添 vs 加 (jiā): both mean 'to add,' but 添 often implies supplementing what's already there, while 加 is more neutral. 添麻烦 (cause trouble), 添乱 (add to confusion) are common set phrases.
memory
氵(water) + 忝 (tiǎn, phonetic). Think of adding water to fill something up.
Three-drop water radical on the left — the indexing radical. The original sense of 添 was 'to add water until a vessel is full,' so adding liquid is the prototypical act of supplementing. From there the meaning generalized to any kind of addition or supplement, even abstract ones.
Right component 忝 supplies the sound: tiǎn → tiān with only a tone shift. The 'disgrace, shame' meaning of 忝 has nothing to do with adding — pure phonetic loan. Same phonetic in 舔 tiǎn (to lick), so the tian/tiǎn family clusters tightly.
