Treasure hunt using complex numbers & an inequality

[Verdict]

Homework Statement

Question 1:
You find an old map revealing a treasure hidden on a small island. The treasure was buried in the following way: the island has one tree and two rocks, one small one and one large one.
Walk from the tree to the small rock, turn 90 to the left and walk the same distance in that direction. Mark the point where you end up by A. Next, do the same for the big rock, but here turn 90 to the right. This way you end up at a point that is marked B. The treasure
is hidden halfway the line AB. You go to the island to find out that the tree no longer exists. How are you going to find the treasure?

Question 2:

Homework Equations

For the first question, I can't really think of any relevant equations. The question is part of a course in complex analysis, so either using the vector representation of complex numbers, a + ib, or the polar coordinates, re^i$\theta$ are bound to be in there somewhere, but how I don't see straight away.

For the second question, again from the complex analysis course, the most important formula that I can think of is the triangle inequality, |z₁+z₂|$\leq$|z₁|+|z₂|

The Attempt at a Solution

Alright, the first question has me most puzzled. After drawing a few random situations, it seems as if the distance from rock one to the middle of AB is the same length as the distance of rock 2 to that same middle. This is all from just drawing mind you, so in no way proven. It also seems as if you walk to the middle of the two rocks and turn 90 degrees, you could dig a trench either down or up, until you find the treasure. Doesn't sound like a solid plan, either. Apart from this I don't really know where to begin. I've also had a lot of trouble approaching this issue not from a geometric side but from a point where I utilize complex functions instead. Could anyone give me a hint on where to start?

The second question I have gotten a bit further, but I somehow feel like either I did something wrong (plausible) or the equality sign in the question is the wrong way around (less plausible, but still possible).

Here's what I did:

I apologize for the images instead of latex, I should and will learn how to use it some time soon.

 

[mfb]

I like question 1 :). It is a puzzle with a surprising result, and complex numbers really help.
Hint: Begin with a coordinate system (for complex numbers). A clever choice of the origin will help to find the coordinates of A and B in a general way. Afterwards, you have to get rid of those coordinates.A general tip for inequalities in complex numbers: Try to avoid them with their components. Usually, this just generates unnecessary complexity.

I think I would multiply the equation with those denominators and try to use the triangle equality afterwards. Not sure if that helps, but it would by my first try.
Edit: On a second thought, you have to include |z1|+|z2| in any inequality in some way.

 

[Verdict]

Hmm. As for coordinate systems, the only ones I know are the Complex (euclidean) plane and the polar representation. I suppose since there isn't a lot of multiplication going on, the euclidean seems like the natural way to go. As for the origin, the only sensible point seems to be the middle between the two rocks. Again, from here on I somehow still start to think in terms of triangles and sides or angles that are equal, which is really not the way to go..

As for the inequality, could you possibly elaborate on that? I could add that they are both smaller than 2 times |z1|+|z2| at the end of the line, but apart from that I don't really see where to go.

 

[mfb]

I suppose since there isn't a lot of multiplication going on, the euclidean seems like the natural way to go.

Right for the wrong reason ;).

As for the origin, the only sensible point seems to be the middle between the two rocks.

I would not do this, but it is possible (just harder to get the main idea probably). The rocks do not have any specific relation to each other here. The position of the treasure has to be independent of the coordinate system - you can use systems which cannot be constructed on the island, too.Concerning the inequality: Hmm, maybe polar coordinates can be useful here. I don't know.

 

[Verdict]

Right for the wrong reason ;).

I would not do this, but it is possible (just harder to get the main idea probably). The rocks do not have any specific relation to each other here. The position of the treasure has to be independent of the coordinate system - you can use systems which cannot be constructed on the island, too.Concerning the inequality: Hmm, maybe polar coordinates can be useful here. I don't know.

Hmm.. Alright then. Another origin that I could pick is the actual middle of the line AB, so the spot where the treasure is buried. But that also seems difficult.. Dang, I'll try and give it some more thought, thanks.

Writing it all out in polar coordinates did not do too much for me, as I don't know how to express |z1 + z2| in them, other than just replacing them with their polar versions, is there anything more I can do?

I suppose picking the treasure as the origin has some merit, as the points a and be will be on opposite sides of a circle with the origin at its center. Which probably isn't the correct way to go either, as that suggests polar again. Meh.

