### Instruction: In the complex plane, what is the length of the diagonal of the square with vertices $4$, $3+5i$, $-2+4i$, and $-1-i$?### Response: To find the length of the diagonal, I need to use the distance formula for complex numbers, which is the modulus of the difference of two numbers.
So, if I call the diagonal $d$, I can write $d = |z_1 - z_2|$, where $z_1$ and $z_2$ are opposite vertices of the square.
It doesn't matter which pair of vertices I choose, since the square is symmetric.
I'll pick $z_1 = 4$ and $z_2 = -2+4i$ for convenience.
Plugging these values into the formula, I get $d = |4 - (-2+4i)| = |6-4i|$.
To simplify this expression, I can use the fact that $|a+bi| = \sqrt{a^2+b^2}$ for any complex numbers $a$ and $b$.
So, $d = |6-4i| = \sqrt{6^2+(-4)^2} = \sqrt{36+16} = \sqrt{52}$.
This is the exact value of the length of the diagonal, but I can also approximate it using a calculator if needed.
Using a calculator, I get $d \approx 7.211$ to three decimal places.