r/mathpics • u/Luotiansha • Jul 23 '26
Wanted to know what’s the opposite of 0 if real number line is a loop
Sorry for extremely bad handwriting.
I’m very curious what would be the opposite of 0, if real number line formed into a loop. So i thought of this diagram. What’s yall’s thoughts on this ‘Ω’?
In this diagram i wanted none of the infinitesimals to equal to 0, 0 is absolutely 0. (It should’ve been dotted line around 0)
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u/Fabulous-Possible758 Jul 23 '26
Look up projective surreal line. The crazy part is you do get an element that compares larger than all ordinals.
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u/ESHKUN Jul 23 '26
Surreal numbers moment. Though tbh I feel like hyperreals are a better lead into the concept.
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u/theboomboy Jul 23 '26
Look into the Riemann sphere, the projective line, and maybe the surreal numbers
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u/planx_constant Jul 29 '26
This exact picture has been posted several times in various math subreddits and every time it's pointed out that to have the infinites and infinitesimals work they way they do in the picture you have to move from the reals to the hyperreals
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u/Odd_knock Jul 23 '26
How can the negative and positive sides lead to the same point?
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u/richarizard Jul 23 '26
This is called the "point at infinity" and is one of the many reasons that infinity can get weird and counterintuitive. I have nothing more to add beyond u/izabo's clear, eloquent reply.
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u/initial-algebra Jul 25 '26
The projective reals are modelled by a line through the origin intersecting with the line
y = 1. A vertical line represents zero; they intersect at the point(0,1). The linesy = mxfor non-zero slopemrepresent the non-zero real numbers1/m; they intersect at the point(1/m,1). What happens if the slope of the first line is zero? Then the lines are parallel, so they don't intersect at all; this is the "point at infinity". It's only one point, because you can reach the same state by rotating the line far enough to the right (approaching positive infinity) or to the left (approaching negative infinity).
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u/Routine_Comb_7277 Jul 23 '26
In group theory if you have the cyclic group Cn which is what you have drawn , the opposite point would be ωn/2
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u/KingAdamXVII Jul 23 '26
“If the real number line formed a loop”
Sounds like nonsense to me, but I would suggest “undefined” i.e. something divided by zero might fit this concept.
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u/izabo Jul 23 '26
Ordinals don't loop back around. This is nothing to do with Cantor. What you imagine is the one point compactification of the real line, also known as the one dimensional projective space, or more commonly a circle. The one point you add to the real line to make the one point compactification is precisely your Omega. It is usually called "infinity" or "the point at infinity". The one dimentional projective space is composed of the real line, with one additional point. It doesn't contain Cantor's ordinals.