If z is a complex number, then the minimum value of |z| + |z − 1| is ______.
Best answer
First, note that |-z|=|z| and |z1 + z2| ≤ |z1| + |z2|
Now |z| + |z-1| = |z| + |1 - z| ≤ |z + (1 - z)|
= |1|
= 1
Hence, minimum value of |z| + |z - 1| is 1.
Answer: 1
Use \(|z - 1| = |1 - z|\) and the triangle inequality \(|z_1| + |z_2| \ge |z_1 + z_2|\):
\[|z| + |z - 1| = |z| + |1 - z| \ge |z + (1 - z)| = 1\]
So the sum is never less than 1.
The value 1 is actually reached: for example \(z = \frac{1}{2}\) gives \(\frac{1}{2} + \frac{1}{2} = 1\). In fact equality holds for every real \(z\) with \(0 \le z \le 1\).
Geometrically, \(|z| + |z-1|\) is the sum of distances of the point \(z\) from 0 and from 1; it is smallest (equal to the distance 1 between them) when \(z\) lies on the segment joining 0 and 1.
Hence the minimum value is 1.
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