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A Closed set in the Complex Field 
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#1
Feb413, 12:14 AM

P: 376

This is elementary but surely this set is closed
 c – i  ≥  c  with c being in ℂ I am trying to picture the set. Is it outside the disc centered at (0,1) with radius equal to modulus c (whatever that is) ? Thanks 


#2
Feb413, 01:22 AM

Sci Advisor
P: 827

Hint: find the boundary first.



#3
Feb413, 07:26 AM

P: 71

Visualize the complex number as a point in the plane. Move it down one unit. (That's what subtracting i does.) Now, what points will be further from the origin when this happens? You can construct a right triangle of legs Re(c) and Im(c) and a right triangle of legs Re(ci) and Im(ci) and see which hypotenuse is longer.
Once you have figured out the region in question, it should be easy to show its closure. 


#4
Feb413, 07:43 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,490

A Closed set in the Complex Field
Corrected thanks to oay. 


#5
Feb413, 08:37 AM

P: 234

You mean: c = x + (1/2)i for any real x. 


#6
Feb413, 02:30 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,490

Right, thanks. And thanks for trailing along behind me cleaning up my mess!



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