"I want half."
Translation:Jag vill ha hälften.
Halv is an adjective (e.g. half a cake) whereas hälft is a noun (the half). You could use either in this context. Either: jag vill ha halva (tårtan) or jag vill ha hälften (av tårtan). There is also a noun halva, -n -or which means pretty much the same thing as hälft.
I see that we are meant to use the definite form of either the adjective halva or the noun hälften. In case of the adjective, I can understand that this is because the implied noun would be a definite one, e.g. "tårtan". Now, if the noun is made explicit, would we insert as usual a definite article in front, e.g. "den halva tårtan"?
As for the noun hälften, I guess we'd just have to accept the definite form as the convention in Swedish.