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Mur vs. mûr

The French words mur and mûr are perfect examples of homophones: they sound identical but have completely different meanings and spellings. The key difference lies in the circumflex accent on the u, which changes the word from a noun to an adjective.

Mur

A1
The word mur is a masculine noun that translates to wall in English. It refers to a physical structure, typically made of brick or stone, that encloses, supports, or divides an area.
Le chat aime dormir sur le mur du jardin.
(The cat likes to sleep on the garden wall.)
Nous allons peindre le mur du salon en bleu.
(We are going to paint the living room wall blue.)
L'artiste a dessiné un graffiti sur le mur de la ville.
(The artist drew a graffiti on the city wall.)
Il y a une fissure dans le mur de la maison.
(There is a crack in the wall of the house.)
Les alpinistes escaladent un immense mur de roche.
(The climbers are scaling a huge rock wall.)

Mûr

A2
The word mûr is an adjective meaning ripe when referring to fruit or mature when referring to a person or a decision. As an adjective, it must agree in gender and number with the noun it describes: mûr (masculine singular), mûre (feminine singular), mûrs (masculine plural), and mûres (feminine plural).
Ce fromage sera mûr dans deux semaines.
(This cheese will be mature in two weeks.)
Cette banane est parfaitement mûre, tu peux la manger.
(This banana is perfectly ripe, you can eat it.)
C'est un jeune homme très mûr pour son âge.
(He is a very mature young man for his age.)
Les abricots sur l'arbre ne sont pas encore mûrs.
(The apricots on the tree are not yet ripe.)
Après mûre réflexion, j'ai décidé d'accepter l'offre.
(After mature consideration, I decided to accept the offer.)
Les fraises sont bien mûres et sucrées cette saison.
(The strawberries are very ripe and sweet this season.)

Summary

In summary, mur is a noun meaning wall, and it never changes its spelling. In contrast, mûr is an adjective meaning ripe or mature, and its spelling changes to mûre, mûrs, or mûres to agree with the noun it modifies. The circumflex accent (^) is the only visual and grammatical clue that separates these two identical-sounding words.