Arithmetik vs. rechnen
While both terms relate to dealing with numbers, Rechnen refers to the practical act of calculating used in daily life and primary education, whereas Arithmetik describes the theoretical and academic branch of mathematics concerning number properties.
Arithmetik
C1This word is used in formal, academic, or scientific contexts. It refers to the theory of numbers and the abstract study of mathematical properties rather than the simple act of adding or subtracting.
Die Arithmetik ist ein grundlegendes Teilgebiet der Mathematik.
(Arithmetic is a fundamental branch of mathematics.)
In diesem Buch werden die Gesetze der Arithmetik erklärt.
(The laws of arithmetic are explained in this book.)
Professor Müller hält eine Vorlesung über höhere Arithmetik.
(Professor Müller is giving a lecture on higher arithmetic.)
Rechnen
A2This is the common, everyday term for doing sums, calculating, or elementary arithmetic. It is used to describe the skill taught in primary school or the practical action of computing numbers in daily situations.
In der ersten Klasse lernen die Kinder lesen und rechnen.
(In the first grade, children learn reading and arithmetic.)
Ich bin leider nicht sehr gut im Kopfrechnen.
(Unfortunately, I am not very good at mental arithmetic.)
Das Rechnen mit großen Zahlen dauert ohne Taschenrechner lange.
(Calculating with large numbers takes a long time without a calculator.)
Er muss rechnen, ob sein Geld für den Urlaub reicht.
(He has to calculate if his money is enough for the vacation.)
Summary
Use Rechnen for everyday calculations, school subjects, and the verb to calculate. Use Arithmetik only when discussing the overarching mathematical theory or in a university context.