## Someone explain continuity principle...

How do the circles still intersect at the bottom, and at 2 points like the top 2 circles?
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 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus 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 $$x^2+y^2=1$$ The blue circle has center at (5,0) and has radius 1. The equation is $$(x-5)^2+y^2=1$$ We can now find the points in the intersection of these two circles. We know from the first equation that $$y^2=1-x^2$$ Substituting that in the second equation gets us $$(x-5)^2 + (1 -x^2 )=1$$ This is an equation that can easily be solved. we get x=5/2. We substitute that in the first equation and get $$y^2=-21/4$$ and thus $$y=\pm i\sqrt{21}/2$$ So the points of intersection are $(5/2,i\sqrt{21}/2)$ and $(5/2,-i\sqrt{21}/2)$.
 Recognitions: Gold Member Science Advisor Staff Emeritus 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!

## Someone explain continuity principle...

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)?
 Recognitions: Gold Member Science Advisor Staff Emeritus $e^{R\theta}$ 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.

 Quote by HallsofIvy 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
 Recognitions: Homework Help Science Advisor 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.