Analysis

This question tests the division operation of complex numbers in algebraic form and the basic concept of complex numbers. It is a fundamental question.

Solution

Since $z= \frac{3-i}{1+i}= \frac{(3-i)(1-i)}{(1+i)(1-i)}= \frac{2-4i}{2}=1-2i$,
the real part of $z$ is $1$.
Therefore, the answer is $\boxed{1}$.