In the magic world of complex numbers, the proof of the Triangle Inequality is somewhat more elegant, and its geometrical interpretation is more straightforward. Note that for a complex number z=x+iy, we denote its distance from the origin by the handy formula sqrt(x2+y2),
and denote this value by |z|.
Theorem: For any w and z in C, |z+w| ≤ |z| + |w|.
Helpful ASCII diagram:
w | w+z (in accordance with the Parallelogram Law)
\ | /\
\ | / \ <-- vector w emanating from z instead of origin
So, the idea is just that any leg of a triangle cannot be
longer than the sum of the other sides.
: Letting z
= x+iy and w
= s+it, we have
|z+w|2 = (x+s)2+(y+t)2
= (|z| + |w|)2.
We finish off the proof by taking the square root of both sides.