Let \( z = 1 + i \sqrt{3} \). Compute \( \frac{z}{1 - z} \) and simplify the result.

Step 1 ：Firstly, we notice that the denominator in the expression is a complex number, so we can't directly compute the division. We need to rationalize the denominator using complex conjugates. The complex conjugate of a complex number \( a + bi \) is \( a - bi \).

Step 2 ：For \( z = 1 + i \sqrt{3} \), the complex conjugate is \( 1 - i \sqrt{3} \).

Step 3 ：We multiply both the numerator and the denominator by the complex conjugate to get: \( \frac{z}{1 - z} \times \frac{1 - i \sqrt{3}}{1 - i \sqrt{3}} = \frac{(1 + i \sqrt{3})(1 - i \sqrt{3})}{(1 - (1 + i \sqrt{3}))(1 - i \sqrt{3})} \)

Step 4 ：Simplify the expressions in the numerator and the denominator to get: \( \frac{1 - 3 + 2i \sqrt{3}}{- i \sqrt{3}} \)

Step 5 ：Simplify further to get: \( \frac{-2 + 2i \sqrt{3}}{- i \sqrt{3}} \)

Step 6 ：Divide the numerator and the denominator by \( -2 \) to get: \( \frac{1 - i \sqrt{3}}{i \sqrt{3/2}} \)

Step 7 ：Finally, simplifying the complex numbers in the denominator, we have: \( \frac{1 - i \sqrt{3}}{i \sqrt{2}} \)