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

Answer

To determine the conjugate and the reciprocal of the complex number \( i \), we will follow these steps:

### Step 1: Find the Conjugate
The conjugate of a complex number \( a + bi \) is given by \( a - bi \). For the complex number \( i \), we can express it as \( 0 + 1i \).

- Here, \( a = 0 \) and \( b = 1 \).
- Therefore, the conjugate of \( i \) is:
\[ \text{Conjugate of } i = 0 - 1i = -i \]

### Step 2: Find the Reciprocal
The reciprocal of a complex number \( z \) is given by \( \frac{1}{z} \). For \( z = i \), we need to find \( \frac{1}{i} \).

- To simplify \( \frac{1}{i} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( -i \):
\[ \frac{1}{i} \cdot \frac{-i}{-i} = \frac{-i}{i \cdot (-i)} = \frac{-i}{-i^2} \]

- Since \( i^2 = -1 \), we have:
\[ -i^2 = -(-1) = 1 \]
- Thus, the reciprocal becomes:
\[ \frac{-i}{1} = -i \]

### Final Answers
- The conjugate of \( i \) is \( -i \).
- The reciprocal of \( i \) is \( -i \).