Determine the conjugate and the reciprocal of each complex number given below:
`3-i`

Answer

To solve the problem of finding the conjugate and the reciprocal of the complex number \(3 - i\), we can follow these steps:

### Step 1: Determine the Conjugate
The conjugate of a complex number \(a + bi\) is given by \(a - bi\). For the complex number \(3 - i\):

\[ \text{Conjugate} = 3 + i \]

### Step 2: Determine the Reciprocal
The reciprocal of a complex number \(z = a + bi\) is given by:

\[ \frac{1}{z} = \frac{1}{a + bi} \]

To find the reciprocal of \(3 - i\), we can multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{3 - i} \cdot \frac{3 + i}{3 + i} = \frac{3 + i}{(3 - i)(3 + i)} \]

### Step 3: Simplify the Denominator
Now we need to simplify the denominator \((3 - i)(3 + i)\):

\[ (3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

### Step 4: Write the Reciprocal
Now we can write the reciprocal:

\[ \frac{3 + i}{10} = \frac{3}{10} + \frac{i}{10} \]

### Final Answer
Thus, the conjugate of \(3 - i\) is \(3 + i\) and the reciprocal is \(\frac{3}{10} + \frac{i}{10}\).

### Summary
- **Conjugate**: \(3 + i\)
- **Reciprocal**: \(\frac{3}{10} + \frac{i}{10}\)