I was wondering if somebody could help me out with the fine differences between annähern and nähern. They are both widely translated as approach, but they do not seem to be interchangeable in all cases. Let me describe the uses as far as I understand the words:
In a physical context, the words are interchangeable, and a matter of style, like steigen and ansteigen, and always used with sich. So the following are equivalent:
Ich nähere mich dem Hund.
Ich nähere mich dem Hund an. I am (physically) approaching the dog.
Annähern has a variety of figurative translations of approach/approximate, which nähern does not:
Unsere Gewinne nähern sich der gewünschten Zahl an.
Our profits are approaching the desired number.
Der ständiger Kontakt mit der Gruppe näherte seine Weltsicht derjenigen in der Gruppe an.
The constant contact with the group made his worldview more similar to those in the group.
As far as I understand, nähern wouldn't work in the second contexts. Am I understanding these differences correctly?