Find the multiplicative inverse of z = 2 - 3i

To find the multiplicative inverse of a complex number, such as \( z = 2 - 3i \), we need to express it in the form \( \frac{1}{z} \). The multiplicative inverse is calculated using the formula:

Step-by-Step Calculation

1. **Identify the complex number**: Here, \( z = 2 - 3i \).

2. **Multiply by the conjugate**: The conjugate of \( z \) is \( 2 + 3i \). We multiply both the numerator and the denominator by this conjugate:

Inverse Calculation:

\( \frac{1}{z} = \frac{1}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{2 + 3i}{(2 - 3i)(2 + 3i)} \)

Calculating the Denominator

The denominator simplifies as follows:

• Using the formula \( (a - b)(a + b) = a^2 - b^2 \):
• Here, \( a = 2 \) and \( b = 3i \):
• So, \( (2)^2 - (3i)^2 = 4 - (-9) = 4 + 9 = 13 \).

Final Result

Now, substituting back into our equation:

\( \frac{1}{z} = \frac{2 + 3i}{13} = \frac{2}{13} + \frac{3}{13}i \)

Thus, the multiplicative inverse of \( z = 2 - 3i \) is:

Answer: \( \frac{2}{13} + \frac{3}{13}i \)