Express the following in the form a+bi, where a and b are rea numbers
`((1-i)/(1+ i))^(2)`

Answer

To express \(\left(\frac{1-i}{1+i}\right)^{2}\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers, we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \left(\frac{1-i}{1+i}\right)^{2} \]

### Step 2: Simplify the fraction
To simplify \(\frac{1-i}{1+i}\), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \(1-i\):
\[ \frac{(1-i)(1-i)}{(1+i)(1-i)} \]

### Step 3: Calculate the numerator
Using the formula \((a-b)^{2} = a^{2} - 2ab + b^{2}\):
\[ (1-i)(1-i) = 1^{2} - 2(1)(i) + (-i)^{2} = 1 - 2i + (-1) = 1 - 2i - 1 = -2i \]

### Step 4: Calculate the denominator
Using the formula \((a+b)(a-b) = a^{2} - b^{2}\):
\[ (1+i)(1-i) = 1^{2} - i^{2} = 1 - (-1) = 1 + 1 = 2 \]

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

### Step 6: Square the result
Next, we need to square the result:
\[ (-i)^{2} = (-1)^{2} \cdot i^{2} = 1 \cdot (-1) = -1 \]

### Step 7: Write in the form \(a + bi\)
Finally, we can express \(-1\) in the form \(a + bi\):
\[ -1 + 0i \]

Thus, the final answer is:
\[ \boxed{-1 + 0i} \]

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