jmc

A 5-digit number can have for its leftmost digit anything from 1 to 9 inclusive, and for each of its next four digits anything from 0 through 9 inclusive. Thus there are $9\times 10\times 10\times 10\times 10=90,000$ 5-digit numbers.

A 5-digit number with no zero as a digit can have for each of its five digits anything from 1 through 9 inclusive. There are $9\times 9\times 9\times 9\times 9=59,049$ such 5-digit numbers. Therefore the number of 5-digit numbers with at least one zero as a digit is $90,000-59,049=\boxed{30,951}.$