Find all complex numbers z that satisfy the equation Z*=2/z+i

We need to solve the equation:  
\[
\bar{z} = \frac{2}{z} + i
\]
where \(\bar{z}\) represents the complex conjugate of \(z\).

---

### Step 1: Represent the complex number \(z\)
Let \(z = x + yi\), where \(x, y \in \mathbb{R}\).  
Its complex conjugate is:
\[
\bar{z} = x - yi.
\]
We substitute these into the equation.

---

### Step 2: Substitute and rewrite the equation
The equation becomes:
\[
x - yi = \frac{2}{x + yi} + i.
\]
To simplify, multiply the numerator and denominator on the right-hand side by the complex conjugate of the denominator:
\[
\frac{2}{x + yi} = \frac{2(x - yi)}{(x + yi)(x - yi)} = \frac{2(x - yi)}{x^2 + y^2}.
\]
Thus, the equation becomes:
\[
x - yi = \frac{2x}{x^2 + y^2} - \frac{2y}{x^2 + y^2}i + i.
\]

---

### Step 3: Separate real and imaginary parts
The left-hand side is:
\[
x - yi.
\]
The right-hand side is:
\[
\left(\frac{2x}{x^2 + y^2}\right) + \left(i - \frac{2y}{x^2 + y^2}i\right).
\]
Simplify the imaginary part:
\[
i - \frac{2y}{x^2 + y^2}i = \left(1 - \frac{2y}{x^2 + y^2}\right)i.
\]
So the equation becomes:
\[
x - yi = \frac{2x}{x^2 + y^2} + \left(1 - \frac{2y}{x^2 + y^2}\right)i.
\]

---

### Step 4: Equate real and imaginary parts
For the real parts:
\[
x = \frac{2x}{x^2 + y^2}.
\]
For the imaginary parts:
\[
-y = 1 - \frac{2y}{x^2 + y^2}.
\]

---

### Step 5: Solve the equations
#### Real part equation:
\[
x(x^2 + y^2) = 2x.
\]
If \(x \neq 0\), we can divide both sides by \(x\):
\[
x^2 + y^2 = 2. \tag{1}
\]

#### Imaginary part equation:
\[
-y = 1 - \frac{2y}{x^2 + y^2}.
\]
Using \(x^2 + y^2 = 2\) from Equation (1):
\[
-y = 1 - \frac{2y}{2}.
\]
\[
-y = 1 - y.
\]
This implies:
\[
0 = 1,
\]
which is a contradiction unless \(y = 0\).

---

### Step 6: Conclusion
Since \(y = 0\), the complex number \(z\) must be purely real. From Equation (1):
\[
x^2 + 0^2 = 2 \quad \Rightarrow \quad x^2 = 2 \quad \Rightarrow \quad x = \pm\sqrt{2}.
\]

Thus, the solutions are:
\[
z = \pm\sqrt{2}.
\]

---

### Final Answer
The complex numbers \(z\) that satisfy the equation are:
\[
z = \sqrt{2} \quad \text{and} \quad z = -\sqrt{2}.
\]

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Do you want further details or have questions?

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**Related Questions:**
1. What is the definition of a complex conjugate?
2. How do you simplify fractions with complex numbers?
3. What is the geometric interpretation of the solutions \(z = \sqrt{2}\) and \(z = -\sqrt{2}\)?
4. Can this method apply to other equations involving complex conjugates?
5. How do we multiply and divide complex numbers?

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**Tip:** Always equate the real and imaginary parts separately when working with complex equations.

Find all complex numbers z that satisfy the equation Z* = 2/z + i.

This page provides a detailed step-by-step solution to the equation Z* = 2/z + i. Learn how to manipulate complex numbers, use complex conjugates, and equate real and imaginary parts to find the solutions z = √2 and z = -√2. Suitable for high school students in Grades 11-12.

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