To express the complex number \(\frac{1 + i}{1 - i}\) in standard form \(a + bi\), we will multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(1 - i\) is \(1 + i\). Let's proceed with the calculation:

\[
\frac{1 + i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}
\]

First, simplify the denominator:

\[
(1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2
\]

Next, simplify the numerator:

\[
(1 + i)(1 + i) = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i
\]

Now, the expression simplifies to:

\[
\frac{2i}{2} = i
\]

Thus, the standard form of the complex number \(\frac{1 + i}{1 - i}\) is \(i\), or \(0 + 1i\).

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. What is the standard form of \(\frac{1 - i}{1 + i}\)?
2. How do you multiply complex numbers?
3. What is the geometric interpretation of multiplying a complex number by its conjugate?
4. How do you divide complex numbers?
5. What is the polar form of the complex number \(i\)?

**Tip:** Multiplying a complex number by its conjugate results in a real number.

Learn how to find the standard form of the complex number \(\frac{1 + i}{1 - i}\) using the conjugate method. Follow a detailed step-by-step solution with explanations on complex number operations. Suitable for advanced high school students and beyond.

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