<Question>
Let $z = 2 + \sqrt{2} - (3 + 3 \sqrt{2})i$, and let $c = 2 - 3i$.  Let $w$ be the result when $z$ is rotated around $c$ by $\frac{\pi}{4}$ counter-clockwise.

[asy]
unitsize(0.6 cm);

pair C, W, Z;

Z = (2 + sqrt(2), -3 - 3*sqrt(2));
C = (2,-3);
W = rotate(45,C)*(Z);

draw(Z--C--W);

dot("$c$", C, N);
dot("$w$", W, SE);
dot("$z$", Z, S);
label("$\frac{\pi}{4}$", C + (0.6,-1));
[/asy]

Find $w.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the complex number w, which is the result of rotating z around the point c by π/4 radians counter-clockwise. Let me start by recalling how rotations work in the complex plane, especially around a point that isn't the origin.

<prompt_1.1> Review the concept of rotating a complex number around a point that isn't the origin. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, I remember that to rotate a point z around another point c by an angle θ, you can translate the system so that c becomes the origin, then apply the rotation, and then translate back. The formula for this is: 

w = (z - c) * e^(iθ) + c

Here, θ is the angle of rotation. Since we're rotating counter-clockwise, θ is positive. In this case, θ is π/4. So, the steps are:
1. Subtract c from z to translate the system so that c is at the origin.
2. Multiply the result by e^(iπ/4) to rotate it by π/4 radians.
3. Add c back to translate the system back to the original position.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me write down the given values to make sure I have them right. 

The complex number z is given as 2 + √2 - (3 + 3√2)i.

<prompt_2.1> Verify the given values of z and c. </prompt_2.1>

So, first, compute z - c.

<prompt_2.2> Calculate z - c using the formula w = (z - c) * e^(iπ/4) + c. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Wait, let me parse that correctly. In the problem statement, z is written as 2 + √2 - (3 + 3√2)i. So, the real part is 2 + √2, and the imaginary part is -(3 + 3√2). So, in complex number terms, z = (2 + √2) - (3 + 3√2)i. 

And c is given as 2 - 3i. So, c is a complex number with real part 2 and imaginary part -3.
</execution_2.1>

<execution_2.2>
Let me parse z and c. 

Wait, z is given as 2 + √2 - (3 + 3√2)i. So, in terms of real and imaginary parts, z is (2 + √2) + (-3 - 3√2)i. And c is 2 - 3i. So, subtracting c from z:

z - c = [(2 + √2) - 2] + [(-3 - 3√2) - (-3)]i

Simplify the real part: (2 + √2 - 2) = √2

Simplify the imaginary part: (-3 - 3√2 + 3) = (-3√2)

So, z - c = √2 - 3√2 i

Alternatively, written as √2(1 - 3i)
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.2:
Now, we need to multiply this by e^(iπ/4).

<prompt_3.1> Multiply z - c by e^(iπ/4) and then add c to find w. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Recall that e^(iθ) = cosθ + i sinθ. So, e^(iπ/4) is cos(π/4) + i sin(π/4) = √2/2 + i√2/2. 

So, multiplying (z - c) by e^(iπ/4):

(√2 - 3√2 i) * (√2/2 + i√2/2)

Let me factor out √2 from the first term:

√2(1 - 3i) * √2/2 (1 + i)

Multiply the constants first: √2 * √2/2 = (2)/2 = 1

So, the multiplication simplifies to:

(1 - 3i)(1 + i)

Now, let's multiply this out:

(1)(1) + (1)(i) - 3i(1) - 3i(i) = 1 + i - 3i - 3i²

Combine like terms:

1 - 2i - 3(-1) [since i² = -1] = 1 - 2i + 3 = 4 - 2i

So, after multiplying by e^(iπ/4), we get 4 - 2i. 

Then, we need to add c back to this result. Recall that c is 2 - 3i. So:

w = (4 - 2i) + (2 - 3i) = 4 + 2 + (-2i - 3i) = 6 - 5i
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, that seems straightforward, but let me verify each step to make sure I didn't make a mistake.

<prompt_4.1> Verify the calculations for z - c, the multiplication by e^(iπ/4), and the addition of c. </prompt_4.1>

But let me think again. Is this correct? Let me visualize the points.

<prompt_4.2> Visualize the points and the rotation to verify the result. </prompt_4.2>

But wait, let me check if the multiplication was done correctly. Let's redo (1 - 3i)(1 + i):

<prompt_4.3> Redo the multiplication of (1 - 3i)(1 + i) to verify the result. </prompt_4.3>

So adding c: 4 - 2i + 2 - 3i = 6 -5i. That seems correct.

But let me verify with another approach to make sure.

<prompt_4.4> Verify the result using an alternative approach. </prompt_4.4>

But let me check if the formula is applied correctly. The formula is w = (z - c) * e^(iθ) + c. Let's confirm each step:

<prompt_4.5> Reconfirm the formula and calculations using complex number properties. </prompt_4.5>

Alternatively, maybe I can represent z, c, and w as points in the plane and use coordinate geometry to verify.

Let me try that.

