Animated Solution for Mathematics - Complex Numbers: $z\ne 0$ is a complex number

A complex number is defined as $z=x+iy$, where $x$ is the real part and $y$ is the imaginary part. Since the problem states $z eq 0$, we are working with a point located anywhere in the complex plane except at the origin.

To investigate the behavior of $z^2$, we expand the expression:

Using the algebraic identity $(a+b{)}^2=a^2+2ab+b^2$, we obtain:

The fundamental property of complex numbers is $i^2=-1$. Applying this to our expansion, we derive the master equation:

This equation explicitly defines the real and imaginary components of $z^2$ in terms of the original coordinates $x$ and $y$.

The Condition $Re(z)=0$

Geometrically, the condition $Re(z)=0$ implies that the point $z$ lies entirely on the vertical imaginary axis, which forces $x=0$. Substituting $x=0$ into our master equation:

Because the resulting expression is purely real, the imaginary part vanishes. Therefore, we conclude that $Im(z^2)=0$.

The Condition $Arg(z)=\frac{\pi }{4}$

An argument of $\frac{\pi }{4}$ corresponds to a $45^{\circ }$ angle, placing the point on the line $y=x$ in the first quadrant. Given $\tan (\theta )=\frac{y}{x}$, the condition $\tan (\frac{\pi }{4})=1$ yields the relation $y=x$.

Substituting $y=x$ into our master equation:

Since the real part of this result is zero, we conclude that $Re(z^2)=0$.

Squaring a complex number is equivalent to doubling its argument. A $45^{\circ }$ angle doubles to $90^{\circ }$, landing the result on the imaginary axis, while a $90^{\circ }$ angle doubles to $180^{\circ }$, landing the result on the negative real axis.

Mastering this rotation of the complex plane is the key to solving complex number problems with speed and precision.