Find the additive inverse and reciprocal of complex number `3-4i`.

Answer

To solve the problem of finding the additive inverse and the reciprocal of the complex number \(3 - 4i\), we will follow these steps:

### Step 1: Find the Additive Inverse
The additive inverse of a complex number \(z\) is the number that, when added to \(z\), results in zero. For a complex number \(z = a + bi\), its additive inverse is \(-a - bi\).

Given the complex number \(z = 3 - 4i\):
- Here, \(a = 3\) and \(b = -4\).
- Therefore, the additive inverse is:
\[ -a - bi = -3 + 4i \]

### Step 2: Find the Reciprocal
The reciprocal of a complex number \(z\) is given by \(\frac{1}{z}\). To find the reciprocal of \(3 - 4i\), we will rationalize the denominator.

1. Write the reciprocal:
\[ \text{Reciprocal} = \frac{1}{3 - 4i} \]

2. Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i} = \frac{3 + 4i}{(3 - 4i)(3 + 4i)} \]

3. Calculate the denominator:
\[ (3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25 \]

4. Now substitute back into the reciprocal:
\[ \frac{3 + 4i}{25} = \frac{3}{25} + \frac{4}{25}i \]

### Final Answers
- The **additive inverse** of \(3 - 4i\) is \(-3 + 4i\).
- The **reciprocal** of \(3 - 4i\) is \(\frac{3}{25} + \frac{4}{25}i\).