 

[mfb]

|z1+z2| does not look nice, of course, but the other parts do.

There is another choice for the origin ;).

 

[Verdict]

Another one? I really can't think of anything else than just the tree at this point.

For the inequality, the other parts do indeed look nicer. I get that (r₁+r₂) multiplied with the absolute value of the angles, which is a maximum of 2, is smaller than or equal to |z1+z2|. Is that what you had in mind?

 

[jbunniii]

I've been playing around with the inequality but haven't found a proof yet. However, I did notice an interesting fact which may prove useful. If you manipulate the left hand side a bit, you can get it equal to this expression:
$$\left|z_1\left(1 + \frac{|z_2|}{|z_1|}\right) + z_2 \left(1 + \frac{|z_1|}{|z_2|}\right)\right|$$
If we put
$$r_1 = 1 + \frac{|z_2|}{|z_1|}$$
and
$$r_2 = 1 + \frac{|z_1|}{|z_2|}$$
note that $r_1$ and $r_2$ have the interesting property that $r_1 r_2 = r_1 + r_2$, or equivalently,
$$\frac{1}{r_1} + \frac{1}{r_2} = 1$$
which is reminiscent of conjugate exponents in $L^p$ spaces.

 

[jbunniii]

Also, note the geometric interpretation:
$$m |u_1 + u_2| \leq |z_1 + z_2|$$
where
$$m = \frac{|z_1|+|z_2|}{2}$$
is the average of the lengths of $z_1$ and $z_2$, and $u_1$ and $u_2$ are unit-length versions of $z_1$ and $z_2$.

 

[mfb]

Another one? I really can't think of anything else than just the tree at this point.

Try it, it is a great choice ;).


@jbunniii: An interesting way to express the left-handed side.

 

[Verdict]

Hmm, I don't really see the geometric interpretation. That the average of the two lengths is smaller than or equal to the sum of the lengths combined?

I thank you all for your great help, I will go to bed now and continue tomorrow!

Edit:
Hah, thanks mfb, I guess I wasn't really playing ball with that hint.. So I use Cartesian coordinates (complex though) with the tree as origin. How do I continue from there on though? Do I just try to write the steps you take initially (when the tree is still there) as vectors?

 

[Verdict]

The professor has given me the following hint regarding the inequality, which gets me even more confused:

if you take two complex numbers z_1=r_1e^{i\theta} and r_2e^{-i\theta}, the l.h.s. is zero, but the r.h.s. is |r_1-r_2|.


Now first of all, I don't see how this is general, as you pick the arguments of the z's to be opposite. on top of that, the lhs does not reduce to zero for me, but to (r1+r2)*|2cos theta |. I don't get the minus sign on the RHS either. It really just made things worse.

 

[mfb]

Ah well, let's simplify the inequality a bit. It is invariant with respect to a common phase of z1 and z2: If you multiply both with $e^{i \theta}$, both sides of the inequality stay the same.
Therefore, without loss of generality, we can assume that the sum of their arguments is zero.

The left side will not be zero (in general), but we get closer to a geometric interpretation.

 

[jbunniii]

While playing around with the inequality yesterday, I made the following simplifications:

The goal is to prove that
$$(|z_1| + |z_2|) \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2|z_1 + z_2|$$
The denominators are not even defined if $z_1 = 0$ or $z_2 = 0$, so we assume that both are intended to be nonzero.

Let us write $z_1 = |z_1|e^{i\theta_1}$ and $z_2 = |z_2|e^{i\theta_2}$.

Without loss of generality, we may rigidly rotate the complex plane by the angle $-\theta_1$, and then scale it by $|z_1|$, to obtain new variables
$$u_1 = \frac{z_1}{|z_1|}e^{i(\theta_1 - \theta_1)} = 1$$
and
$$u_2 = \frac{z_2}{|z_1|}e^{i(\theta_2 - \theta_1)}$$
Then the stated inequality will be true if and only if this equivalent inequality holds:
$$(1 + |u_2|) \left| 1 + \frac{u_2}{|u_2|}\right| \leq 2 |1 + u_2|$$
We are free to assume either $|u_2| \leq 1$ or $|u_2| \geq 1$, whichever makes this work out properly. This corresponds to whether the original unscaled numbers satisfied $|z_2| \leq |z_1|$ or $|z_2| \geq |z_1|$, respectively.

