Arithmétique vs. calcul
While both words relate to numbers, the main difference lies in theory versus practice. Arithmétique refers to the branch or science of mathematics concerning numbers, whereas calcul refers to the practical act or process of computing and performing operations.
Arithmétique
C1This word functions as a noun or an adjective referring to the theoretical branch of mathematics that deals with the properties of numbers. It describes the subject being studied or the logical nature of a sequence.
L'arithmétique est la base de toutes les mathématiques.
(Arithmetic is the foundation of all mathematics.)
Les enfants apprennent les règles de l'arithmétique élémentaire.
(The children are learning the rules of elementary arithmetic.)
Cette suite de nombres suit une progression arithmétique.
(This sequence of numbers follows an arithmetic progression.)
Il est un expert en arithmétique modulaire.
(He is an expert in modular arithmetic.)
Calcul
B1This noun defines the action of counting, computing, or figuring out a result. In the context of arithmetic, it is often translated as calculation or computation and refers to the doing of the math rather than the theory.
Le calcul mental est un excellent exercice pour le cerveau.
(Mental arithmetic is an excellent exercise for the brain.)
Il y a une erreur dans votre calcul.
(There is an error in your calculation.)
Faites le calcul vous-même pour vérifier le prix.
(Do the math yourself to check the price.)
La machine effectue le calcul en une fraction de seconde.
(The machine performs the calculation in a fraction of a second.)
Summary
Use arithmétique when referring to the science, the school subject, or the theoretical properties of numbers. Use calcul when referring to the action of processing operations, doing sums, or mental arithmetic.