Express the following in the form a+ bi
`(i-1)/(i+1)`

Answer

To express \(\frac{i-1}{i+1}\) in the form \(a + bi\), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \frac{i - 1}{i + 1} \]

### Step 2: Rationalize the denominator
To eliminate the imaginary part in the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(i - 1\):
\[ \frac{(i - 1)(i - 1)}{(i + 1)(i - 1)} \]

### Step 3: Expand the numerator
Now, we expand the numerator:
\[ (i - 1)(i - 1) = i^2 - 2i + 1 \]
Since \(i^2 = -1\), we substitute:
\[ -1 - 2i + 1 = -2i \]

### Step 4: Expand the denominator
Next, we expand the denominator:
\[ (i + 1)(i - 1) = i^2 - 1^2 = -1 - 1 = -2 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{-2i}{-2} = i \]

### Step 6: Write in the form \(a + bi\)
We can express \(i\) in the form \(a + bi\):
\[ 0 + 1i \]
Thus, we have \(a = 0\) and \(b = 1\).

### Final Answer
The expression \(\frac{i - 1}{i + 1}\) in the form \(a + bi\) is:
\[ 0 + 1i \]