Question

Find all complex numbers $z$ satisfying the equation
Z / z−1 =i.

Answer 1

To solve the equation Z / (z-1) = i, we can start by multiplying both sides of the equation by (z-1). Then, we can distribute the i to both terms on the right side. Finally, we can solve for Z by dividing both sides of the equation by (1 - i) and simplify the expression.

Step-by-step explanation:

To solve the equation Z / (z-1) = i, we can start by multiplying both sides of the equation by (z-1). This will eliminate the denominator on the left side. The equation becomes:

Z = i(z-1)

Next, we can distribute the i to both terms on the right side:

Z = iz - i

After that, we can subtract iz from both sides:

Z - iz = -i

Then, we can factor out Z on the left side:

Z(1 - i) = -i

Finally, we can solve for Z by dividing both sides of the equation by (1 - i):

Z = -i / (1 - i)

To simplify this further, we can multiply the numerator and denominator by the conjugate of the denominator, which is (1 + i):

Z = (-i)(1 + i) / (1 - i)(1 + i)

Using the distributive property, we get:

Z = (-i -i²) / (1 - i + i - i²)

Since i² is equal to -1, we can simplify further:

Z = (-i + 1) / (1 + 1)

Z = (-i + 1) / 2

So, the complex numbers that satisfy the equation are Z = (-i + 1) / 2.