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How does arithmetic differ from number theory?

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arithmetic A page from a first-grade workbook typical of “new math” might state: “Draw connecting lines from triangles in the first set to triangles in the second set. Are the two sets equivalent in number?”

Arithmetic and number theory are closely related branches of mathematics, though they differ in scope and focus.12

Arithmetic is generally defined as the elementary branch of mathematics concerned with numerical computation—such as addition, subtraction, multiplication, division, and the extraction of roots—as well as the study of relations among numbers and basic mensuration. It serves as the foundational practice of working with numbers to solve practical problems.1

Number theory, sometimes referred to as "higher arithmetic," is a more specialized branch focused on the properties of positive integers. While arithmetic deals with the mechanics of calculation, number theory explores deeper theoretical questions, such as the nature of prime numbers, unique factorization, and congruence.23

Historically, the distinction has not always been rigid; for example, Carl Friedrich Gauss used the term "arithmetic" in his seminal work Disquisitiones Arithmeticae to encompass advanced theoretical topics that are now classified under modern number theory. Today, number theory is recognized as a broad, sophisticated field that utilizes various methods—including algebraic, analytic, and geometric approaches—to investigate the fundamental properties of integers.132