Algebra, branch of mathematics in which symbols (usually letters) represent unknown numbers in mathematical equations. Algebra allows the basic operations of arithmetic, such as addition, subtraction, and multiplication, to be performed without using specific numbers. People use algebra constantly in everyday life, for everything from calculating how much flour they need to bake a certain number of cookies to figuring out how long it will take to travel by car at a certain speed to a destination that is a specific distance away.
Arithmetic alone cannot deal with mathematical relations such as the Pythagorean theorem, which states that the sum of the squares of the lengths of the two shorter sides of any right triangle is equal to the square of the length of the longest side. Arithmetic can only express specific instances of these relations. A right triangle with sides of length 3, 4, and 5, for example, satisfies the conditions of the theorem: 32 + 42 = 52. (32 stands for 3 multiplied by itself and is termed “three squared.”) Algebra is not limited to expressing specific instances; instead it can make a general statement that covers all possible values that fulfill certain conditions—in this case, the theorem: a2 + b2 = c2.
This article focuses on classical algebra, which is concerned with solving equations, uses symbols instead of specific numbers, and uses arithmetic operations to establish ways of handling symbols. The word algebra is also used, however, to describe various modern, more abstract mathematical topics that also use symbols but not necessarily to represent numbers. Mathematicians consider modern algebra a set of objects with rules for connecting or relating them. As such, in its most general form, algebra may fairly be described as the language of mathematics.