Find the multiplicative inverse of each of the following complex numbers when it exists.
`(i)/(1+i)`

Answer

To find the multiplicative inverse of the complex number \( \frac{i}{1+i} \), we will follow these steps:

### Step 1: Write the expression for the multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by \( \frac{1}{z} \). Here, we have:
\[ z = \frac{i}{1+i} \]
Thus, the multiplicative inverse is:
\[ \frac{1}{z} = \frac{1}{\frac{i}{1+i}} = \frac{1+i}{i} \]

### Step 2: Simplify the expression
Now we simplify \( \frac{1+i}{i} \):
\[ \frac{1+i}{i} = \frac{1}{i} + \frac{i}{i} = \frac{1}{i} + 1 \]

### Step 3: Simplify \( \frac{1}{i} \)
To simplify \( \frac{1}{i} \), we multiply the numerator and denominator by \( i \):
\[ \frac{1}{i} = \frac{1 \cdot i}{i \cdot i} = \frac{i}{-1} = -i \]

### Step 4: Combine the results
Now, substituting back into our expression:
\[ \frac{1+i}{i} = -i + 1 = 1 - i \]

### Conclusion
Thus, the multiplicative inverse of \( \frac{i}{1+i} \) is:
\[ \boxed{1 - i} \]

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