I still didn't manage to prove it, but perhaps you'll find the above useful.

 

[DrClaude]

Can't you just make use of
$$\left| z_1 + z_2 \right| \leq \left| z_1 \right| + \left| z_2 \right|$$
in the LHS of the inequality?

 

[Verdict]

Hmm. The inequality is not really going anywhere for me. The simplifications / changes you make I do understand, but indeed, it does not give the proof just yet.

What I have done for the treasure now, is write out the coordinates of the interesting points, as follows:
The origin (0,0) = 0 + 0i = the tree = z1
Rock 1 = z2 = (a,b) = a + ib
Rock 2 = z3 = (c, d) = c + id
Point A = z4 = z2 plus z2 turned by 90 degrees to the left = (a + ib) + (-b + ia) = (a-b, b+a)
Point B = z5 = z3 plus z3 turned by 90 degrees to the right = (c + id) + (d - ic) = (c+d, d-c)
The treasure = z6 = in the middle of point A and B = (a-b+c+d / 2, b+a+d-c /2) = (a-b+c+d)/2 + i(b+a+d-c)/2

Is this useful, or is it just redundant? I understand that I don't have these coordinates when looking for the treasure, but something about the distances between them or whatever might be extractable from the situation..

 

[jbunniii]

Since $u_2/|u_2|$ is a unit vector,
$$\left| 1 + \frac{u_2}{|u_2|} \right|$$
cannot be larger than 2. Unfortunately, it is not true that $(1 + |u_2|) \leq |1 + u_2|$; indeed, the inequality goes in the opposite direction. So obviously we have to combine the two factors on the left side and rewrite them in a clever way if we are going to succeed. I didn't manage to find the trick yet.

 

[jbunniii]

Can't you just make use of
$$\left| z_1 + z_2 \right| \leq \left| z_1 \right| + \left| z_2 \right|$$
in the LHS of the inequality?

How do you suggest using it?

 

[jbunniii]

The professor has given me the following hint regarding the inequality, which gets me even more confused:

if you take two complex numbers z_1=r_1e^{i\theta} and r_2e^{-i\theta}, the l.h.s. is zero, but the r.h.s. is |r_1-r_2|.


Now first of all, I don't see how this is general, as you pick the arguments of the z's to be opposite. on top of that, the lhs does not reduce to zero for me, but to (r1+r2)*|2cos theta |. I don't get the minus sign on the RHS either. It really just made things worse.

I agree that his hint seems wrong. While it may be useful to orient the plane as he describes, you don't get the simplifications he claims.

 

[jbunniii]

Also, note the geometric interpretation:
$$m |u_1 + u_2| \leq |z_1 + z_2|$$
where
$$m = \frac{|z_1|+|z_2|}{2}$$
is the average of the lengths of $z_1$ and $z_2$, and $u_1$ and $u_2$ are unit-length versions of $z_1$ and $z_2$.

I interpret this geometrically as follows: if we strip away the magnitude information from $z_1$ and $z_2$ but don't change their direction, then we get unit vectors $u_1$ and $u_2$. Multiplying by $m = (|z_1| + |z_2|)/2$ reassigns them both the same new length, namely the average of the two original lengths. The inequality states that doing this will never make the magnitude of the sum larger than it was to begin with.

This doesn't make a proof any more obvious, but the statement at least seems plausible.

After reading about your instructor's nonsensical hint, I decided it would be a good idea to get some confidence that the inequality is in fact true, so I wrote a Matlab script to test it with 10,000 random complex values. Fortunately, it passed:

numTrials = 10000;

for n = 1:numTrials
    
    z_1 = 10*randn(1) + i*10*randn(1);
    z_2 = 10*randn(1) + i*10*randn(1);
    
    lhs = (abs(z_1) + abs(z_2)) * abs(z_1/abs(z_1) + z_2/abs(z_2));
    rhs = 2*abs(z_1 + z_2);
    
    if (lhs > rhs)
        error('Hey! The inequality is false!');
    end
end

 

[DrClaude]

$$\left| z_1 + z_2 \right| \leq \left| z_1 \right| + \left| z_2 \right|$$
therefore
$$\left| z_1 + z_2 \right| \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq \left| z_1 \right| + \left| z_2 \right| \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right|$$
Going back to the original inequality, we can write
$$\left| z_1 + z_2 \right| \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2|z_1 + z_2|$$
If $\left| z_1 + z_2 \right| = 0$ we have an equality, otherwise, since $\left| z_1 + z_2 \right| > 0$
$$\left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2$$
which we can rewrite as
$$\left|e^{i \theta_1} + e^{i \theta_2} \right| \leq 2$$
QED

[Edit: Forget about this, it's wrong.]

