bar |y| = |x|c
  35. S

    How a Absolute value Function can be continuous?

    Hello friends, I am quite confused how an absolute function is called a continuous one. f(x) = |x| has no limit at x=0 , that is when x > 0 it has a limit +1 {+.1, +.01, +.001} and -1 when x <0 {-.1, -.01, -.001} that is the reason it's not differentiable (left and right side limits are not the...
  36. S

    Double integrals with an absolute value

    Homework Statement Im getting very confused with working out how to integrate the following double integral with an absolute value: \int^{2}_{-2}\int^{2}_{-2}\left|x^{2}+y^{2}-1\right|dxdy Homework Equations The Attempt at a Solution I know you have to split it down into where it...
  37. russ_watters

    Derivative of Absolute Value Confusion

    Quick question (hopefully): I'm helping my girlfriend with some calculus homework over the phone and it looks like the book has a different answer for the derivative of an absolute value than I'm seeing in other places. I'm seeing it as (and I derived it myself to be) x/abs(x) whereas the book...
  38. W

    Divergence involving absolute value

    Homework Statement Part of a problems asks us to show that for a general function f(|r1-r2|)=f(R) that Div[f] only with respect to r1 is the same as Div[f] only with respect to R Homework Equations f(|r1-r2|)=f(R) The Attempt at a Solution I'm really unsure where to start, but I...
  39. K

    I hope that will get you started.

    Homework Statement Prove that, if f is continuous on [a,b] and ∫ab= l f(x) l dx = zero then f(x) = 0 for all x in [a ,b]. Homework Equations Hint- from book- Section 2.4 Exercise 50 Let f and g be continuous at c. Prove that if: (a) f(c) > o, then there exists delta > o such...
  40. M

    Whats the derivative of the absolute value to the power of p?

    Homework Statement Let p>0. What is the derivative of |x|^{p}? Homework Equations The Attempt at a SolutionI know that if p is even, then the derivative is just px^{p-1} . But what if x is odd? Would it turn out to be some piecewise function, such as px^{p-1},\: if x \geq 0 and -px^{p-1}...
  41. M

    Absolute value of riemann sums

    Homework Statement I'm trying to prove that Sp|f| - sp|f| \leq Spf - spf Where P is a partition of [a,b] and f is function that is riemann integrable. Homework Equations The Attempt at a Solution So I get to a point where M = supf(x) and m = inff(x) then |M|(b-a) - |m|(b-a)...
  42. X

    Absolute Value with a number in front?

    Alright, I'm completely confused. I was given a study guide for my Algebra 2 exam(this isn't homework...) and one of the questions was 2|3x+5| < 16 I've seen the absolute value, but never seen a number in front of it...How do I solve this ?
  43. G

    Understanding RMS vs. Absolute Value for Calculating Averages in Data Analysis

    I never really understood why using the Root Mean Square of negative data is the preferred method of finding the average of the data as opposed to taking the absolute value of the data and taking the average (arithmetic mean) of that. The example that recently made me wonder about this is...
  44. A

    Find the absolute value of theta 650-theta 420

    Two rays of light of wavelength 650 nm and 420 nm are normally incident on a different grating. If the grating has 5000 slits/cm, what is the angular separation of of the two light rays' second order maximum. -Need to find the absolute value of theta 650-theta 420 I know I need to use...
  45. R

    Absolute value for a sequence.

    Hey. I have | tn - t | < M How can i show that t - M < tn thanks
  46. P

    Complex number equation - absolute value assumption

    I have a problem in understanding the procedures of a solved example. It goes like this. \left ( \frac{z+i}{z-i} \right )^4 = -1 Therefor we can write: \left | \frac{z+i}{z-i} \right | = 1 From that we can see that z is a real number because: \left | z+i \right | = \left | z-i...
  47. P

    Equations with multiple absolute value operators

    Hello there. I'm having some problems with absolute values when they contain multiple "abs" operators and some other numbers outside the "abs"-es. For example: \left | x+2 \right | - \left | x \right | > 1 If i check it for the positive scenario, the result is true for all x-es. x+2...
  48. L

    Can absolute value functions be considered polynomial functions?

    Are lx3l or lxl3 polynomials? If not, what would be a good example of a cubic polynomial function (R \rightarrow R) that doesn't cover all real numbers in its codomain?
  49. T

    Arctan absolute value problem, calculus 1

    Homework Statement prove |arctan(x)-arctan(y)| \leq |x-y| The Attempt at a Solution I tried separating cases: 1. both x and y are in the same quadrant; 2. x and y are not in the same quadrant; 3. x and y are zero (trivial); 4. either x or y is zero. Now, i can see why each of these is...
  50. H

    How Do You Solve Absolute Value Inequalities with Double Variables?

    how do you go about solving abs value inequalities with double variables when the abs value bars are on both the variables? eg; |x| + or - |y| =, >, <, a
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