
die Annäherung
approach
The movement in which the distance to a person, place, or object decreases.
Die Annäherung des Fuchses an den Hühnerstall blieb unbemerkt.
The fox's approach to the henhouse went unnoticed.
approach / rapprochement / approximation
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approach
The movement in which the distance to a person, place, or object decreases.
Die Annäherung des Fuchses an den Hühnerstall blieb unbemerkt.
The fox's approach to the henhouse went unnoticed.

rapprochement
A process in which differing opinions, positions, or parties move toward each other in order to reach an agreement or improve relations.
Nach langen Verhandlungen gab es eine erste Annäherung zwischen den Tarifpartnern.
After long negotiations, there was an initial rapprochement between the collective bargaining parties.

approximation
In mathematics or science, a value or method that is not exact but is sufficiently accurate for a particular purpose; an approximation.
Für diese schnelle Berechnung reicht eine Annäherung an den exakten Wert.
For this quick calculation, an approximation of the exact value is sufficient.
'Annäherung' (approach/rapprochement/approximation) is a noun derived from the separable verb 'sich annähern' (to approach). The ending '-ung' turns the verb into a feminine noun.
Die Kinder beobachten die Annäherung der Katze an den Vogel.
The children watch the cat's approach toward the bird.
The noun 'Annäherung' is almost always used with the preposition 'an' followed by the accusative.
Die Annäherung an das schwierige Thema erfordert viel Sensibilität.
The approach to the difficult topic requires a lot of sensitivity.
The context shows the meaning: If it is about places or living beings, spatial movement is meant. If it is about opinions, countries, or parties to a conflict, the figurative meaning of reaching understanding is meant.
Die Annäherung des Schiffes an die Küste war schwierig.
The ship's approach to the coast was difficult.
In political or social contexts, people often speak of a 'schrittweise Annäherung' (gradual rapprochement) or of 'eine Annäherung erzielen' (to achieve a rapprochement).
Die Diplomaten arbeiten an einer Annäherung der beiden Länder.
The diplomats are working on a rapprochement between the two countries.
The meaning as 'Approximation' (approximation) is used almost exclusively in scientific or mathematical texts.
In der Physik ist die Verwendung von Annäherungen oft notwendig.
In physics, the use of approximations is often necessary.
reference / reliance
Annäherung describes a state reducing its distance from another state and seeking a better relationship. With Anlehnung, the focus is more on seeking support and backing from that state. Annäherung fits a mutual improvement in relations, while Anlehnung is more suitable when the focus is on seeking protection.

Die Annäherung an den Nachbarstaat soll nach Jahren des Streits das Verhältnis verbessern.
The rapprochement with the neighboring state is intended to improve relations after years of disputes.
Die Anlehnung an den mächtigen Nachbarstaat soll dem kleinen Land zusätzlichen Rückhalt geben.
Relying on the powerful neighboring state is intended to give the small country additional backing.