Algebra

Summary

Algebra is a branch of applied mathematics that goes beyond the practical and theoretical applications of the numbers of arithmetic. Algebra has a definitive structure with specified elements, defined operations, and basic postulates. Such abstractions identify algebra as a system, so there are algebras of different types, such as the algebra of sets, the algebra of propositions, and Boolean algebra. Algebra has connections not only to other areas of mathematics but also to the sciences, engineering, technology, and other applied sciences. For example, Boolean algebra is used in electronic circuit design, programming, database relational structures, and complexity theory.

Definition and Basic Principles

Algebra is a branch of mathematics. The word “algebra” is derived from an Arabic word that links the content of classical algebra to the theory of equations. Modern algebra focuses on laws of operations on symbolic forms and provides a systematic way to examine relationships between such forms. The concept of a basic algebraic structure arises from understanding an important idea. That is, with the traditional definition of addition and multiplication, the identity, associative, commutative, and distributive properties characterize these operations with not only real numbers and complex numbers but also polynomials, certain functions, and other sets of elements. Even with modifications in the definitions of operations on other sets of elements, these properties continue to apply. Thus, the concept of algebra is extended beyond a mere symbolization of arithmetic. It becomes a definitive structure with specified elements, defined operations, and basic postulates. Such abstractions identify algebra as a system, and therefore, there are algebras of many different types, such as the algebra of sets, the algebra of propositions, and Boolean algebra.

The algebra of sets, or set theory, includes such fundamental mathematical concepts as set cardinality and subsets, which are a part of the study of various levels of mathematics, from arithmetic to calculus and beyond. The algebra of propositions (logic or propositional calculus) was developed to facilitate the reasoning process by providing a way to symbolically represent statements and to perform calculations based on defined operations, properties, and truth tables. Logic is studied in philosophy, as well as various areas of mathematics, such as finite mathematics. Boolean algebra is the system of symbolic logic used primarily in computer science applications; it is studied in areas of applied mathematics, such as discrete mathematics.

Boolean algebra can be considered a generalization of the algebra of sets and the algebra of propositions. Boolean algebra can be defined as a nonempty set B together with two binary operations, sum (symbol +) and product (symbol ×). There is also a unary operation, complement (symbol ¢). In set B, there are two distinct elements, a zero element (symbol 0) and a unit element (symbol 1), and certain laws or properties hold. The laws and properties table shows how laws and properties used in the algebra of sets and the algebra of propositions relate to those of Boolean algebra.

Background and History

The Algebra of Sets. In 1638, Italian scientist Galileo published Discorsi e dimostrazioni matematiche: Intorno à due nuove scienze attenenti alla mecanica e i movimenti locali (Dialogues Concerning Two New Sciences, 1900). In this work, Galileo recognized the basic concept of equivalent sets and the distinguishing characteristics of infinite sets. During the nineteenth century, Bohemian mathematician Bernhard Bolzano studied infinite sets and their unique properties; English mathematician George Boole took an algebraic approach to the study of set theory. However, it was German mathematician Georg Cantor who developed a structure for set theory that later led to the modernization of the study of mathematical analysis.

Cantor had a strong interest in the arguments of medieval theologians concerning continuity and the infinite. With respect to mathematics, Cantor realized that not all infinite sets were the same. In 1874, his controversial work on infinite sets was published. After additional research, he established set theory as a mathematical discipline known as Mengenlehre (theory of assemblages) or Mannigfaltigkeitslehre (theory of manifolds).

The Algebra of Propositions and Boolean Algebra. During the nineteenth century, George Boole, English mathematician Charles Babbage, German mathematician Gottlob Frege, and Italian mathematician Giuseppe Peano tried to formalize mathematical reasoning by an “algebraization” of logic.

Boole, who had clerical aspirations, regarded the human mind as God's greatest accomplishment. He wanted to mathematically represent how the brain processes information. In 1847, his first book, The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning, was published with limited circulation. He rewrote and expanded his ideas in an 1854 publication, An Investigation of the Laws of Thought: On Which Are Founded the Mathematical Theories of Logic and Probabilities. Boole introduced the algebra of logic and is considered the father of symbolic logic.

