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: $\frac{1-i\sqrt{3}}{i\sqrt{2}}$