Question

Which of the following statements are true about the simplified form of the expression (2 + 2i) / (1 - I)? Select all that apply.

The simplified form is 2i
The simplified form is 4i
The simplified form is 2 + 2i
The simplified form is 4 + 4i
The simplified form is a complex number because complex numbers are closed under division.
The simplified form is not a complex number because complex numbers are not closed under division.

Answer 1

To simplify the expression \((2 + 2i) / (1 - i)\), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + i\).

So, we have:

\[
\frac{(2 + 2i)(1 + i)}{(1 - i)(1 + i)}
\]

Calculating the denominator:

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

Now, calculating the numerator:

\[
(2 + 2i)(1 + i) = 2(1) + 2(i) + 2i(1) + 2i(i) = 2 + 2i + 2i - 2 = 2 + 4i - 2 = 4i
\]

So, combining the results, we have:

\[
\frac{4i}{2} = 2i
\]

Now let's analyze the statements:

1. **The simplified form is 2i** - **True** (This is what we found)
2. **The simplified form is 4i** - **False** (We found it to be 2i)
3. **The simplified form is 2 + 2i** - **False** (We found it to be 2i)
4. **The simplified form is 4 + 4i** - **False** (We found it to be 2i)
5. **The simplified form is a complex number because complex numbers are closed under division.** - **True** (The result is indeed a complex number)
6. **The simplified form is not a complex number because complex numbers are not closed under division.** - **False** (That statement is incorrect; complex numbers are closed under division, except when dividing by zero)

In conclusion, the true statements are:
- The simplified form is 2i.
- The simplified form is a complex number because complex numbers are closed under division.