It seems you're asking about the expression $|z - i| = \sqrt{2}$. This represents a complex number problem involving distances in the complex plane.

Understanding the Problem:

The expression $|z - i| = \sqrt{2}$ can be interpreted geometrically as follows:

• $z$ is a complex number that we can represent as $z = x + iy$, where $x$ and $y$ are real numbers (i.e., $z = x + iy \in \mathbb{C}$).
• $|z - i|$ represents the distance between the complex number $z$ and $i$ in the complex plane. The number $i$ corresponds to the point $(0, 1)$ in the complex plane.

Geometrical Interpretation:

The equation $|z - i| = \sqrt{2}$ describes a circle in the complex plane with a center at $i$ (which is $(0,1)$) and a radius of $\sqrt{2}$.

Detailed Solution:

1. Let $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part.

2. The distance between $z$ and $i$ is given by:

   $|z - i| = |(x + iy) - i| = |x + i(y - 1)|$

3. By definition of modulus of a complex number, we have:

   $|x + i(y - 1)| = \sqrt{x^2 + (y - 1)^2}$

4. The equation $|z - i| = \sqrt{2}$ becomes:

   $\sqrt{x^2 + (y - 1)^2} = \sqrt{2}$

5. Squaring both sides:

   $x^2 + (y - 1)^2 = 2$

This is the equation of a circle with center $(0,1)$ and radius $\sqrt{2}$ in the complex plane.

Summary:

The equation $|z - i| = \sqrt{2}$ represents a circle in the complex plane, centered at $(0,1)$ with a radius of $\sqrt{2}$.

Would you like further details or have any other questions?

Related Questions:

1. What is the geometric interpretation of a complex modulus in general?
2. How do you solve for specific points $z$ that satisfy $|z - i| = \sqrt{2}$?
3. What other conic sections can be represented using complex numbers?
4. How can this problem be extended to three-dimensional space using quaternions?
5. What is the relationship between the modulus and the argument of a complex number?

Tip: Visualizing complex numbers on the complex plane can help you understand relationships like distances and rotations more clearly.