# Complex Numbers: Finding the least value of |z-w|

## Homework Statement

The complex numbers, z and w satisfy the inequalities |z-3-2i|<=2 and |w-7-5i|<=1

Find the least possible value of |z-w|

No clue at all.

## The Attempt at a Solution

Since its |z-w| i figured that the least possible value will only be when both are max. I tried finding the maximum distance of each complex number by using $$\sqrt{}(x^2+y^2)$$+r and came up with a Z=$$\sqrt{}13$$+2 and W being $$\sqrt{}74$$+1

Both of which are incorrect as z-w gives 4 while the answer is 2

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jbunniii
Homework Helper
Gold Member
The triangle inequality gives you

$$|z-w| \leq |z| + |w|$$

This gives you an upper bound for the distance.

You can also use the triangle inequality to obtain a lower bound, as follows:

$$|z| = |(z - w) + w|$$

$$|w| = |(w - z) + z|$$

Now apply the triangle inequality to the right hand sides of both of the above equalities and rearrange to get a lower bound for $|z - w|$.

LCKurtz
Homework Helper
Gold Member
They describe two circular discs. The closest points will be on the line between their centers.

Could you kindly explain this a little bit:

I first get |z| by using |z|<= 2+|3+2i| and then put z= 2+|3+2i| in the formula for |z| that u have given?

jbunniii
Homework Helper
Gold Member
OK, to keep the notation simpler, let $z_0 = 3 + 2i$ and $w_0 = 7 + 5i$.

Then you can write

$$z - w = (z - z_0) - (w - w_0) + (z_0 - w_0)$$

Therefore

$$|z - w| = |(z - z_0) - (w - w_0) + (z_0 - w_0)|$$

You can apply the triangle inequality in reverse to obtain

$$|(z - z_0) - (w - w_0) + (z_0 - w_0)| \geq | |z_0 - w_0| - |(z - z_0) - (w - w_0)||$$

Now, $|z_0 - w_0|$ is just a positive constant (call it $c$), so the minimization of the right hand side is easy. The task is to mimimize

$$|c - |(z - z_0) - (w - w_0)||$$

If there are $z,w$ that make this zero, then the minimum is zero. Otherwise, the minimum is achieved by maximizing $|(z - z_0) - (w - w_0)|$, which you can easily do by using the triangle inequality again.

By the way, this procedure will give you a lower bound. You still have to justify why there are $z,w$ that achieve the lower bound. (Think: under what condition does the triangle inequality become an equality?)

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They describe two circular discs. The closest points will be on the line between their centers.

Do u mean it will be |3+2i - ( 7+5i)|?? but that gives the answer 5 while it is 2 in the book

LCKurtz
Homework Helper
Gold Member
Do u mean it will be |3+2i - ( 7+5i)|?? but that gives the answer 5 while it is 2 in the book

No. |z - (3+2i)| is the distance from x + yi to 3 + 2i. If that is equal to 2 it says the point (x,y) is distance 2 from the point (3,2) in the xy plane. That describes a circle and the inequality describes the interior of that circle. Similarly for the other inequality. Draw a picture.

No. |z - (3+2i)| is the distance from x + yi to 3 + 2i. If that is equal to 2 it says the point (x,y) is distance 2 from the point (3,2) in the xy plane. That describes a circle and the inequality describes the interior of that circle. Similarly for the other inequality. Draw a picture.

hmm..I get this part. You are saying that Z and W lie on the circumference of the circle with center 3,2 and radius 2 and center 7,5 with radius 1 right?

I cannot get beyond this point. The question is asking for the minimum value of |z-w| so this means that both Z and W should be at maximum distance from the origin?

OK, to keep the notation simpler, let $z_0 = 3 + 2i$ and $w_0 = 7 + 5i$.

Then you can write

$$z - w = (z - z_0) - (w - w_0) + (z_0 - w_0)$$

Therefore

$$|z - w| = |(z - z_0) - (w - w_0) + (z_0 - w_0)|$$

You can apply the triangle inequality in reverse to obtain

$$|(z - z_0) - (w - w_0) + (z_0 - w_0)| \geq | |z_0 - w_0| - |(z - z_0) - (w - w_0)||$$

Now, $|z_0 - w_0|$ is just a positive constant (call it $c$), so the minimization of the right hand side is easy. The task is to mimimize

$$|c - |(z - z_0) - (w - w_0)||$$

If there are $z,w$ that make this zero, then the minimum is zero. Otherwise, the minimum is achieved by maximizing $|(z - z_0) - (w - w_0)|$, which you can easily do by using the triangle inequality again.

By the way, this procedure will give you a lower bound. You still have to justify why there are $z,w$ that achieve the lower bound. (Think: under what condition does the triangle inequality become an equality?)

Thanks!! Although i would like to find a simpler way to do it as well.

Ok by drawing a picture the shortest distance between Z and W is coming out too be$$\sqrt{}74$$-$$\sqrt{}13$$-3=1.9996

I did this by calculating W by $$\sqrt{}74$$- 1 and Z by $$\sqrt{}13$$ + 2, then subtracted the distance W from Z to get its smallest value.

I would be greatly obliged if someone can check my method, or better yet do the whole question.

LCKurtz
Homework Helper
Gold Member
hmm..I get this part. You are saying that Z and W lie on the circumference of the circle with center 3,2 and radius 2 and center 7,5 with radius 1 right?

I cannot get beyond this point. The question is asking for the minimum value of |z-w| so this means that both Z and W should be at maximum distance from the origin?

Why would you think that? It asks for the minimum distance between z and w, nothing about how far from the origin they are. As I said in my first post, the points closest to each other on the circles will be on the line joining the centers. Draw a picture. Look at it. It is simple geometry. You can do it in your head.

jbunniii