For two given complex numbers a and b, is it true that |a-b| = |a|-|b| ?

For two given complex numbers a and b, is it true that |a-b| = |a|-|b| ?

Student X

Generally, it isn't true. However, in specific instances, |a-b|  can be equal to  |a|-|b|.  This happens when arg(a)= arg(b). Bearing in mind |a-b| always represents geometrically the physical distance between the two complex numbers a and b, here is a visual representation for this unique case:

Another instance worth noting is when arg(b)= arg(a) - π, then |a-b| actually becomes |a|+|b|:

Hope the above clarifies. Peace.

Best Regards,

Mr Koh