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February 17th, 2013, 02:55 PM   #1
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Negative Radius Circle

Good day,

Just joined this forum and am curious if there are some here who might like to help extend the thinking of one of my young gifted students. We got into a discussion about the radius of circles and he asked what a circle with a radius of -1 would look like? Of course, I resisted the impulse to squash his curiosity and have instead asked him to explore the topic until we meet again. I find young mathematicians gain a deeper understanding when they explore these topics for themselves, so I'm not asking for an "answer" but some trails he might follow to satisfy his own curiosity. Perhaps into topology(?)

If you had suggestions for theories / references / applets etc. that might help him explore this topic, I would love to collate them and present them back to him (and you).

Thanks in advance,
Tom
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February 17th, 2013, 06:18 PM   #2
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Re: Negative Radius Circle

There is no circle with a negative radius,because negative numbers do not exist in real life.

You cannot say I have -1 pen,but you can say I got -1 pen meaning that you lost one.

So "a circle having radius -1" is not a good statement.You can say the circle's radius increased by -1 meaning it was reduced by 1.You cannot possess a negative number,neither can anything.You can either get it or lose it.
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February 18th, 2013, 04:10 AM   #3
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Re: Negative radius circle

Quote:
Originally Posted by tfullerton
We got into a discussion about the radius of circles and he asked what a circle with a radius of -1 would look like?
The circle of radius 1 centred at is . The circle of radius ?1 centred at the same point is . So the two circles are the same.
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February 18th, 2013, 04:17 AM   #4
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Re: Negative Radius Circle

...if you kick the ball in your own net during a soccer game, did you score -1 goals ? :P
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February 18th, 2013, 04:51 AM   #5
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Re: Negative Radius Circle

Quote:
Originally Posted by mathmaniac
There is no circle with a negative radius,because negative numbers do not exist in real life. You cannot say I have -1 pen,but you can say I got -1 pen meaning that you lost one. So "a circle having radius -1" is not a good statement.You can say the circle's radius increased by -1 meaning it was reduced by 1.You cannot possess a negative number,neither can anything.You can either get it or lose it.
Thank you for your response. I'll ask my student where in life he might find negative numbers. I suspect he will respond with references to temperature, elevation and so on.
Similar discussion around the concept of zero. Is it a number? A value? Or can we only ever achieve a closeness? But that would best start a new thread I think.....

What I'm trying to do is to inspire in this young gifted student a curiosity to explore mathematical concepts, not to tell him "the" answer (in part because I don't think there is one in this case, but more because I want to encourage in him an inquiry disposition).
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February 18th, 2013, 04:52 AM   #6
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Re: Negative Radius Circle

Quote:
Originally Posted by Denis
...if you kick the ball in your own net during a soccer game, did you score -1 goals ? :P
Ha! Thanks, I'll pass that one along
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February 18th, 2013, 04:53 AM   #7
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Re: Negative radius circle

Luckily for sports fans soccer wasn’t invented by mathematicians.
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February 18th, 2013, 05:46 AM   #8
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Re: Negative radius circle

Quote:
Originally Posted by Crimson Sunbird
Quote:
Originally Posted by tfullerton
We got into a discussion about the radius of circles and he asked what a circle with a radius of -1 would look like?
The circle of radius 1 centred at is . The circle of radius ?1 centred at the same point is . So the two circles are the same.
Very helpful, thank you!
I think I'll ask him what happens when he graphs this using Geonext. I'm hoping he can begin to visualize algorithms and move from theoretical to applied (and back again!).
Appreciate your help inspiring this young mathematician,
Tom
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February 18th, 2013, 06:00 AM   #9
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Re: Negative Radius Circle

Hy,

Well, the radius of the circle is the distance from the center of the circle to its points. A metric (or distance) "d" in metric spaces is defined as a function from the used set A, and the set of real numbers R, that satisfies the following conditions :

1. d(x, y) = 0 if and only if x = y (identity of indiscernibles, or coincidence axiom)
2. d(x, y) = d(y, x) (symmetry)
3. d(x, z) ? d(x, y) + d(y, z) (subadditivity / triangle inequality).

now I will demonstrate that 0 ? d(x,y) for any x and y in the set A:

assume that there exists x and y such as : d(x,y) < 0
cond2 gives us d(x,y) + d(y,x) < 0
cond3 gives us d(x,x) ? d(x,y) + d(y,x) < 0
cond1 gives us 0 = d(x,x) ? d(x,y) + d(y,x) < 0
then 0 < 0 which is contradiction, then with the said conditions, we can say that d(x, y) >= 0 for any x and y in the set A.

Your student should either think of a way to redefine the circle, or to redefine the distance, to make a generalization.

For me, the most simple generalization would be redefining the circle, and taking into account the 3rd dimension, by saying that a circle has an orientation, it is either looking "up" or "down" following whether the radius is positif or negatif, in an oriented plan. And don't tell me that "up" and "down" are not defined and corresponds to from where we look !!! I said oriented plan.

cheers
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February 18th, 2013, 06:18 AM   #10
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Re: Negative Radius Circle

Or maybe a circle with positif radius would be looking inside, and with negatif radius would be looking outside, so the area of a circle with R > 0 would be Pi*R², but if R < 0 it would be +infinity
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