 

[jbunniii]

$$\left| z_1 + z_2 \right| \leq \left| z_1 \right| + \left| z_2 \right|$$
therefore
$$\left| z_1 + z_2 \right| \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq \left| z_1 \right| + \left| z_2 \right| \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right|$$
Going back to the original inequality, we can write
$$\left| z_1 + z_2 \right| \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2|z_1 + z_2|$$

But now you've replaced the original inequality with an easier one. You made the left hand side smaller without changing the right hand side. Of course this will be easier to prove, but it's not equivalent to the original. To use an analogy, we are trying to prove that $b \leq 2$. You found an expression $a$ that satisfies $a \leq b$, and found that $a \leq 2$. But this does not imply $b \leq 2$.

 

[DrClaude]

You made the left hand side smaller without changing the right hand side. Of course this will be easier to prove, but it's not equivalent to the original.

Of course. I'm tired.

 

[Verdict]

Alright, so I'm almost there for the treasure. I'll write it out after I have dinner, but basically you show that the distance between the two rocks is 2 times as big as the distance between the middle of the two rocks and the treasure. Now I just need to show that the angle between the line between the two rocks and the middle of the rocks to the treasure is 90 degrees, and then I'm there!

 

[mfb]

There is an easier way to express the coordinates with multiplication, but this one works as well: If you shift the position of the tree by (x,y), how does the position of the treasure shift?

You can use this to simplify the problem a lot.

Spoiler

If the treasure position does not depend on the tree position, choose any tree position you like to find it.

 

[Verdict]

What I did was prove that the distance between the two rocks is 2 times as large as the distance from the middle point of the two rocks to the treasure. I then showed that the dot product of the direction vector of the line connecting the two rocks with the direction vector of the middle point of the two rocks to the treasure is 0, which means that they are orthogonal. So just measure the distance between the rocks, walk halfway, and turn 90 degrees to either left or right, move the measured distance and dig. If its not there, go to the other side instead, and there it is!

Thanks a lot for the help, your tips were very solid indeed. The inequality however.. That one still eludes me..

 

[jbunniii]

The inequality is pretty tough! I posted a request for assistance on the homework helper's forum, and micromass and BruceW were able to come up with two different solutions, both of which involving quite a bit of algebra and trig.

BruceW's is slightly more straightforward, so I'll show how his proof begins. We wish to prove the following:
$$(|z_1| + |z_2|) \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \leq 2|z_1 + z_2|$$
Both sides are nonnegative, so the inequality holds if and only if the square is true:
$$(|z_1| + |z_2|)^2 \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right|^2 \leq 4|z_1 + z_2|^2$$
Now express everything in polar coordinates: put $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$. When you plug this back into the squared inequality and simplify, you should get something that looks like this:
$$(r_1^2+r_2^2+2r_1r_2)(2+2cos(\theta_1-\theta_2)) \leq 4(r_1^2+r_2^2+2r_1r_2cos(\theta_1-\theta_2))$$
Now simplify both sides and eventually you should be able to reduce it to
$$cos(\theta_1-\theta_2) \leq 1$$
which is of course true. I don't know if there is a cleaner, more insightful way to do it, perhaps using some geometric observation. I would hope so, but I haven't found it.

 

[TSny]

Now that you've worked the treasure problem, you might enjoy reading this version of the problem on pages 35 -37 of the classic: 1 2 3 ∞

 

[Verdict]

Hmm, alright, let's see. I see what is done, and apart from the squaring I had tried that too. However, I don't understand the simplification that is made, to for example 2 + cos (theta 1 - theta 2) on the left hand side.

