Is |1−i||1-i| larger than |1||1|?

I am confused about complex numbers. Does 1i lie outside the unit circle? How do I show that the absolute value of 1i is larger than that of 1?

Answer

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

Note that 1i 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;1i.

The diagonal running from 1+i to 1i is a straight line passing through 0. Its length, by the Pythagoras theorem, is 22=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:

complex plane

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

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