Animated Solution for Mathematics - Complex Numbers: If $z$ be a complex number satisfying $|\text{Re}(z)|+|\text{Im}(z)|=4$, then $|z|$ cannot be:

We are given the condition $|\text{Re}(z)|+|\text{Im}(z)|=4$. By representing the complex number $z$ as $x+iy$, the condition transforms into the Cartesian equation:

This equation describes a geometric shape in the Argand plane. Rather than a circle, this locus represents a square.

By analyzing the equation across the four quadrants, we identify the boundary lines: First quadrant ($x,y\ge 0$): $x+y=4$ Second quadrant ($x<0,y\ge 0$): $-x+y=4$ Third quadrant ($x,y<0$): $-x-y=4$ Fourth quadrant ($x>0,y<0$): $x-y=4$

The resulting figure is a square with vertices at $(4,0)$, $(0,4)$, $(-4,0)$, and $(0,-4)$.

The term $|z|$ represents the Euclidean distance from the origin $(0,0)$ to the point $z$ on the perimeter of the square. As $z$ traverses the boundary of the square, this distance varies continuously.

Our objective is to determine the range of this distance, specifically identifying the minimum and maximum values of $|z|$.

The maximum distance occurs at the points furthest from the origin, which are the vertices of the square. Calculating the distance for any vertex, such as $(4,0)$:

The minimum distance occurs at the points closest to the origin. Geometrically, this is the perpendicular distance from the origin to any of the four sides of the square.

Using the perpendicular distance formula $d=\frac{|A{x}_0+B{y}_0+C|}{\sqrt{A^2+B^2}}$ for the line $x+y-4=0$:

We have established that the modulus $|z|$ is constrained to the interval $[\sqrt{8},4]$.

Any value of $|z|$ must satisfy the inequality:

Consequently, values such as $\sqrt{10}$ are valid, as they fall within this range. However, a value like $\sqrt{7}$ is strictly less than $\sqrt{8}$ and therefore lies in the "forbidden zone" inside the square.