arithmetic progression: see progression.

arithmetic, branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with Numbers for practical purposes, e.g., computation of areas, proportions, costs, and the like. The four fundamental operations of this study are addition, subtraction, multiplication, and division. In advanced study the concept of number is greatly generalized to include not only complex numbers, but also quaternions, tensors, and abstract entities with no other meaning than that they obey certain laws (see algebra). The division of arithmetic into the practical and the theoretical dates back to classical Greek times, when the term logistic referred to elementary arithmetic and the term arithmetic was reserved for the theory.

Branch of mathematics concerned with properties of and relations among integers. It is a popular subject among amateur mathematicians and students because of the wealth of seemingly simple problems that can be posed. Answers are much harder to come up with. It has been said that any unsolved mathematical problem of any interest more than a century old belongs to number theory. One of the best examples, recently solved, is Fermat's last theorem.

Learn more about number theory with a free trial on Britannica.com.

Branch of mathematics that deals with the properties of numbers and ways of combining them through addition, subtraction, multiplication, and division. Initially it dealt only with the counting numbers, but its definition has broadened to include all real numbers. The most important arithmetic properties (where math.a and math.b are real numbers) are the commutative laws of addition and multiplication, math.a + math.b = math.b + math.a and math.amath.b = math.bmath.a; the associative laws of addition and multiplication, math.a + (math.b + math.c) = (math.a + math.b) + math.c and math.a(math.bmath.c) = (math.amath.b)math.c; and the distributive law, which connects addition and multiplication, math.a(math.b + math.c) = math.amath.b + math.amath.c. These properties include subtraction (addition of a negative number) and division (multiplication by a fraction).

Learn more about arithmetic with a free trial on Britannica.com.