Someone explain continuity principle

Click For Summary

Discussion Overview

The discussion revolves around the continuity principle in the context of circle intersections, particularly focusing on the implications of imaginary solutions in algebraic equations representing circles. Participants explore the nature of intersections in both real and complex planes.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents algebraic equations for two circles and finds that their intersection points yield imaginary values for y, questioning the validity of claiming they intersect in the traditional sense.
  • Another participant suggests the possibility of plotting these circles in the complex plane, with real and imaginary axes, to visualize the intersection.
  • A different viewpoint emphasizes that while the intersection points are imaginary, it does not negate the existence of intersections outside the Euclidean plane.
  • One participant asserts that the principle stating equations like X^2 = t always have two solutions supports the original assertion about intersections, implying a connection to the continuity principle.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of intersections involving imaginary numbers, with some arguing that such intersections are valid in a broader mathematical context, while others challenge the generalization of the term "intersect" in this scenario. No consensus is reached.

Contextual Notes

The discussion highlights the limitations of applying traditional geometric interpretations to equations that yield complex solutions, as well as the dependence on definitions of intersection in different mathematical frameworks.

Physics news on Phys.org
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).
 
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!
 
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)?
 
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.
 
HallsofIvy said:
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
 
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.
 

Similar threads

Undergrad Circumference of a circle on a spherical surface
  • · Replies 7 ·
Replies
7
Views
6K
Undergrad Geometry and algebraic equations
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Holographic principle in continuous spacetime?
  • · Replies 8 ·
Replies
8
Views
4K
High School How to draw a plane intersecting a cylinder at a "compound angle"
  • · Replies 8 ·
Replies
8
Views
3K
Continuity principle in practice
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Understanding the Equivalence Principle
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Properties of the initial topology from a topological manifold
  • · Replies 8 ·
Replies
8
Views
2K
Map of a 4D planet
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad It's still not clear to me what's the limit of light propagation
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Please Explain (actually explain) The Monty Hall Problem
  • · Replies 131 ·
5
Replies
131
Views
10K