Suppose I want to find the maximum value of |a + 3+ 4i| where a is any complex number with |a| = 5. Now I assume b = 3+4i, resulting in the following inequality:
|a + b| ≤ |a| + |b|
which gives the maximum value as 10.
But I considered another way of looking at this. Suppose we assume some complex number p= 3 + 0i and q = 0 + 4i. Then we get another inequality:
|a + p + q| ≤ |a| + |p| + |q|
which when computed gives maximum value as 12.
Now as per my understanding, the correct answer is 10 because if we see consider this graphically, we get a and b as points on a circle of radius 5 units, whose maximum separation is equal to the diameter of the circle, that is, 10 units. But I am unable to clearly point out what is wrong in the second approach. I did not think there is any mistake in the second approach, yet it is tending towards an incorrect result. I would appreciate a well explained answer, clearly pointing out the errors.