One More Point Than a Line

Date: 07/05/98 at 17:51:47
From: Noel Pascual
Subject: Circle and line

Our professor asked us the question "Why does a circle have exactly 1 
more point than a line?"  In terms of 1-1 correspondence, he said that 
a circle has 1 more point than a line.

Date: 07/06/98 at 13:03:42
From: Doctor Floor
Subject: Re: Circle and line

Hi Noel,

Thank you for sending your question to Dr. Math!

I think that your professor wants to hear from you that a circle is 
closed, and a line is not, because it has two open ends. A line would 
need to have one more point, "the point at infinity," to be closed 
too. You can formulate this in terms of topology (a line is an open 
set and a circle is a closed set), but I don't know whether your 
professor wants to hear that.

In terms of 1-1-correspondence you can see this nicely with the help 
of a picture I made for you: 

     

In this picture you see a circle, with center A, and a line (I will 
refer to it as "the line"). On the circle point B is chosen in such a 
way that AB is perpendicular to the line, and B is at the opposite 
side of A seen from the line. 

Now each point X on the circle except B itself can be projected to the 
line by drawing BX and intersecting it with the line, producing X'. 
The other way around can be done in the reverse way: If you have a 
point Y' on the line, draw BY' and intersect it with the circle, 
producing Y. 

So there is a 1-1-correspondence for almost all points. You could take 
any point on the line, and find an image on the circle. And you could 
take any point but B on the circle, and find an image on the line.

However, for point B itself we created no image. There is no image for 
point B in this projection, because you should then intersect 'BB' 
with the line. What is 'BB'? 'BB' is the line through B tangent to the 
circle. You can see this from the fact that if you take X closer and 
closer to B, the line XB gets closer and closer to the tangent line 
through B (drawn in the picture). So 'BB' is the tangent line through 
B, and thus is perpendicular to the radius AB. But then 'BB' is 
parallel to the line also, so that it cannot intersect the line. So 
B has no image in this correspondence. So the circle has one point 
more, as your professor says.

I hope this makes it clear.

Best regards,

- Doctor Floor, The Math Forum
Check out our web site! http://mathforum.org/dr.math/