Yes, you can cast spells without your spellbook. And yes, your list of prepared spells will stay prepared until you prepare a new list. In the Preparing and Casting spells section, only preparing a new list of spells actually requires your spellbook.

Does a wizard need a spell book?

If you want to imagine that your character gets a tattoo for each spell then that is fine, but a Wizard must still carry an object called a Spellbook in order to scribe and prepare spells. If that object is lost then no amount of tattoos can replace it.

What if a wizard loses his spellbook?

They can find or buy spell scrolls to transcribe. ... But, the spellbook also creates a limit on the wizard's power, and a danger to the use of that power: if a wizard loses his spellbook, then he's basically stuck with only the spells he had prepared when the book was lost.