Modulus inequalities of complex numbers

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.

• Hi and welcome to Math.SE. It would be preferable to use MathJax for mathematical expressions. You can get started here, and a more complete reference can be found here. Mar 29, 2021 at 8:14

Let $$c$$ and $$d$$ be the real and imaginary parts of $$a$$. We want to maximize the square root of $$(c+3)^{2}+(d+4)^{2}$$. Expand the squares. Since $$c^{2}+d^{2}=25$$ it is enough to maximixe $$6c+8d$$. By Cauchy-Schwarz inequality the maximum value is attained when $$c=6t$$ and $$d=8t$$ for some $$t$$ and $$t$$ is determined by the condition $$c^{2}+d^{2}=25$$. I will let you finish.
$$|a+b+c|\le|a|+|b|+|c|$$ is proved by induction with the base case of $$|a+b|\le|a|+|b|$$, so we have $$|a+b+c|\le|a|+|b+c|\le|a|+|b|+|c|$$.
The inequality is generic, so in your case with $$|p+q|\le|p|+|q|$$, equality is when the arguments of $$\vec p$$ and $$\vec q$$ are the same.
You have used a specific $$\vec p$$ and $$\vec q$$, and thus equality is not reached, and there is an over-estimation.