MathFoundations28: What exactly is a circle?

Moving beyond points and lines, circles are the next geometrical objects we encounter. Here we address the question of how best to introduce this important notion, strictly in the setting of rational numbers, and without metaphysical waffling about `infinite sets.'

Category:

Science & Technology

Tags:

• Mathematics
• geometry
• foundations
• circles
• rational
• numbers
• Wildberger
• quadraance

[simpeltree]

Hi !

We see at time 3:59 that the cirkel equation is given by:

(1) x^2+y^2-2ax-2by+c=0

At time 4:21 we see that the cirkel equation can be rewritten as:

(2) (x-a)^2+(y-b)^2=k

Now I expand (2) and get: (3) x^2-2ax+a^2+y^2-2by+b^2=k

Here in (3) I have two new terms a^2 and b^2 that does not appear in (1). Further I have no c term in (3) which appear in (1). Does it mean that c term is given implicitly in (3) to be: c=a^2+b^2 ? Thanks.

[njwildberger]

Hi simpeltree,

You're almost correct. Comparing (1) and (3) we must have c=a^2+b^2-k.

[njwildberger]

Hi mattmoss,

This series is trying to build mathematics completely logically. Although you are used to talking about circles over the real numbers, it turns out that this doesn't make proper sense, due to foundational difficulties with irrational numbers.

However, over the rational numbers we can still do geometry. And talk about circles, hyperbolas etc. This way arithmetical questions connect to geometrical ones.

Thanks for your question!

[simpeltree]

Hi njwildberger, Thanks.

[simpeltree]

@simpeltree Hi again ! I think I self have the answer to my question. in (3) c is:

c=a^2+b^2-k, where a, b, and k all are constants.

[mattmoss]

Like the series! (and have the book) Question: I am accustomed, from H.S. and college, to the circle over the irrational numbers and speaking about infinite sets. Putting the latter aside for the moment, what is the purpose/goal of examining the circle over the rational set? I understand just fine that some lines may not meet a circle when described over the rationals, but am unsure as to the reasons why one would do so. Thanks!