2008 HMMT November Guts Round Problem 28

Johnny the grad student is typing all the integers from 1 to \infty, in order. The 2 on his computer is broken however, so he just skips any number with a 2. What's the 2008th number he types?

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2008 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.