arithmetics

Arithmetic is the branch of mathematics that studies numbers and the basic operations of addition, subtraction, multiplication, and division. It also considers the properties and rules that govern these operations, including how numbers combine, compare, and approximate. The standard term is arithmetic; arithmetics is a less common variant used historically.

Core operations: Addition combines two numbers to produce a sum; subtraction removes a quantity from another; multiplication is repeated addition; division partitions a quantity into equal parts or finds how many times a number fits into another. Properties: Addition is commutative and associative with 0 as the additive identity; multiplication is commutative and associative with 1 as the multiplicative identity and is distributive over addition. Subtraction and division are not generally commutative, and division by zero is undefined.

Notation and development: The Hindu-Arabic numeral system, with positional notation and zero, enables efficient arithmetic and the expression of large and fractional quantities. Ancient civilizations such as the Babylonians and Egyptians practiced arithmetic, while mathematical thinkers in India and the Islamic world contributed methods and notation that influenced modern arithmetic. Algorithms for long addition, subtraction, multiplication, and division are taught in schools and extended in computer arithmetic.

Scope and applications: In addition to real numbers, arithmetic concepts extend to integers, fractions, decimals, and rational numbers, and to more general number systems in higher mathematics. Arithmetic supports science, engineering, finance, statistics, and daily activities such as budgeting and measurement. It also leads to abstract branches of mathematics, such as number theory and algebra, reflecting its central role in mathematical reasoning.