Boole's algebra was further developed between 1864 and 1895 through the contributions of British mathematician Augustus De Morgan, British economist William S. Jevons, American logician Charles Sanders Peirce, and German mathematician Ernst Schröder. In 1904, American mathematician Edward V. Huntington's Sets of Independent Postulates for the Algebra of Logic developed Boolean algebra into an abstract algebraic discipline with different interpretations. With the additional work of American mathematician Marshall Stone and Polish American logician Alfred Tarski in the 1930s, Boolean algebra became a modern mathematical discipline with connections to several other branches of mathematics, including topology, probability, and statistics.

In his 1940 Massachusetts Institute of Technology master's thesis, Claude Elwood Shannon used symbolic Boolean algebra to analyze relay and switching circuits. Boole's work thus became the foundation for the development of modern electronics and digital computer technology.

Outside the realm of mathematics and philosophy, Boolean algebra has found applications in such diverse areas as anthropology, biology, chemistry, ecology, economics, sociology, and especially computer science. For example, in computer science, Boolean algebra is used in electronic circuit design, programming, database relational structures, and complexity theory.

How It Works

Boolean algebra achieved a central role in computer science and information theory that began with its connection to set theory and logic. Set theory, propositional logic, and Boolean algebra all share a common mathematical structure that becomes apparent in the properties or laws that hold.

Set Theory. The language of set theory is used in the definitions of nearly all mathematical elements, and set theory concepts are integrated throughout the mathematics curriculum from the elementary to the college level. In primary school, basic set concepts may be introduced in sorting, combining, or classifying objects even before the counting process is introduced. Operations such as set complement, union, and intersection can be easily understood in this context.

For example, let the universal set U consist of six blocks, each of which is a different color. A block may be red, orange, yellow, violet, blue, or green. Using set notation, U = {red, orange, yellow, violet, blue, green}. Let four of the six blocks be sorted into two subsets, A and B, such that A = {red, yellow} and B = {blue, green}. The complement of set A is the set of blocks that are neither red nor yellow, A¢ = {orange, violet, blue, green}. The union of sets A and B is the set that contains all of the blocks in set A or set B or both if there were any colors in common: A È B = {red, yellow, blue, green}. The intersection of sets A and B is the set of blocks that are in set A and in set B, any color that both sets have in common. Because the two sets of blocks have no color in common, A Ç; B = Æ.

Above the primary level, the concepts of logic are introduced. Daily life often requires that one construct valid arguments, apply persuasion, and make meaningful decisions. Thus, the development of the ability to organize thoughts and explain ideas in clear, precise terms makes the study of reasoning and the analysis of statements most appropriate.

Logic. In propositional algebra, statements are either true or false. A statement may be negated by using “not.” Statements can be combined in a variety of ways by using connectives such as “and” and “or.” The resulting compound statements are either true or false, based on given truth tables.

A compound statement such as “The First International Conference on Numerical Algebra and Scientific Computing was held in 2006 and took place at the Institute of Computational Mathematics of the Chinese Academy of Sciences in New York” can thus be easily analyzed, especially when written symbolically. The “and” connective indicates that the compound statement is a conjunction. Let p be “The First International Conference on Numerical Algebra and Scientific Computing was held in 2006,” a true statement; let q be “(it) took place at the Institute of Computational Mathematics of the Chinese Academy of Sciences in New York,” a false statement because the institute is in Beijing. The truth table for the conjunction indicates that the given compound statement is false: T Ù F º F.

Compound symbolic statements may require multistep analyses, but established properties and truth tables are still used in the process. For example, it is possible to analyze the two symbolic compound statements ∼(p Ú q) and ∼p Ù ∼q and also to verify that they are logically equivalent. The truth tables for each compound statement can be combined in one large table to facilitate the process. The first two columns of the table show all possibilities for the truth values of two statements, p and q. The next three columns show the analysis of each of the parts of the two given compound statements, using the truth tables for negation, disjunction, and conjunction. The last two columns of the table have exactly the same corresponding T and F entries, showing that the truth value will be the same in all cases. This verifies that the two compound statements are logically equivalent. Note that the equivalence of these two propositions is one of De Morgan's laws: ∼(p Ú q) º ∼p Ù ∼q.

