insight / access
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insight
The understanding or recognition of a situation, often also of one's own mistake. It describes the moment when you grasp something.
Nach langem Nachdenken kam er zur Einsicht, dass sein Verhalten falsch war.
After long reflection, he came to the insight that his behavior had been wrong.
The word is made up of 'ein' (in/into) and 'Sicht' (view). An 'Einsicht' is therefore literally a 'look inside' a situation, which explains the meaning 'understanding' or 'realization' well.
Seine Erklärung gab mir eine neue Einsicht in das Problem.
His explanation gave me a new insight into the problem.
'Die Einsicht' (the insight/realization) is the noun for the verb 'einsehen' (to realize/admit). When you acknowledge a mistake, you 'kommt man zur Einsicht' (come to the realization).
Ich sehe ein, dass ich einen Fehler gemacht habe.
I realize that I made a mistake.
In the meaning of 'understanding', 'Einsicht' is often used in fixed expressions. The most common are 'zur Einsicht kommen' (to come to the realization) or 'Einsicht zeigen' (to show insight).
Nach dem Gespräch zeigte er endlich Einsicht.
After the conversation, he finally showed insight.
In an administrative or legal context, the fixed expression 'Einsicht nehmen in' (+ accusative) is very common. It means that you inspect official documents.
Die Journalistin wollte Einsicht in die geheimen Dokumente nehmen.
The journalist wanted to inspect the secret documents.
The meaning 'Akteneinsicht' (access to files/inspection of files) belongs to administrative and legal language. In everyday speech, you would more likely say that you are looking at papers or documents.
Der Antrag auf Einsicht in die Baupläne wurde genehmigt.
The request for access to the building plans was approved.
understanding
'das Verständnis' and 'die Einsicht' can both mean that someone understands something. 'Verständnis' describes more an ability or a state: you understand a topic, a situation, or another person. 'Einsicht' places more emphasis on the moment when you realize something, often also your own mistake. That is why you say, for example, 'Verständnis für jemanden haben' (to have understanding for someone), but 'zur Einsicht kommen' (to come to the realization). Learners may confuse the words because both have to do with understanding. When it is about thinking along or empathy, 'Verständnis' is more natural; when someone suddenly realizes what was right or wrong, 'Einsicht' fits better.
Nach dem Streit kam Mina zu der Einsicht, dass sie sich entschuldigen sollte.
After the argument, Mina came to the realization that she should apologize.
Hast du Verständnis dafür, dass ich morgen den Haushalt nicht schaffe?
Do you understand that I won't be able to get the housework done tomorrow?