### Instruction: Gracie and Joe are choosing numbers on the complex plane. Joe chooses the point $1+2i$. Gracie chooses $-1+i$. How far apart are Gracie and Joe's points?### Response: To find the distance between two points on the complex plane, I need to use the distance formula, which is based on the Pythagorean theorem.
The distance formula is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points.
In this problem, the coordinates are given in terms of real and imaginary parts, so I can rewrite them as $(1,2)$ and $(-1,1)$.
Plugging these values into the distance formula, I get $d=\sqrt{(-1-1)^2+(1-2)^2}$.
Simplifying the expression inside the square root, I get $d=\sqrt{(-2)^2+(-1)^2}$.
Squaring the terms, I get $d=\sqrt{4+1}$.
Adding the terms, I get $d=\sqrt{5}$.
This is the exact answer, but I can also approximate it using a calculator if needed.