Computer Circuits. Shannon showed how logic could be used to design and simplify electric circuits. For example, consider a circuit with switches p and q that can be open or closed, corresponding to the Boolean binary elements 0 and 1. A series circuit corresponds to a conjunction because both switches must be closed for electric current to flow. A circuit where electricity flows whenever at least one of the switches is closed is a parallel circuit; this corresponds to a disjunction. Because the complement for a given switch is a switch in the opposite position, this corresponds to a negation table. When a circuit is represented in symbolic notation, its simplification may use the laws of logic, such as De Morgan's laws. The simplification may also use tables in the same way as the analysis of the equivalence of propositions, with 1 replacing T and 0 replacing F. Other methods may use Karnaugh maps, the Quine-McCluskey method, or appropriate software.

Computer logic circuits are used to make decisions based on the presence of multiple input signals. The signals may be generated by mechanical switches or by solid-state transducers. The various families of digital logic devices, usually integrated circuits, perform various logic functions through logic gates. Logic gates are the basic building blocks for constructing digital systems. The gates implement the hardware logic function based on Boolean algebra. Two or more logic gates may be combined to provide the same function as a different type of logic gate. This process reduces the total number of integrated circuit packages used in a product.

Boolean expressions can direct computer hardware and also be used in software development by programmers managing loops, procedures, and blocks of statements.

Boolean Searches. Boolean algebra is used in information theory. Online queries are input in the form of logical expressions. The operator “and” is used to narrow a query, and “or” is used to broaden it. The operator “not” is used to exclude specific words from a query.

For example, a search for information about “algebra freeware” may be input as “algebra or freeware,” “algebra and freeware,” or perhaps “algebra and freeware not games.” The amount of information received from each query will be different. The first query will retrieve many documents because it will select those that contain “algebra,” those that contain “freeware,” and those that contain both terms. The second query will retrieve fewer documents because it will select only those documents that contain both terms. The last query will retrieve documents that contain both “algebra” and “freeware” but will exclude items containing the term “games.”

Applications and Products

Logic Machines, Calculating Machines, and Computers. The “algebraization” of logic, primarily the work of De Morgan and Boole, was important to the transformation of Aristotelian logic into modern logic and to the automation of logical reasoning. Several machines were built to solve logic problems, including the Stanhope demonstrator, Jevons's logic machine, and the Marquand machine. In the mid-nineteenth century, Jevons's logic machine, or logic piano, was among the most popular; it used Boolean algebra concepts. Harvard undergraduates William Burkhardt and Theodore Kalin built an electric version of the logic piano in 1947.

In the 1930s, Boolean algebra was used in wartime calculating machines. It was also used in the design of the first digital computer by John Atanasoff and his graduate student Clifford Berry. During 1944–45, John von Neumann suggested using the binary mathematical system to store programs in computer memory. In the 1930s and 1940s, British mathematician Alan Turing and American mathematician Shannon recognized that binary logic was well suited to developing digital computers. Just as Shannon's work served as the basis for the theory of switching and relay circuits, Turing's work became the basis for the field of automata theory, the theoretical study of information processing and computer design.

By the end of World War II, it was apparent that computers would soon replace logic machines. Later, computer software and hardware developments confirmed that the logic process could be mechanized. Although research work continues to provide theoretical guidelines, automated reasoning programs, such as those used in robotics development, are in demand by researchers to resolve questions in mathematics, science, engineering, and technology.

Integrated Circuit Design. Boolean algebra became indispensable in the design of computer microchips and integrated circuits. It is among the fundamental concepts of digital electronics that are essential to understanding the design and function of different types of equipment.

Many integrated circuit manufacturers produce complex logic systems that can be programmed to perform a variety of logical functions within a single integrated circuit. These integrated circuits include gate array logic (GAL), programmable array logic (PAL), the programmable logic device (PLD), and the complex programmable logic device (CPLD).

Engineering approaches to the design and analysis of digital logic circuits involves applications of advanced Boolean algebra concepts, including algorithmic state and machine design of sequential circuits, as well as digital logic simulation. The actual design and implementation of sizeable digital design problems involves the use of computer-aided design (CAD).

