When would you use one over the other, or are they interchangeable?
My understanding is that 半 is only for time, i.e. 二時半 (half past two), and 半分 is used for saying half of something. Hope a more experienced Japanese speaker can help confirm this!
半 usually appears as a prefix or suffix. 半分 is usually used when you want to express the word “half” on its own. When a concept can be expressed in both ways, usually one or the other will be more common, but both will probably be understood.
For instance if you want to tell someone “I’m going to take a half day off,” the most natural expression would probably be 半休をとる. But if you said 一日の半分休む, it wouldn’t be misunderstood.
半日 - half day
半年 - half year
半生 - half of a lifetime
１時半 - one thirty
半身 - half of one’s body
半音 - semitone (in music)
一日の半分 - half of one day
人生の半分 - half of a lifetime
クラスの半分 - half of the class
ご飯の半分 - half of a meal
議員の半分 - half of the diet members
距離の半分 - half of the distance
体の半分 - half of one’s body
By the way you can see the same pattern with 温 and 温度.
温度 is the generic noun for “temperature” (温 warmth - 度 degree), like 半分 is the generic noun for “half”. The kanji 温, like 半, is not used on its own, but as a prefix or suffix to generate all kind of specialized temperature / half thing. 水温 is water temperature, 体温 is body temperature, 地温 is ground temperature, 室温 is room temperature, 気温 is air temperature etc.
I get the feeling the Chinese and Japanese often add another word to express the same or similar idea. Is this so? My guess would be that since the languages have a lot of soundalike words, this helps establish more context. Am I correct in my guess?