The multiplicative inverse of 3+4i is . . .

Answer

To find the multiplicative inverse of the complex number \(3 + 4i\), we will follow these steps:

### Step 1: Define the complex number
Let \( z = 3 + 4i \).

### Step 2: Write the formula for the multiplicative inverse
The multiplicative inverse of \( z \) is given by \( \frac{1}{z} \).

### Step 3: Substitute the value of \( z \)
We have:
\[ \frac{1}{z} = \frac{1}{3 + 4i} \]

### Step 4: Multiply by the conjugate
To simplify \( \frac{1}{3 + 4i} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 3 - 4i \):
\[ \frac{1}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} = \frac{3 - 4i}{(3 + 4i)(3 - 4i)} \]

### Step 5: Calculate the denominator
Now, calculate the denominator:
\[ (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 6: Substitute back into the expression
Now we substitute back into our expression:
\[ \frac{3 - 4i}{25} \]

### Step 7: Split into real and imaginary parts
This can be written as:
\[ \frac{3}{25} - \frac{4}{25}i \]

### Final Answer
Thus, the multiplicative inverse of \( 3 + 4i \) is:
\[ \frac{3}{25} - \frac{4}{25}i \]

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