Computer Algebra Systems. During the 1960s and 1970s, the first computer algebra systems (CASs) emerged and evolved from the needs of researchers. Computer algebra systems are software that enables users to do tedious and sometimes difficult algebraic tasks, such as simplifying rational functions, factoring polynomials, finding solutions to a system of equations, and representing information graphically in two or three dimensions. The systems offer a programming language for user-defined procedures. Computer algebra systems have not only changed how algebra is taught, but also provided a convenient tool for mathematicians, scientists, engineers, and technicians worldwide.

Among the first popular computer algebra systems were Reduce, Scratchpad, Macsyma (later Maxima), and Mu-Math. Later popular systems include MATLAB, Mathematica, Maple, and MathCAD.

In 1987, Hewlett-Packard introduced HP-28, the first handheld calculator series with the power of a computer algebra system. In 1995, Texas Instruments released the TI-92 calculator with advanced CAS capabilities based on Derive software. Manufacturers continued to offer devices such as these with increasingly powerful functions. With the advent of the smartphone in 1995, companies began releasing scientific and algebraic calculator applications or apps. Some, such as Photomath, even allowed users to photograph a problem for the app to solve.

Careers and Course Work

The applications of algebra are numerous, which means that those interested in algebra can pursue jobs and careers in a wide range of fields, including business, engineering, and science, particularly computer science.

Data Analyst or Data Miner. Data mining is a broad mathematical area that involves discovering patterns and hidden information in large databases using algorithms. In applications of data mining, career opportunities emerge in e-commerce, security, forensics, medicine, bioinformatics and genomics, astrophysics, and chemical and electric power engineering. Coursework should include a focus on higher-level mathematics in such areas as combinatorics, topology, and algebraic structures.

Materials Engineer. Materials science is the study of the properties, processing, and production of such items as metallic alloys, liquid crystals, and biological materials. There are many career opportunities in research, manufacturing, and development in aerospace, electronics, biology, and nanotechnology. The design and analysis of materials depend on mathematical models and computational tools. Course work should include a focus on applied mathematics, including differential equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, and probability.

Computer Animator or Digital Artist. Computer animation encompasses many areas, including mathematics, computer science, physics, biomechanics, and anatomy. Career opportunities arise in medical diagnostics, multimedia, entertainment, and fine arts. The algorithms for computer animation come from scientific relationships, statistics, signal processing, linear algebra, control theory, and computational geometry. Recommended mathematics coursework includes statistics, discrete mathematics, linear algebra, geometry, and topology.

Financial Analyst. As quantitative methods transform the financial industry, banking, insurance, investment, and government regulatory institutions are among those relying on mathematical tools and computational models. Such tools and models are used to support investment decisions, develop and price new securities, manage risk, and guide portfolio selection, management, and optimization. Coursework should include a focus on the mathematics of finance, linear algebra, linear programming, probability, and descriptive statistics.

Social Context and Future Prospects

Algebra is part of two broad, rapidly growing fields: applied mathematics and computational science. Applied mathematics is the branch of mathematics that develops and provides mathematical methods to meet scientific, engineering, and technological needs. It includes discrete mathematics, linear algebra, numerical analysis, operations research, and probability. Computational science integrates applied mathematics, science, engineering, and technology to create a multidisciplinary field that develops and uses innovative problem-solving strategies and methodologies.

Applied mathematics and computational science are used in almost every area of science, engineering, and technology. Business also relies on applied mathematics and computational science to research, design, and manufacture products that include aircraft, automobiles, computers, communication systems, and pharmaceuticals. Research in applied mathematics, therefore, often leads to the development of new mathematical models, theories, and applications that contribute to diverse fields. There are many possibilities for the future prospects of algebra, both within mathematics specifically and in interdisciplinary studies. As the twenty-first century progressed, algebra continued to play a role in developing theoretical physics, especially relating to quantum physics and string theory. It also continued to play a prominent role in computer science. The theoretics and methodology of algebra also continued to develop, and with these developments came educational advances. 

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