What I came to is
(r1²+r2²+2r1r2)(e^i$\theta1$+e^i$\theta2)$)² is smaller than or equal to 4(r1e^i$\theta1$+r2e^i$\theta2$)²

Writing out the squares is not the issue, but rewriting them to what you have is. Could you enlighten me? :)

Oh, and I can't thank you (all) enough for all the effort you have put into this!

@Tsny: It does seem like a very good read, from the description, and from the fact that Gamow wrote it! (Not that he is necessarily a good writer, but still)

 

[DrClaude]

I don't understand the simplification that is made, to for example 2 + cos (theta 1 - theta 2) on the left hand side.

Don't forget the absolute value. What you have is
$$\left| e^{i \theta_1} + e^{i \theta_2} \right|^2$$
Use Euler's formula, rewrite the products of cos and sin as cos of a sum of angles, then take the absolute value squared, and you'll find
$$2 + 2 \cos (\theta_1 - \theta_2)$$

 

[mfb]

What I did was prove that the distance between the two rocks is 2 times as large as the distance from the middle point of the two rocks to the treasure. I then showed that the dot product of the direction vector of the line connecting the two rocks with the direction vector of the middle point of the two rocks to the treasure is 0, which means that they are orthogonal. So just measure the distance between the rocks, walk halfway, and turn 90 degrees to either left or right, move the measured distance and dig. If its not there, go to the other side instead, and there it is!

There is no need to check both sides.

Here is my solution:
With the tree as origin and rock positions r and R, A=(1+i)*r and B=(1-i)*R.
Therefore, the treasure is at (A+B)/2 = r*(1+i)/2 + R(1-i)/2.
If you shift the tree by an amount -x, this is equivalent to shifting both rocks by x, and the treasure is located at (r+X)(1+i)/2 + (R+x)(1-i)/2 = (A+B)/2 + x. Therefore, the treasure is shifted by the same amount as the rocks, and the tree position does not matter for their relative orientation. Choose R as tree position, and the location of the treasure is easy to get (45° to the left of the direction to r, with 1/sqrt(2) its distance).

 

[Verdict]

Don't forget the absolute value. What you have is
$$\left| e^{i \theta_1} + e^{i \theta_2} \right|^2$$
Use Euler's formula, rewrite the products of cos and sin as cos of a sum of angles, then take the absolute value squared, and you'll find
$$2 + 2 \cos (\theta_1 - \theta_2)$$

Hmm, what I thought was that it was defined as ((z1+z2)²)^1/2, so that if you square that, the square root just disappears. Am I using the wrong definition? I'll try and work it out your way :)Edit: Working out the absolute value in terms of euler's formula, I get

-2 sin(θ₁) sin(θ₂)+2 cos(θ₁) cos(θ₂)+i (2 sin(θ₁) cos(θ₂)+2 cos(θ₁) sin(θ₂)+2 sin(θ₁) cos(θ₁)+2 sin(θ₂) cos(θ₂))-sin[sup2[/sup](θ₁)+cos²(θ₁)-sin²(θ₂)+cos²(θ₂)Simplifying that with the angle formula's, I get 2cos(theta1+theta2) -2, which is different from what you have, but also I am left with a whole bunch of imaginary stuff. What am I doing wrong?
@mfb: That is very clever, thanks a lot for all your help, I would not have figured it out on my own.

 

[jbunniii]

Now that you've worked the treasure problem, you might enjoy reading this version of the problem on pages 35 -37 of the classic: 1 2 3 ∞

Ha, I knew this problem sounded familiar, but I couldn't remember where I had seen it before. I was thinking it was in a Sherlock Holmes episode, but that was a different map puzzle.

 

[mfb]

Hmm, what I thought was that it was defined as ((z1+z2)²)^1/2, so that if you square that, the square root just disappears. Am I using the wrong definition? I'll try and work it out your way :)

This does not work with complex numbers. As an example, |z1+z2| and its square are always positive, while (z1+z2)^2 can be complex.

 

[Verdict]

This does not work with complex numbers. As an example, |z1+z2| and its square are always positive, while (z1+z2)^2 can be complex.

Hmm alright. So then is it.. (z1 + z2)(z1* + z2*)? As in, the star is the complex conjugate. If not, I am just not familiar with the formula I guess..