Animated Solution for Mathematics - Complex Numbers: If the four complex numbers $z,\bar{z},z-2\text{Re}(z)$ and $\bar{z}-2\text{Re}(\bar{z})$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to :

Welcome, fellow traveler of the complex plane! Today, we are going to unravel a problem that might look like a dry algebraic exercise, but is actually a beautiful dance of symmetry and geometry. We are given four complex numbers that form the vertices of a square in the Argand plane.

Deconstructing the Vertices

First, let's define our complex number $z$ as $z=x+iy$. In the Argand plane, this is just a point $(x,y)$.

The second vertex is the conjugate, $\bar{z}=x-iy$, which is a reflection of $z$ across the real axis.

The third vertex is $z-2\text{Re}(z)$. Since $\text{Re}(z)=x$, this becomes:

This is a reflection of $z$ across the imaginary axis. Finally, the fourth vertex is $\bar{z}-2\text{Re}(\bar{z})=(x-iy)-2x=-x-iy$.

If you plot these, you get the coordinates $(x,y)$, $(x,-y)$, $(-x,y)$, and $(-x,-y)$.

Visualizing the Square

Look at these four points. They form a rectangle centered at the origin.

The horizontal distance between $(x,y)$ and $(-x,y)$ is $|x-(-x)|=2|x|$. The vertical distance between $(x,y)$ and $(x,-y)$ is $|y-(-y)|=2|y|$.

The problem states this is a square of side 4. This means both the horizontal and vertical sides must be 4 units long:

This simplifies to $|x|=2$ and $|y|=2$.

The Final Calculation

Now, we need to find $|z|$. The modulus of $z$ is the distance from the origin to the point $(x,y)$, given by:

Substituting our values, we get:

Simplifying this, we find the final result:

Isn't it elegant? By translating the algebraic operations into geometric reflections, we turned a potentially confusing problem into a simple visualization.

The square was always there, symmetric and centered at the origin. Keep this geometric intuition in your toolkit, and you will find that even the most complex problems become clear.