TITLE: Is $|1-i|$ larger than $|1|$?
QUESTION [49 upvotes]: I am confused about complex numbers. Does $1-i$ lie outside the unit circle? How do I show that the absolute value of $1-i$ is larger than that of $1$?

REPLY [2 votes]: This is a small variant of Asaf Karagila's geometric proof:  If $a$ and $b$ are two points in the complex plane, the distance between them is $|a-b|$, so if you draw the (right isosceles) triangle with $a=1$, $b=i$ and $c=0$, you see that $|1-i|\gt1$.