Animated Solution for Mathematics - Complex Numbers: If $z$ is a complex number such that $|z|\ge 2$, then the minimum value of $|z+1/2|$

Imagine you are standing in the vast, open expanse of the complex plane. You are restricted to points $z$ such that your distance from the origin is at least two units, defined by the condition $|z|\ge 2$.

Picture a circle of radius $2$ centered at the origin. You are forbidden from entering the interior of this circle and are free to roam anywhere on the boundary or in the infinite space outside it.

We aim to minimize the expression $|z+\frac{1}{2}|$. In the language of complex numbers, the modulus of a difference $|z_1-z_2|$ represents the distance between two points. Thus, $|z+\frac{1}{2}|$ is the distance between your current position $z$ and the fixed point $P(-\frac{1}{2},0)$.

Our goal is to find the point $z$ in the allowed region that is closest to the fixed point $P$. The point $P$ lies at $-\frac{1}{2}$ on the real axis, which is clearly inside the forbidden zone.

To reach the allowed region as quickly as possible from $P$, you must walk in a straight line directly away from the origin until you hit the boundary of the circle. This path lies entirely on the real axis.

The point on the boundary that you hit is $Q(-2,0)$. The distance between $P(-\frac{1}{2},0)$ and $Q(-2,0)$ is the minimum distance we seek.

Calculating this distance:

We can confirm this result using the Triangle Inequality, a cornerstone of complex analysis. The theorem states:

By setting $z_1=z$ and $z_2=\frac{1}{2}$, we obtain:

Given that $|z|\ge 2$, the smallest value $|z|$ can take is $2$. Substituting this into our inequality yields:

The algebra perfectly mirrors our geometric intuition. The minimum value is $\frac{3}{2}$ (or $1.5$), which sits comfortably in the interval $(1,2)$.