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?

**Answer**

Let me do something *very* different for me. Let me give a geometric proof:

Note that 1−i is the bottom-right corner of the square whose center is 0 and whose edges have length 2. The other corners are 1+i;−1+i;−1−i.

The diagonal running from −1+i to 1−i is a straight line passing through 0. Its length, by the Pythagoras theorem, is 2√2=√8. Therefore the distance between 0 and each of the corners is exactly half, i.e. √2.

And it is trivial to see that √2>1.

Here is a drawing:

**Attribution***Source : Link , Question Author : phil12 , Answer Author : aduh*