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Стало понятно, что ...
It seems that when the verb "стать" is used in the sense of "become ...", the instrumental case follows. According to Wiki, however, the instrumental case of the adjective "понятный" is listed as "поня́тным", and "понятно" seems to be its shortened form, whatever its case is.
So I wonder if this shortened form "понятно" applies to all cases of the adjective "понятный"?
And is it more common to use "понятно" than "поня́тным" in conversation?
This is not a shortened adjective here, this is an adverb.
Usually adverbs like this are conveyed using adjectives or "to feel + adv." in English (ему одиноко "he's feeling lonely", ему холодно "he's cold") etc., however English does not have a passive version of "to understand", so it might not indeed be obvious that понятно here is an adverb.
Adverbs with close meanings, like стало ясно "it became clear", стало очевидно "it became obvious" etc. might make this pattern more clear.
So понятно here is an adverb, and as an adverb it does not decline or agree in number and gender.
There are similar constructs, where понятно is a true adjective: математика стала ему понятной (понятна). Here, the full form adjective is in instrumental and the short form one is in nominative, and they both agree in gender with the noun they define.
"Понятно" is translated as understandabe, so стало понятно is simply '[It] became understandable'.
Понятно is either a shortened form of adjective or an adverb (there is support for both theories), but neither adverbs nor shortened adjectives have cases.
Стало понятным is rarely used in everyday speech, but it is a valid expression and you may find it in written texts.
The word "понятно" in your example is called "категория состояния"(category of state).It's a certain word class, which has similar semantic, syntactic and morphological qualities. The main of them are : 1. In the sentence it functions as an impersonal predicate. 2. The meaning isn't an action, but a state. 3.It doesn't change (no gender, no case,etc.) 4. And in the form it looks like a noun, an adjective or an adverb.
Грустно и тяжело было видеть ее в столь нелегком положении.
The verbs "делаться, становиться, стать, быть" show us that this is "категория состояния"
Когда я ее слушал, мне становилось грустно; Мне сделалось неловко.
Such words don't modify (don't belong to) any other word class.
1.Она глядела грустно (an adverb –здесь наречие определяет глагол). 2. Лицо девушки было грустно (a short adjective, can also be "грустным"-- в данном случае краткое прилагательное определяет существительное) 3.Девушке было грустно (category of state категория состояния)
You can read more here: Наречие или краткая форма
