Let z be a complex such that:

1<=|z|<=6.

Then what are the minimum and maximum values for the complex (z+3).

I suspect that:

0<=|z+3|<=9, but I can't find a way to prove it.

Please help!

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- Mar 15th 2008, 07:40 AMtombrowningtonmodulus of a comples
Let z be a complex such that:

1<=|z|<=6.

Then what are the minimum and maximum values for the complex (z+3).

I suspect that:

0<=|z+3|<=9, but I can't find a way to prove it.

Please help! - Mar 15th 2008, 07:57 AMMoo
Hello,

I don't understand well, are you asked to find the minimum and maximum values for the complex (z+3) or the minimum and maximum values for the modulo of (z+3) ? - Mar 15th 2008, 09:40 AMtombrownington
Hi Moo,

Sorry I didn't put it well.

It's the maximum value for the modulus of (z+3) that I was looking for. - Mar 15th 2008, 09:54 AMMoo
I'll try, but i find the question weird :

Replace z by a+ib.

Hence

So

(squared the previous inequations)

z+3 would be (a+3)+ib

=>

So

And that's what i find strange : there is no information about a :( - Mar 15th 2008, 10:02 AMtombrownington
Moo, I already tried that method and came to the same result.

I thought of it in a different way - the proof isn't at all rigorous; in fact it's just an insight not a proof.

By adding 3 to z we are shifting the point representing z (on the complex plane) 3 units to the right.

Now since z can be -3, this gives us |z+3|=0. So this has to be the minimum - the smallest possible value for a modulus being 0.

Also if z = 6, then |z+3| = 9, and you will find that whatever other point z you take, the modulus |z+3| will always be smaller than 9.

But as I said, this is only an insight. I can't get any further.

Do you see something that I don't? - Mar 15th 2008, 10:16 AMMoo
Well, i need to think again about it. There must be a way to put a restriction to a and to make it fit with your supposition (which seems correct ! ;))

- Mar 15th 2008, 10:50 AMwingless
This is what the problem wants to say:

Find such |z| that and that maximizes (or minimizes) |z+3|.

Now this is only an optimization question =)

will make minimum. So,

For maximum,

The maximum x and y values that satisfy and maximize are 6 and 0.

This is how to find it:

Maximize:

Maximize:

There is no term of y in 6x + 9 so x must be maximum. Then and

- Mar 15th 2008, 10:57 AMwingless
- Mar 15th 2008, 05:49 PMThePerfectHacker