Someone explain continuity principle

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  • #1
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  • #2
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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].
 
  • #3
HallsofIvy
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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!
 
  • #4
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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)?
 
  • #5
HallsofIvy
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[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.
 
  • #6
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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!
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
 
  • #7
mathwonk
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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.
 

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