solving polynomial equations with one or more variables; the solutions to such equa-
tions are often obtained by the operations of addition, subtraction, multiplication,
division, raising to a power, and extracting a root. (For more information about all
these operations, see “Math Basics.”) This also includes determining the properties of
functions and graphs.
But mathematicians use the word “algebra” most often in reference to the
abstract study of number systems and operations within them, such as groups, rings,
and invariant theory. This is called abstract algebra.
How else is the word algebra used?
Algebra may be defined as the subjects of arithmetic and abstract algebra, but there
are other meanings. These algebras involve vectors and matrices, the algebra of real
numbers, complex numbers, and quaternions (an operator or factor that changes one
vector into another). There are also those exotic algebras “invented” by mathemati-
cians—and usually named after the inventor—with the majority not truly understood
except, perhaps, by their creators.
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When most people think of “algebra,” their thoughts turn to the polynomial equations they learned in school.
But mathematicians have a much broader definition of the term, which includes concepts of “abstract algebra.”
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