max and min (complex) (1 Viewer)

[jnney]

If z is a complex number such that |z-4-3i|=1, find algebraically [i.e. by triangular inequality], the max and min value of |z|.

 

[SpiralFlex]

Max value 6.










Min value = 4.

 

[jnney]

how did you get from line 3 to 4? :/

 

[jnney]

OH

 

[SpiralFlex]

Always remember the power of modulus.

 

[bleakarcher]

spiral, how is mod(z1-z2) greater than or equal to the mod(z1-z2)?

 

[D94]

spiral, how is mod(z1-z2) greater than or equal to the mod(z1-z2)?

Think of a triangle. The third side can't be longer than the combination of the other two sides.

 

[bleakarcher]

Ah ok, I understand now actually.

 

[bleakarcher]

Wait, no, wikipedia says the opposite.
http://en.wikipedia.org/wiki/Triangle_inequality

 

[D94]

Wait, no, wikipedia says the opposite.
http://en.wikipedia.org/wiki/Triangle_inequality

Yeah, my bad, I meant "can't be longer". But yes, that concept.

 

[bleakarcher]

If so, then I think spiralflex made a mistake?

 

[bleakarcher]

Can someone clear up what spiralflex did for me?

 

[D94]

If so, then I think spiralflex made a mistake?

This one: http://en.wikipedia.org/wiki/Triangle_inequality#Example_norms

 

[bleakarcher]

D94, do u mind explaining how spiral got the inequality in line 2?

 

[deterministic]

rearrange

which comes from the triangle inequality

 

[math man]

you are forgetting that |z1-z2| also forms one of the sides of a triangle when you use the parallelogram of addition for vectors, hence the triangle inequality can be used for |z1-z2| as well as |z1+z2|

 

[tohriffic]

Are you okay with the idea that the sum of any two sides of a triangle will be greater than the other side?

Similarly with the other triangle inequalities,

<img src="http://latex.codecogs.com/gif.latex?|z_{2}| + |z_{1} - z_{2}| > |z_{1}|" title="|z_{2}| + |z_{1} - z_{2}| > |z_{1}|" />

Rearranging that will give us:

<img src="http://latex.codecogs.com/gif.latex?|z_{1} - z_{2}| > |z_{1}| - |z_{2}|" title="|z_{1} - z_{2}| > |z_{1}| - |z_{2}|" />

And we know that equality holds true when the points are collinear. And we end up with what Spi provided:

<img src="http://latex.codecogs.com/gif.latex?|z_{1} - z_{2}| \geq |z_{1}| - |z_{2}|" title="|z_{1} - z_{2}| \geq |z_{1}| - |z_{2}|" />

Hurr I attached a picture just for the sake of visualising.

 

[largarithmic]

You actually need another step to solve a "maximum" or "minimum" problem: you need to show the value is achieved. What I mean is this; you can define a maximum as follows:
"A function f has a maximum M, if for all x, f[x] <= M, and for some a, f[a] = M"

So what Spiralflex and others have shown in this problem, is that 4 <= |z| <= 6 for instance. But that is actually not enough to show that, for instance, 6 is the maximum value of |z|. Because isn't it also true that |z| <= 9, since everything that is less than or equal to 6 is also less than or equal to nine - but clearly, 9 is not the maximum! So you need the second part of the definition to actually have it make sense. Here it's sufficient, I think, to just draw a diagram and do it geometrically since that makes it clear that there are cases where equality is reached in the 4 <= |z| <= 6 bound.