<prompt_4.6> Verify the result using coordinate geometry and trigonometric calculations. </prompt_4.6>

But let me check once more for calculation errors.

<prompt_4.7> Perform a final verification of calculations and conclude the result. </prompt_4.7>
</Plan_4>

<Execution_4>
<execution_4.1>
First, z - c:

z = (2 + √2) + (-3 - 3√2)i

c = 2 - 3i

Subtracting c from z:

Real part: (2 + √2) - 2 = √2

Imaginary part: (-3 - 3√2)i - (-3)i = (-3 - 3√2 + 3)i = (-3√2)i

So, z - c = √2 - 3√2 i. Correct.

Then, multiplying by e^(iπ/4) which is √2/2 + i√2/2. So:

(√2 - 3√2 i)(√2/2 + i√2/2)

Factor out √2 from the first term and √2/2 from the second term:

√2*(1 - 3i) * √2/2*(1 + i) = (√2 * √2/2)*(1 - 3i)(1 + i)

√2 * √2/2 = (2)/2 = 1, so that's 1. Then (1 - 3i)(1 + i):

Multiply out:

1*1 = 1

1*i = i

-3i*1 = -3i

-3i*i = -3i² = 3 (since i² = -1)

So, adding up: 1 + i - 3i + 3 = (1 + 3) + (i - 3i) = 4 - 2i. Correct.

Then, adding c: 4 - 2i + 2 - 3i = 6 - 5i. So, w = 6 - 5i.
</execution_4.1>

<execution_4.2>
The point z is (2 + √2, -3 - 3√2). Wait, no. Wait, in the Asymptote code, they plot Z as (2 + sqrt(2), -3 - 3*sqrt(2)), but according to the problem statement, z is 2 + √2 - (3 + 3√2)i. So, the real part is 2 + √2, and the imaginary part is -(3 + 3√2). So, when plotted in the complex plane, z is at (2 + √2, -3 - 3√2). Then c is 2 - 3i, which is (2, -3). So, the point c is (2, -3). 

So, the vector from c to z is z - c, which would be (2 + √2 - 2, -3 - 3√2 - (-3)) = (√2, -3√2). So, that's the vector we're rotating by π/4 radians counter-clockwise. 

So, the original vector is (√2, -3√2). Let me represent this as a vector in R²: [√2, -3√2]. Rotating this vector by π/4. The rotation matrix is:

[cosθ  -sinθ]
[sinθ   cosθ]

So, plugging θ = π/4:

[√2/2  -√2/2]
[√2/2   √2/2]

Multiply this matrix by the vector [√2, -3√2]:

First component:

√2/2 * √2 + (-√2/2) * (-3√2) 

= (2/2) + (3*2/2) 

= 1 + 3 = 4

Second component:

√2/2 * √2 + √2/2 * (-3√2)

= (2/2) + (-3*2/2)

= 1 - 3 = -2

So, the rotated vector is [4, -2]. Then, adding back the point c, which is (2, -3), gives the new point (2 + 4, -3 + (-2)) = (6, -5). Which corresponds to the complex number 6 - 5i.
</execution_4.2>

<execution_4.3>
First term: 1*1 = 1

Outer term: 1*i = i

Inner term: -3i*1 = -3i

Last term: -3i*i = -3i² = -3*(-1) = 3

So, adding up the real parts: 1 + 3 = 4

Imaginary parts: i - 3i = -2i

So, yes, 4 - 2i.
</execution_4.3>

<execution_4.4>
Alternatively, perhaps I made a mistake in translating the system. Let me re-examine the steps.

Given z = (2 + √2) - (3 + 3√2)i. 

c = 2 - 3i

So, z - c = (2 + √2 - 2) + (-3 - 3√2 + 3)i = √2 + (-3√2)i. Yes, that's correct. So, z - c = √2 - 3√2 i. 

Then, multiplying by e^(iπ/4) which is √2/2 + i√2/2. So:

(√2 - 3√2 i)(√2/2 + i√2/2)

Let me compute this multiplication step by step:

First, multiply √2 by √2/2: (√2)(√2)/2 = 2/2 = 1

Then, √2 multiplied by i√2/2: √2 * i√2 /2 = i*(2)/2 = i

Next, -3√2 i multiplied by √2/2: -3√2 * √2 /2 * i = -3*2/2 * i = -3i

Finally, -3√2 i multiplied by i√2/2: -3√2 * i * √2 /2 * i = -3*(2)/2 * i^2 = -3*(-1) = 3

So, adding all these parts together: 1 + i -3i + 3 = (1 + 3) + (i - 3i) = 4 - 2i. 

Yes, that's the same result as before. Then adding c: 4 - 2i + 2 - 3i = 6 -5i.
</execution_4.4>

<execution_4.5>
1. z - c: z is (2 + √2) - (3 + 3√2)i, c is 2 -3i. Subtracting c from z:

Real part: (2 + √2) - 2 = √2

Imaginary part: [-(3 + 3√2)] - (-3) = -3 -3√2 +3 = -3√2

So, z - c is √2 -3√2 i, which is correct.

