
sich annähern
to approach
To move spatially toward someone or something and reduce the distance.
Langsam näherte sich die Katze dem Vogel an.
Slowly, the cat approached the bird.
to approach / approximate
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to approach
To move spatially toward someone or something and reduce the distance.
Langsam näherte sich die Katze dem Vogel an.
Slowly, the cat approached the bird.

to come closer
To become more similar in opinions, positions, or characteristics; to reach an agreement.
Nach langer Diskussion näherten sich die beiden Parteien in ihren Standpunkten an.
After a long discussion, the two parties came closer in their positions.

to approximate
In mathematics, to replace or calculate an exact value using a simpler, nearby value or expression.
Man kann die Kreiszahl Pi durch den Bruch 22/7 annähern.
You can approximate the number pi with the fraction 22/7.
'Annähern' (to approach / approximate) is a separable verb. The prefix 'an-' comes at the end in main clauses. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Das Schiff nähert sich dem Hafen an.
The ship is approaching the harbor.
The verb 'annähern' (to approach / approximate) belongs to the word family of 'nah' (near) and 'die Nähe' (closeness / proximity). All of these words have to do with a short distance, whether spatially or figuratively.
Die Kinder wollten sich den Tieren im Zoo annähern.
The children wanted to approach the animals at the zoo.
The verb 'sich annähern' (to approach / come closer to) is often used reflexively. The target that one approaches is always in the dative.
Der Wanderer nähert sich dem Gipfel an.
The hiker is approaching the summit.
The context makes the meaning clear: If it is about places or objects, spatial movement is meant. If it is about opinions or people, the figurative meaning is meant. If it is about numbers, the mathematical meaning is meant.
Die Fronten im Konflikt haben sich angenähert.
The opposing sides in the conflict have come closer.
approach / rapprochement / approximation
Die Annäherung and sich annähern can describe the same process, in which positions become more similar. Die Annäherung names this process as a thing or result; sich annähern describes what the people involved do as a verb. The choice therefore depends on the sentence structure here.

Die beiden Gruppen näherten sich nach mehreren Gesprächen deutlich an.
The two groups clearly moved closer together after several discussions.
Die Annäherung der beiden Gruppen wurde nach mehreren Gesprächen deutlich.
The rapprochement between the two groups became clear after several discussions.