A translation of the plane takes $5 + 5i$ to $6 + 8i.$  Find the complex number that the translation takes $-6$ to.
To solve this problem, I need to find the rule that the translation follows. One way to think of a translation is as adding a constant complex number to every point in the plane. So, if the translation takes $5 + 5i$ to $6 + 8i,$ then it must have added $1 + 3i$ to it. This means that the rule of the translation is to add $1 + 3i$ to every complex number. Therefore, to find the complex number that the translation takes $-6$ to, I just need to add $1 + 3i$ to $-6.$ I can do this by adding the real and imaginary parts separately: $-6 + 1 + (-0 + 3)i = -5 + 3i.$ 