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How does arithmetic differ from algebra?

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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 algebra are both fundamental branches of mathematics, but they differ primarily in their approach to numerical quantities and operations.12

Arithmetic is the study of numbers and the relationships between them. It focuses on specific numerical values and the application of computational processes, such as addition, subtraction, multiplication, division, and the extraction of roots.1

Algebra, by contrast, applies these same operations and formal manipulations to abstract symbols rather than specific numbers. Its defining characteristic is the use of variables—typically letters like x, y, or z—to represent unknown values or arbitrary quantities. This shift from concrete numbers to symbolic representation allows for the creation of general formulas and the solution of equations, providing a flexible framework that is essential to fields ranging from computer science to economics.23456

While arithmetic deals with the direct calculation of known quantities, algebra provides the tools to model relationships and solve for unknowns, making it a powerful extension of arithmetical thought.64