What is the $1000^{\rm th}$ positive integer with an odd number of digits?

9 numbers have one digit (1-9)
900 have three digits (100-999)
90000 have five digits (10000-99999)

Clearly the 1000th positive integer with an odd number of digits will have five digits. Since there are 909 integers with up to three digits, the 91st integer in the third range will be the one you want. That number is 10090.