# Thread: Complex number

1. ## Complex number

a) modulus of Z-3i<3. Find the greatest possible value of modulus of z. Modulus means square bracket.

I have no idea about finding the greatest possible value.

2. Originally Posted by kingkaisai2
a) modulus of Z-3i<3. Find the greatest possible value of modulus of z. Modulus means square bracket.

I have no idea about finding the greatest possible value.
The region of the complex plain such that $\displaystyle |z-3i|<3$ is the interior of the circle of
radius $\displaystyle 3$ centred at $\displaystyle 3i$. If the region was in fact closed (that is $\displaystyle |z-3i|\le 3$)
then $\displaystyle z=6i$ would give the maximum modulus of $\displaystyle |z|=6$, but the region is open,
so there is no $\displaystyle z$ with maximum modulus in the region.

RonL

3. Originally Posted by kingkaisai2
a) modulus of Z-3i<3. Find the greatest possible value of modulus of z. Modulus means square bracket.

I have no idea about finding the greatest possible value.
I will explain differently.
Let,
$\displaystyle z=x+iy$
Then,
$\displaystyle |z-3i|=|x+iy-3i|=|x+i(y-3)|=\sqrt{x^2+(y-3)^2}$
But,
$\displaystyle |z-3i|<3$
Thus,
$\displaystyle \sqrt{x^2+(y-3)^2}<3$
Square,
$\displaystyle x^2+(y-3)^2<9$
This is a circle center at $\displaystyle (0,3)$
Now look upon CaptainBlank's post.
He said that is the region is closed meaning,
$\displaystyle x^2+(y-3)^2\leq 9$ then you can find a maximum value, namely, 9. Which would give 6.
If the region is open meaning,
$\displaystyle x^2+(y-3)^2<9$ then you cannot find a maximum value.