arithmetic

Arithmetic is a branch of mathematics that studies numbers and the basic operations of addition, subtraction, multiplication, and division. It concerns counting, measuring, and comparing quantities, and it provides the numerical foundation for most practical calculations in daily life and the sciences.

Core concepts include the properties of the operations. Commutativity and associativity describe how order and grouping affect results for addition and multiplication, while distributivity links multiplication and addition. Place value in positional numeral systems, especially decimal, enables efficient representation and calculation of large and small numbers. Arithmetic commonly handles integers, fractions, decimals, and real numbers, with methods for rounding and estimation.

History and development: Arithmetic arose in ancient civilizations for trade, measurement, and astronomy. The Hindu-Arabic numeral system, with zero and place-value notation, greatly facilitated computation and spread worldwide. Over time, formal algorithms and, later, axiomatic approaches increased rigor and scope.

Applications and methods: Arithmetic underpins education, finance, science, engineering, and computing. It is taught at early levels as elementary arithmetic and expanded in higher courses with topics such as fractions, percentages, exponents, and roots. In computing, arithmetic operations are performed by processors in binary form, and modular arithmetic plays a key role in number theory and cryptography.

Limitations and scope: Division by zero is undefined, and some real numbers cannot be represented exactly in decimal form. Arithmetic serves as the foundation for more advanced mathematics, including algebra, analysis, and number theory, while also supporting practical problem solving in daily life.