Animated Solution for Mathematics - Complex Numbers: If $|z^2-1|=|z{|}^2+1$, then $z$ lies on

Welcome, future engineers! Today, we are going to embark on a journey through the Argand plane. We are looking at the equation $|z^2-1|=|z{|}^2+1$.

At first glance, this might look like a daunting algebraic mess, but I want you to see it as a beautiful dance of symmetry. When you encounter modulus signs in complex numbers, do not panic; they are simply telling you about distances. Our goal is to translate this geometric constraint into a clear, solvable algebraic path.

Squaring to Simplify

Our first instinct, and our most powerful one, is to eliminate those modulus bars by squaring both sides. This transforms the equation into:

Now, we invoke the most important identity in complex number theory: $|w{|}^2=w\bar{w}$. This identity is the key that unlocks the door, allowing us to replace the modulus of a complex number with the product of that number and its conjugate.

Applying this to the left-hand side, where our $w$ is $z^2-1$, the expression becomes:

Now, let us expand both sides. On the left, we have $(z^2-1)({\bar{z}}^2-1)$, which expands to:

On the right, we have the square of a binomial:

Look closely at the first term on the left: $z^2{\bar{z}}^2$. Since $z\bar{z}=|z{|}^2$, we know that $z^2{\bar{z}}^2=(z\bar{z}{)}^2=|z{|}^4$.

This is the "Aha!" moment. When we equate the two sides, the $|z{|}^4$ terms and the constant $1$ cancel out perfectly. We are left with a beautifully simple equation:

To see the geometry, let us switch to Cartesian coordinates. Let $z=x+iy$, which implies $\bar{z}=x-iy$.

When we calculate $z^2+{\bar{z}}^2$, the imaginary parts cancel out, leaving us with $2(x^2-y^2)$. We also know that $|z{|}^2=x^2+y^2$.

Substituting these into our equation, we get:

Dividing by $2$, we obtain $-(x^2-y^2)=x^2+y^2$, which simplifies to $-x^2+y^2=x^2+y^2$. The $y^2$ terms vanish, leaving us with $-x^2=x^2$, or $2x^2=0$. This forces $x=0$.

The Imaginary Axis

Since $x=0$, the real part of our complex number $z$ is zero. In the Argand plane, any complex number with a real part of zero lies on the vertical axis.

We have proven that $z$ must lie on the imaginary axis. You have successfully navigated the algebra to reveal the underlying geometry. Keep practicing, keep visualizing, and keep falling in love with the elegance of mathematics!