The number of complex numbers $$z$$ such that $$|z - 1| = |z + 1| = |z - i|$$ equals

1. Understand the Geometric Definition

The equation represents distances to three distinct points.

The points are $$(1, 0)$$, $$(-1, 0)$$, and $$(0, 1)$$.

These three points form a unique triangle.

The solution point is the circumcentre.

Any triangle has exactly one circumcentre.

2. Set Up the Algebraic Equations

Let $$z = x + iy$$.

Here, $$x$$ and $$y$$ are real numbers.

3. Solve the First Equality

Consider the first equation part:

$$|z - 1| = |z + 1|$$

Substitute $$z = x + iy$$ into the modules:

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

Square both sides to remove the radicals:

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

Expand and cancel out the shared terms:

$$x^2 - 2x + 1 = x^2 + 2x + 1$$

$$-2x = 2x$$

$$4x = 0 \implies x = 0$$

4. Solve the Second Equality

Substitute $$x = 0$$ into the remaining relationship:

$$|z + 1| = |z - i|$$

$$|1 + iy| = |iy - i|$$

Write out the algebraic modulus forms:

$$\sqrt{1^2 + y^2} = \sqrt{0^2 + (y - 1)^2}$$

Square both sides to clear the roots:

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

$$1 + y^2 = y^2 - 2y + 1$$

Cancel out the shared algebraic terms:

$$0 = -2y \implies y = 0$$

5. Find the Final Complex Number

Combine the evaluated coordinates together:

$$z = 0 + 0i = 0$$

There is only one unique solution.

Final Answer

The total number of such complex numbers is $$1$$.