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

Answer

To express \((1+i)/(1-i)\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers, we can follow these steps:

### Step 1: Multiply by the Conjugate
We start with the expression:
\[ \frac{1+i}{1-i} \]
To simplify this, we multiply both 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 2: Simplify the Denominator
Now, we simplify the denominator:
\[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 3: Simplify the Numerator
Next, we simplify the numerator:
\[ (1+i)(1+i) = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i \]

### Step 4: Combine the Results
Now, we can combine the results from the numerator and the denominator:
\[ \frac{2i}{2} = i \]

### Step 5: Write in the Form \(a + bi\)
Finally, we express \(i\) in the form \(a + bi\):
\[ i = 0 + 1i \]
Thus, we have \(a = 0\) and \(b = 1\).

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

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