# Find the least possible value of ##|z-w|## -Complex numbers

• chwala
In summary: As for the summary, here it is:In summary, the conversation discusses finding the least possible value of |z-w| for two complex numbers z and w satisfying certain inequalities. The distance between the two centers is given by L = sqrt((7-3)^2+(5-2)^2) = 5 and the least possible value |z-w| is 5-(2+1) = 2. The conversation also explores the possibility of finding the greatest distance, which is determined to be 8. The conversation refers to a 2010 post on Physics Forums but does not provide a link.
chwala
Gold Member
Homework Statement
Two complex numbers ##z## and ##w## satisfy the inequalities ##|z-3-2i|≤2## and ##|w-7-5i|≤1##. By drawing an argand diagram, find the least possible value of ##|z-w|##

There is a similar post to this posted in 2010 on physicsforums and the OP did not seem to have posted his working to solution.(I wanted to make some comments on that but the post is not open to further replies)
Relevant Equations
Complex numbers.
OK, here once a sketch is done, we have two circles ##c_1## and ##c_2## with centre's ##c_1 (3,2)## and ##c_2 (7,5)## having radius ##2## and ##1## respectively. It follows that the distance between the the two centre's is given by ##L=\sqrt {(7-3)^2+(5-2)^2}##=##5##
Now, the least possible value ##|z-w|=5-(2+1)=2##

Supposing, just to explore this further, they want us to find the greatest distance, then we may say that the greatest ditance of ##|z-w|=5+1+2=8##

I would appreciate your thoughts on this...cheers guys

Last edited:
No thoughts, except: 'where's the picture?'
Well, perhaps one small second thought: what 2010 thread ?

##\ ##

chwala
BvU said:
No thoughts, except: 'where's the picture?'
Well, perhaps one small second thought: what 2010 thread ?

##\ ##
Bvu...you can tell from my working that i know how the pic looks like...This is the 2010 post;

chwala said:
Homework Statement:: Two complex numbers ##z## and ##w## satisfy the inequalities ##|z-3-2i|≤2## and ##|w-7-5i|≤1##. By drawing an argand diagram, find the least possible value of ##|z-w|##

There is a similar post to this posted in 2010 on physicsforums and the OP did not seem to have posted his working to solution.(I wanted to make some comments on that but the post is not open to further replies)

I think it's pretty standard to not care if a thread was made twelve years ago, and just make a new one instead.

chwala said:
This is the 2010 post;
Haha, as if I care for a screen shot. My angle was: if you refer to something, don't let others search for it but provide a link. @SammyS understood.

chwala said:
you can tell from my working that i know how the pic looks like
Same difference: yes I can, but maybe others can not.

Office_Shredder said:
I think it's pretty standard to not care if a thread was made twelve years ago, and just make a new one instead.
Especially the bad and messy ones

##\ ##

chwala
chwala said:
Supposing, just to explore this further, they want us to find the greatest distance, then we may say that the greatest distance of ##|z-w|=5+1+2=8##
Yes, of course.

Thanks haruspex...cheers mate...

## 1. What is the definition of the least possible value of |z-w| in complex numbers?

The least possible value of |z-w| is the smallest distance between two complex numbers, z and w, on the complex plane. It represents the shortest distance between the two numbers and can be found by calculating the absolute value of the difference between the real and imaginary components of z and w.

## 2. How do you find the least possible value of |z-w| in complex numbers?

To find the least possible value of |z-w|, first calculate the absolute value of the difference between the real and imaginary components of z and w. Then, take the square root of this value to get the distance between the two numbers on the complex plane. The resulting value is the least possible value of |z-w|.

## 3. Can the least possible value of |z-w| be negative?

No, the least possible value of |z-w| cannot be negative. Since it represents the distance between two complex numbers, it must always be a positive value. If the calculated value is negative, it means there was an error in the calculation.

## 4. How does the least possible value of |z-w| relate to the distance formula in the Cartesian plane?

The least possible value of |z-w| is essentially the distance formula in the Cartesian plane, but applied to the complex plane. It follows the same principles of calculating the distance between two points, but takes into account the real and imaginary components of complex numbers.

## 5. Are there any real-life applications for finding the least possible value of |z-w| in complex numbers?

Yes, there are many real-life applications for finding the least possible value of |z-w| in complex numbers. For example, it can be used in navigation systems to calculate the shortest distance between two locations on a map. It can also be used in physics and engineering to determine the minimum distance between two points in a complex system.

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