Example 3: In front of each number in $1, 2, 3, \cdots, 1989$, add a “+” or “-” sign to make their algebraic sum the smallest non-negative number, and write out the equation.
(1989, All-Russian Mathematical Olympiad)
Proof: First, we prove that the algebraic sum is an odd number.
Consider the simplest case:
If all are filled with “+”, then at this moment
$$
1+2+\cdots+1989=995 \times 1989
$$

is an odd number.
For the general case, it only requires adjusting some “+” to “-”.

Since $a+b$ and $a-b$ have the same parity, the parity of the algebraic sum remains unchanged with each adjustment, meaning the total sum is always odd. Therefore, 
$$
1+(2-3-4+5)+(6-7-8+9)+\cdots+
$$
$$
(1986-1987-1988+1989)=1 \text {, }
$$

Thus, the minimum value is 1.