Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Someone explain continuity principle

  1. Aug 29, 2012 #1
  2. jcsd
  3. Aug 29, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You can show this algebraically. Let's take our circle with radius 1.
    Then the red circle has center at (0,0) and has radius 1. The equation for such a circle is
    [tex]x^2+y^2=1[/tex]
    The blue circle has center at (5,0) and has radius 1. The equation is
    [tex](x-5)^2+y^2=1[/tex]

    We can now find the points in the intersection of these two circles. We know from the first equation that

    [tex]y^2=1-x^2[/tex]

    Substituting that in the second equation gets us

    [tex](x-5)^2 + (1 -x^2 )=1[/tex]

    This is an equation that can easily be solved. we get x=5/2. We substitute that in the first equation and get
    [tex]y^2=-21/4[/tex]

    and thus

    [tex]y=\pm i\sqrt{21}/2[/tex]

    So the points of intersection are [itex](5/2,i\sqrt{21}/2)[/itex] and [itex](5/2,-i\sqrt{21}/2)[/itex].
     
  4. Aug 30, 2012 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    But the y values are imaginary numbers while the numbers defining the coordinate system must be real numbers- so to say the circles "intersect" there is generalizing "intersect" a heck of a lot!
     
  5. Aug 30, 2012 #4
    is it possible to plot the circles with y-axis having the imaginary part and x axis having the real part(on the complex plane)?
     
  6. Aug 30, 2012 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    [itex]e^{R\theta}[/itex] gives a circle with center at 0 and radius R in the complex plane. You cannot plot an equation like y= f(x) with y and x complex numbers because you would have to have real and complex axes for both x and y- and that requires 4 dimensions.
     
  7. Sep 2, 2012 #6
    It is my understanding that the intersection does exist, just not in the euclidian plane. So it can be said that the circles intersect without changing the meaning of intersection
     
  8. Sep 17, 2012 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    this principle in its simplest form says that the equations X^2 = t always have two solutions no matter what t is. if you believe that, then you must also believe the original assertion, as micromass showed.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Someone explain continuity principle
  1. Continuous on a curve? (Replies: 1)

  2. Continuous function (Replies: 8)

  3. Topology continuity (Replies: 1)

Loading...