2. e^(iπ/4) is indeed √2/2 + i√2/2. Multiplying that by (z - c):

(√2 -3√2 i)(√2/2 + i√2/2). As before, factor √2:

√2*(1 -3i) * √2/2*(1 + i) = (√2*√2/2)*(1 -3i)(1 +i) = (2/2)*(1 -3i)(1 +i) = 1*(1 -3i)(1 +i). Then expanding gives 1 +i -3i -3i² = 1 -2i +3 = 4 -2i. Then adding c: 4 -2i +2 -3i =6 -5i. 

Wait, but let me check if the angle is indeed π/4. The problem says rotated by π/4 counter-clockwise, so yes, e^(iπ/4) is correct. 

But let me verify the multiplication step again. Let's compute (√2 -3√2 i)(√2/2 + i√2/2):

First, expand the product:

√2 * √2/2 + √2 * i√2/2 -3√2 i * √2/2 -3√2 i * i√2/2

Compute each term:

1. √2 * √2/2 = (2)/2 = 1

2. √2 * i√2/2 = i*(2)/2 = i

3. -3√2 i * √2/2 = -3i*(2)/2 = -3i

4. -3√2 i * i√2/2 = -3i²*(2)/2 = -3*(-1)*1 = 3

So, adding all terms: 1 + i -3i +3 = (1 +3) + (i -3i) = 4 -2i. Correct.

Then adding c: 4 -2i +2 -3i =6 -5i.
</execution_4.5>

<execution_4.6>
First, plot the points. 

Point c is at (2, -3). Point z is at (2 + √2, -3 - 3√2). So, relative to c, z is at (√2, -3√2). So, the vector from c to z is (√2, -3√2). Then, we need to rotate this vector by π/4 radians counter-clockwise and then add it back to c. 

Alternatively, the rotation can be done using the formula for rotating a vector (x, y) by an angle θ:

The new vector (x', y') is given by:

x' = x cosθ - y sinθ

y' = x sinθ + y cosθ

In this case, θ = π/4, so cosθ = sinθ = √2/2. 

So, applying this to the vector (√2, -3√2):

x' = √2 * √2/2 - (-3√2) * √2/2

y' = √2 * √2/2 + (-3√2) * √2/2

Let's compute x':

√2 * √2/2 = (2)/2 = 1

-(-3√2) * √2/2 = 3√2 * √2/2 = 3*(2)/2 = 3

So, x' = 1 + 3 = 4

Now, y':

√2 * √2/2 = 1

(-3√2) * √2/2 = -3*(2)/2 = -3

So, y' = 1 - 3 = -2

Therefore, the rotated vector is (4, -2). Then, adding this back to point c (2, -3):

w = (2 + 4, -3 + (-2)) = (6, -5)

Which corresponds to the complex number 6 - 5i. 

So, this confirms the earlier result. Therefore, w = 6 - 5i.
</execution_4.6>

<execution_4.7>
Original z: (2 + √2, -3 - 3√2)

c: (2, -3)

z - c: (2 + √2 - 2, -3 -3√2 - (-3)) = (√2, -3√2)

So, as a complex number, that's √2 - 3√2 i. 

Multiplying by e^(iπ/4) which is (√2/2 + √2/2 i). So, in complex form, (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

Let me apply this formula:

Let me denote:

First complex number: √2 - 3√2 i → a = √2, b = -3√2

Second complex number: √2/2 + √2/2 i → c = √2/2, d = √2/2

Compute real part: ac - bd = (√2)(√2/2) - (-3√2)(√2/2)

= (2/2) - (-3*2/2)

= 1 - (-3)

= 1 + 3 = 4

Compute imaginary part: ad + bc = (√2)(√2/2) + (-3√2)(√2/2)

= (2/2) + (-3*2/2)

= 1 + (-3)

= -2

Therefore, the result is 4 - 2i, which matches our previous result. Then adding c: 4 -2i + 2 -3i = 6 -5i.

So, the final answer is 6 -5i, which in the box is \boxed{6-5i}.

Wait, but let me just confirm once more. Because sometimes when dealing with rotations, especially when translating, it's easy to make a sign error or something. Let me check the translation steps again.

Original z: (2 + √2) - (3 + 3√2)i

c: 2 -3i

So, z - c is (2 + √2 - 2) + [ - (3 + 3√2) - (-3) ]i

Simplify real part: √2

Imaginary part: -3 -3√2 +3 = -3√2

So, z - c = √2 -3√2 i. Correct.

Multiply by e^(iπ/4):

√2 -3√2 i multiplied by √2/2 + √2/2 i.

As we did earlier, which gives 4 -2i. Then adding c: 4 -2i +2 -3i =6 -5i. Correct.

Therefore, the answer is 6 -5i.
</execution_4.7>
</Execution_4>

<Plan_5>
Based on execution_4.7:
**Final Answer**
\boxed{6-5i}
</Plan_5>

<Execution_5>
No prompt to execute.
</Execution_5>

<Final_answer>
\[w = \boxed{6-5i}.
\]
</Final_answer>
</think>