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October 1st, 2016, 07:57 AM   #1
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graphing set

How do I graph a set that is defined as z given that |z -[complex number]| <= 1
(apologies: I am approaching an online class with an undeniably weak background, but with determination....).

Last edited by skipjack; October 22nd, 2016 at 10:31 AM.
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October 1st, 2016, 08:06 AM   #2
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You'll see the expression $|z - z_0|$ a lot.

The region

$|z-z_0| \leq \rho$ is a disk on the complex plane of radius $\rho$, centered at the point $z_0$.
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Last edited by skipjack; October 22nd, 2016 at 10:32 AM.
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October 1st, 2016, 08:27 AM   #3
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Thanks!

Last edited by skipjack; October 22nd, 2016 at 10:31 AM.
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October 22nd, 2016, 09:38 AM   #4
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Specifically, in the "complex" plane z= x+ iy. Writing point $\displaystyle z_0$ as a+ iy, then $\displaystyle |z- z_0|= |x+ iy- (a+ ib)|= |(x- a)+ i(y- b)|= \sqrt{(x-a)^2+ (y- b)^2}$ so that $\displaystyle |z- z_0|\le \rho$ is $\displaystyle \sqrt{(x- a)^2+ (y- b)^2}\le \rho$ and squaring both sides, $\displaystyle (x- a)^2+ (y- b)^2\le \rho^2$.

$\displaystyle (x- a)^2+ (y- b)^2= \rho^2$ is the circle, in the complex plane with center $\displaystyle z_0= a+ ib$ and radius $\displaystyle \rho$.
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