### Instruction: Matt and Marie are choosing numbers on the complex plane. Matt chooses the point $1-4i$. Marie chooses $-4+2i$. How far apart are Marie and Matt's points?### Response: To find the distance between two points on the complex plane, I can use the distance formula, which is the same as the Pythagorean theorem for right triangles.
The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, where $x_1$ and $y_1$ are the real and imaginary parts of the first point, and $x_2$ and $y_2$ are the real and imaginary parts of the second point.
In this case, the first point is $1-4i$, so $x_1 = 1$ and $y_1 = -4$.
The second point is $-4+2i$, so $x_2 = -4$ and $y_2 = 2$.
Plugging these values into the distance formula, I get $d = \sqrt{(-4 - 1)^2 + (2 - (-4))^2} = \sqrt{(-5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61}$.
The distance is the square root of 61, which is an irrational number.
I can use a calculator to get a decimal approximation, or leave it as a radical.