Algebra Games

[Niklas Terki]

Algebrais the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.

Elementary algebra is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field investigating variables that appear in several linear equations, so-called systems of linear equations. It tries to discover the values that solve all equations at the same time.

Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several binary operations defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow. Universal algebra constitutes a further level of generalization that is not limited to binary operations and investigates more abstract patterns that characterize different classes of algebraic structures.

Algebraic methods were first studied in the ancient period to solve specific problems in fields like geometry. Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries, when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and algebraic structures. Algebra is relevant to many branches of mathematics, like geometry, topology, number theory, and calculus, and other fields of inquiry, like logic and the empirical sciences.

Algebra is the branch of mathematics that studies algebraic operations[a] and algebraic structures.[2] An algebraic structure is a non-empty set of mathematical objects, such as the real numbers, together with algebraic operations defined on that set, such as addition and multiplication.[3] Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it studies the use of variables in equations and how to manipulate these equations.[4][b]

Algebra is often understood as a generalization of arithmetic.[8] Arithmetic studies arithmetic operations, like addition, subtraction, multiplication, and division, in a specific domain of numbers, like the real numbers.[9] Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers.[10] A higher level of abstraction is achieved in abstract algebra, which is not limited to a specific domain and studies different classes of algebraic structures, like groups and rings. These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them.[11] Universal algebra is still more abstract in that it is not limited to binary operations and not interested in specific classes of algebraic structures but investigates the characteristics of algebraic structures in general.[12]

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.[14] When used as a countable noun, an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation.[15] Depending on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra.[16]

The word algebra comes from the Arabic term الجبر (al-jabr), which originally referred to the surgical treatment of bonesetting. In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it as the title of a treatise on algebra, also known by the name The Compendious Book on Calculation by Completion and Balancing. The word entered the English language in the 16th century from Italian, Spanish, and medieval Latin.[17] Initially, the meaning of the term was restricted to the theory of equations, that is, to the art of manipulating polynomial equations in view of solving them. This changed in the course of the 19th century[c] when the scope of algebra broadened to cover the study of diverse types of algebraic operations and algebraic structures together with their underlying axioms.[20]

Elementary algebra, also referred to as school algebra, college algebra, and classical algebra,[21] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on the use of variables and examines how mathematical statements may be transformed.[22]

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithm. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7 \displaystyle 2+5=7 .[9]

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation 2 3 = 3 2 \displaystyle 2\times 3=3\times 2 belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combinations of numbers, like the commutative property of multiplication, which is expressed in the equation a b = b a \displaystyle a\times b=b\times a .[22]

Systems of linear equations are often expressed through matrices[f] and vectors[g] to represent the whole system in a single equation. This can be done by moving the variables to the left side of each equation and moving the constant terms to the right side. The system is then expressed by formulating a matrix that contains all the coefficients of the equations and multiplying it with the column vector made up of the variables.[35] For example, the system of equations

Whether a consistent system of equations has a unique solution depends on the number of variables and the number of independent equations. Several equations are independent of each other if they do not provide the same information and cannot be derived from each other. A unique solution exists if the number of variables is the same as the number of independent equations. Underdetermined systems, by contrast, have more variables than equations and have an infinite number of solutions if they are consistent.[38]

On a geometric level, systems of equations can be interpreted as geometric figures. For systems that have two variables, each equation represents a line in two-dimensional space. The point where the two lines intersect is the solution. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to graphically look for solutions by plotting the equations and determining where they intersect.[42] The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space and the points where all planes intersect solve the system of equations.[43]

Abstract algebra, also called modern algebra,[44] studies different types of algebraic structures. An algebraic structure is a framework for understanding operations on mathematical objects, like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as groups, rings, and fields.[45]

Besides groups, rings, and fields, there are many other algebraic structures studied by abstract algebra. They include magmas, semigroups, monoids, abelian groups, commutative rings, modules, lattices, vector spaces, and algebras over a field. They differ from each other in regard to the types of objects they describe and the requirements that their operations fulfill. Many of them are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements.[62] For example, a magma becomes a semigroup if its operation is associative.[63]

Another tool of comparison is the relation between an algebraic structure and its subalgebra.[72] The algebraic structure and its subalgebra use the same operations,[p] which follow the same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure.[q] All operations in the subalgebra are required to be closed in its underlying set, meaning that they only produce elements that belong to this set.[72] For example, the set of even integers together with addition is a subalgebra of the full set of integers together with addition. This is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra since adding two odd numbers produces an even number, which is not part of the chosen subset.[73]

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