1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Approximation to sqrt(1+(d^2)/(x^2))

  1. May 27, 2016 #1
    using binomial theorem can I write sqrt(1+(d^2)/(x^2)) = 1+ .5(d^2)/(x^2)?
    d is a variable. X known constant.
  2. jcsd
  3. May 27, 2016 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    If ##d^2 < x^2##, then you can indeed write an approximate equality

    [tex]\sqrt{1+ \frac{d^2}{x^2}}\approx 1 + \frac{1}{2} \frac{d^2}{x^2}[/tex]

    Whether this approximation is good enough depends entirely on the specifics.

    If ##d^2 = x^2##, then it's a boundary case. You can still write the above approximaton, but this is very sensitive to errors. For example, if you measured ##d## or ##x## a bit incorrectly, then you could be in trouble easily.
  4. May 28, 2016 #3


    User Avatar
    2016 Award

    Staff: Mentor

    @micromass: the sensitivity to d and x individually should not depend on the approximation made.
    If d2=x2, then the left side is about 1.41 and the right side is 1.5, so we have some notable deviation.
  5. May 28, 2016 #4


    User Avatar
    Science Advisor

    For a first approximation, yes. But you can do better:

    1. Calculate [itex]a=1+\frac{d^{2}}{x^{2}} [/itex]
    2. Let [itex]z_{0}=1+\frac{1}{2}\frac{d^{2}}{x^{2}}=\frac{1+a}{2} [/itex]
    3. Then [itex]z_{n+1}=\frac{1}{2}(z_{n}+\frac{a}{z_{n}}) [/itex] is a better approximation.
    Stop when zn+1 is sufficiently close to zn.

    In the example above (d=x); a = 2 and z0 = 1.5. Then z1 = 1.417 and z2 = 1.414
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Approximation to sqrt(1+(d^2)/(x^2))
  1. Graphing sqrt(x^2 + 1) (Replies: 12)

  2. Sqrt(x^2) and (Sqrt x)^2 (Replies: 14)