"That is half."
Translation:Das ist die Hälfte.
‘(die) Hälfte’ is a noun, which means it stands alone and doesn't need to modify a noun. Examples: ‘meine bessere Hälfte’ (‘my better half’) or ‘das ist meine Hälfte des Geldes’ (‘this is my half of the money’).
‘halb’ is an adjective, used to modify nouns: ‘eine halbe Stunde’ (‘half an hour’), ‘das halbe Geld’ (‘half the money’). As all German adjectives, the form without endings can also be used as an adverb, for example: ‘halb voll’ (also ‘halbvoll’, ‘half full’), ‘das ist nur halb wahr’ (‘that is only half true’).
Why not use 'the' in the english translation, since the article is used in the german translation?
Because that would not be as natural in English.
"I am giving you half."
"I ate half of the cake."
We would not say "I am giving you the half" or "I ate the half of the cake".
Correct, you wouldn't say that, but not because it sounds wrong, because it has a different meaning.
"I am giving you half" is actually a shortening of "I am giving you a half of it".
"I am giving you the half" is specific to a particular half and would be used, for instance, after saying "I have 3 and a half apples and ...".
How would you say this in German, or is there no way of differentiating between a generic thing and a specific thing (given that the 'the' is always required)?
German divides this up differently.
If you contrast two halves, you can talk about (die) eine Hälfte und die andere Hälfte, using the indefinite article eine.
But if you're just talking about half of something, it's always definite, die Hälfte -- perhaps because two halves are (by definition) identical.
why not say in English "that is the half".
Because that's not how we usually say it in English.
We say "I slept for half the day" or "Half of my grades are A's" -- not "I slept for the half of the day" or "The half of my grades are A's".
The fact that German says Ich habe die Hälfte des Tages geschlafen or Die Hälfte meiner Noten sind Einsen is irrelevant for how English says it.
So this is one place where you have to learn that English and German use the definite article differently.