Arithmetic is an elementary branch of mathematics in which real numbers and relations among real numbers are studied and used to solve quantitative problems in finance, engineering, geometry (mensuration), and such fields. The basic arithmetic operations are addition, subtraction, multiplication, division, raising to integral powers, and extraction of integral roots. The term is derived from the Greek word άριθμητικη (arithmètikè, "the art of reckoning"). In the Middle Ages arithmetica was one of the seven Artes Liberales (together with geometria, astronomia, and musica, arithmetica constituted the quadrivium, four of the seven artes).
The arithmetic rules coincide with the rules valid on the field ℝ of real numbers. It will not be attempted here to formulate a set of rigorous rules. The natural numbers and their notation 1,2,3, … are pre-assumed and also the fact that they belong to ℝ. The arithmetic rules are then briefly the following:
- There is a commutative binary operation, called addition (denoted by +) and the field ℝ is closed under this operation, that is, the result of addition of any two numbers belongs again to ℝ. For example, 3 + 5 ≡ 5 + 3 (= 8) belongs to ℝ.
- The field ℝ has a number zero (denoted by 0), which has the property that a + 0 = a for all a in ℝ.
- Every number a is uniquely associated with another number, designated by −a ("minus a") such that a+(−a) = 0. It is common to write this as a−a = 0 and to refer to this process as subtraction.
- Every two numbers of ℝ can be subtracted (addition of minus the second number to the first number). This process augments the set of positive numbers with negative numbers.
- There is a commutative binary operation, called multiplication (denoted by ×) and the field ℝ is closed under this operation. For example, 3 × 8 = 8 × 3 (= 24) belongs to ℝ.
- Multiplication by the zero number gives zero: a × 0 = 0, for all a in ℝ.
- The field ℝ has a number unity (denoted by 1), which has the property a × 1 = a for all a in ℝ.
- Every number a in ℝ, except a = 0, is uniquely associated with another number, designated by 1/a (the inverse of a) such that a × (1/a) = 1. It is common to write this as a / a = 1, and to refer to this process as division.
- Every two elements of ℝ can be divided (multiplication of the first by the inverse of the second) except if the second number is equal to zero. By division the set of integral numbers is augmented with the rational numbers (quotients of two integral numbers).
- Addition and multiplication satisfy the distributive law a×(b+c) = a×b+a×c.
- Natural numbers (used for counting) make it possible to write a+a = 2×a, (2×a)+a = 3×a, and so on.
- Using natural numbers, one may introduce integral powers, e.g., a×a×a×a may be written in short-hand notation as a4 (count the number of factors, in this case 4).
- The question of the inverse of integral powers leads to the concept of integral roots. For instance, what is x if it is given that x2 = 2? The answer is conventionally written as x=± √2. However, this question leads us outside arithmetic proper into abstract algebra and number theory. In the latter branches of mathematics it is studied under which conditions equations as x2 = 2 may have a solution.