$2 \cdot 46$ Insert "+" or "-" between $1,2,3 \cdots, 1989$, what is the smallest non-negative number that the sum expression can obtain?
[Solution] Except for 995, all numbers from $1,2,3, \cdots, 1989$ can be divided into 994 pairs: $(1, 1989)$, $(2, 1988)$, ..., $(994, 996)$. Since the parity of the two numbers in each pair is the same, the result of the operation is always even regardless of how the “+” and “-” signs are placed before each pair; and 995 is an odd number, so the value of the sum of the numbers $1,2,3, \cdots, 1989$ is always odd regardless of how the “+” and “-” signs are placed. Therefore, the smallest non-negative number that can be obtained is not less than 1.
On the other hand, the number 1 can be obtained in the following way:
$1=1+(2-3-4+5)+(6-7-8+9)+\cdots+(1986-1987-1988+1989)$.
Thus, the smallest non-negative number that the sum can achieve is 1.