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.

'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.
to estimate / appreciate
'schätzen' and 'annähern' can both have to do with a value that is not completely exact. 'schätzen' is used in everyday language when you give an approximate number without exact measurement or calculation. In this meaning, 'annähern' is more mathematical: you replace an exact value with a simpler value that is close to it. Learners may confuse the words because both do not emphasize the exact result. But 'schätzen' fits better in everyday situations, while 'annähern' is more suitable for calculations, formulas, or technical explanations.

Im Unterricht nähern wir den Wert der Wurzel mit einer einfachen Zahl an.
In class, we approximate the value of the root with a simple number.
Mein Nachbar schätzt, dass wir für den Zaun zehn Bretter brauchen.
My neighbor estimates that we need ten boards for the fence.