Crosswords5 mins ago
The circle with center infinity with infinite radius is the point zero on the complex plane?yes,no,why?
8 Answers
something my prof asked me..maybe has something to do with the one point compactification of the complex plane to define the infinity..not sure how the centre can be zero once you specify its at infinity though...sigh!
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You might find this useful.
http://www.liv.ac.uk/...a-giblin-isophote.pdf
http://www.liv.ac.uk/...a-giblin-isophote.pdf
This was the answer i got back though i couldnt quite see how the compactification proved it..
''if you are considering the one-point compactification of the complex plane. A glance at the stereographic projection of the sphere onto the plane makes this immediately clear.''
anyway ill check the material you gave..but i would appreciate if you could explain the above..thanks in advance
''if you are considering the one-point compactification of the complex plane. A glance at the stereographic projection of the sphere onto the plane makes this immediately clear.''
anyway ill check the material you gave..but i would appreciate if you could explain the above..thanks in advance
This is interesting too.
http://math.fullerton...FunReciprocalMod.html
It seems you have posted this query on some other forums too- have you had any responses there?
http://math.fullerton...FunReciprocalMod.html
It seems you have posted this query on some other forums too- have you had any responses there?
The main point here is to make a distinction between infinity and the point at infinity. What is relevant in your case is the point AT infinity, which is DEFINED to be the point corresponding to z=0 (the origin) under the transformation 1/z. This means that an EXTENDED complex plane is now defined which includes this point at infinity.
This is nothing mysterious, just a convenience which makes sense and allows mathematicians to say that under the transformation 1/z circles are transformed to circles, where the definition of circle is now extended to in clude straight lines.
Any circle in the complex plane which passes through the origin (z=0) is transformed by 1/z into a line, and all other circles are transformed to circles.
Compactification is most easily understood in terms of the projection of the complex plane onto a sphere by stereographic projection:
The complex plane WITHOUT the point at infinity transforms to a sphere (usually called the Riemann sphere) with the north pole point missing. In the case of the EXTENDED complex plane then this north pole point corresponds to the point at infinity of the extended complex plane and so the Riemann sphere now has no holes in it and so is compact.
I know this doesn't answer your question yet but it may help to clarify your question. Can you perhaps ask your prof. to clarify his question and then we can come up with an answer.
This is nothing mysterious, just a convenience which makes sense and allows mathematicians to say that under the transformation 1/z circles are transformed to circles, where the definition of circle is now extended to in clude straight lines.
Any circle in the complex plane which passes through the origin (z=0) is transformed by 1/z into a line, and all other circles are transformed to circles.
Compactification is most easily understood in terms of the projection of the complex plane onto a sphere by stereographic projection:
The complex plane WITHOUT the point at infinity transforms to a sphere (usually called the Riemann sphere) with the north pole point missing. In the case of the EXTENDED complex plane then this north pole point corresponds to the point at infinity of the extended complex plane and so the Riemann sphere now has no holes in it and so is compact.
I know this doesn't answer your question yet but it may help to clarify your question. Can you perhaps ask your prof. to clarify his question and then we can come up with an answer.
He told me the answer at last..thanks for the help!
basically the Riemann sphere is for the compaction of the two real R manifold. Thats why I specified the complex plain so that point infinity would be one point. Next we using the definition of a circle as the locus of equidistant points from a specific point. when you have infinite radius the only point that has maximum distance from this point would be the origin that is infinite radius. Its not that the origin and the point infinity are the same.
basically the Riemann sphere is for the compaction of the two real R manifold. Thats why I specified the complex plain so that point infinity would be one point. Next we using the definition of a circle as the locus of equidistant points from a specific point. when you have infinite radius the only point that has maximum distance from this point would be the origin that is infinite radius. Its not that the origin and the point infinity are the same.
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