Question

Express in the form a + ib:z=2+3i/1-i

Answer 1

To express the complex number \( z = \frac{2 + 3i}{1 - i} \) in the form \( a + bi \), we multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of \( 1 - i \) is \( 1 + i \). Thus, we have:

\[
z = \frac{(2 + 3i)(1 + i)}{(1 - i)(1 + i)}
\]

Let's start with the denominator:

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

Now for the numerator:

\[
(2 + 3i)(1 + i) = 2(1) + 2(i) + 3i(1) + 3i(i) = 2 + 2i + 3i - 3 = (-1) + 5i
\]

Now we can substitute back into our expression for \( z \):

\[
z = \frac{-1 + 5i}{2}
\]

We can separate this into real and imaginary parts:

\[
z = \frac{-1}{2} + \frac{5}{2}i
\]

Thus, expressed in the form \( a + bi \), we have:

\[
z = -\frac{1}{2} + \frac{5}{2}i
\]