The second one

Hi Cindy, the second one is correct. For the first, you could say: I don't know exactly what number it is. Hope this helps! :) I'd be happy to help you improve your English. Please check out my profile and feel free to book a lesson :)

Hello, "I don't know the exact number" is correct. I hope this helps. If you’d like to practice speaking. I’d be happy to help you develop a study plan. Feel free to view my teaching profile and class offerings and send me a message if you think I can help you. Regards.

#2 is correct. You can also fix #1 as: "I don't know exactly the number". As you wrote it, #1 is wrong because nouns can't stand all alone. Rather than "number", you need to say "a number" or "the number" or "that number" or "this number" or "any number" or "every number". Or, you can use a possessive pronoun: "my number", "your number", "its number", "our number", etc. But a noun sitting all by itself doesn't work in English.

The second one is correct because exact is used here as an adjective which describes the noun number.