1. Insert “+” or “- -” between $1,2,3, \cdots, 1989$, what is the smallest non-negative number that the sum can achieve?
Except for 995, the numbers $1,2,3, \cdots, 1989$ can all be divided into 994 pairs: $(1,1989),(2,1988)$, $\cdots$, (994,996). Since the parity of the two numbers in each pair is the same, the result of the operation, regardless of how “+” or “-” signs are placed before each pair, can only be an even number. And 995 is an odd number. Therefore, no matter how “+” or “-” signs are placed before the numbers $1,2, \cdots, 1989$, the value of the sum is always odd. Thus, the smallest non-negative number sought is no less than 1. The number 1 can be obtained in the following way:
$$
\begin{aligned}
1 & =1+(2-3-4+5)+(6-7-8+9) \\
+\cdots+ & (1986-1987-1988+1989)
\end{aligned}
$$