**Algebra**,
branch of mathematics in which letters are used to represent basic arithmetic
relations. As in arithmetic,
the basic operations of algebra are addition, subtraction, multiplication, division,
and the extraction of roots.
Arithmetic, however, cannot generalize mathematical relations such as the Pythagorean
theorem, which states that the sum of the squares of the sides of any right
triangle is also a square. Arithmetic can only produce specific instances of
these relations (for example, 3, 4, and 5, where 32 + 42 = 52). But algebra
can make a purely general statement that fulfills the conditions of the theorem:
*a*2 + *b*2 = *c*2. Any number multiplied by itself is termed
*squared* and is indicated by a superscript number 2. For example, 3 ×
3 is notated 32; similarly, *a* × *a* is equivalent to *a*2 (*see*
Exponent;
Power;
Root).

Classical
algebra, which is concerned with solving equations, uses symbols instead of
specific numbers and uses arithmetic operations to establish ways of handling
symbols (*see* Equation;
Equations,
Theory of).
Modern algebra has evolved from classical algebra by increasing its attention
to the structures within mathematics.
Mathematicians consider modern algebra to be 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.

**History **

The
history of algebra began in ancient Egypt and Babylon, where people learned
to solve linear (*ax* = *b*) and quadratic (*ax*2 + *bx*
= *c*) equations, as well as *indeterminate equations* such as *x*2
+ *y*2 = *z*2, whereby several unknowns are involved. The ancient
Babylonians solved arbitrary quadratic equations by essentially the same procedures
taught today. They also could solve some indeterminate equations.

The
Alexandrian mathematicians Hero
of Alexandria
and Diophantus
continued the traditions of Egypt and Babylon, but Diophantus' book *Arithmetica*
is on a much higher level and gives many surprising solutions to difficult indeterminate
equations. This ancient knowledge of solutions of equations in turn found a
home early in the Islamic world, where it was known as the "science of
restoration and balancing." (The Arabic word for restoration, *al-jabru,*
is the root of the word *algebra.*) In the 9th century, the Arab mathematician
Al-Khwarizmi
wrote one of the first Arabic algebras, a systematic exposé of the basic theory
of equations, with both examples and proofs. By the end of the 9th century,
the Egyptian mathematician Abu Kamil had stated and proved the basic laws and
identities of algebra and solved such complicated problems as finding *x,
y,* and *z* such that *x* + *y* + *z* = 10, *x*2
+ *y*2 = *z*2, and *xz* = *y*2.

Ancient
civilizations wrote out algebraic expressions using only occasional abbreviations,
but by medieval times Islamic mathematicians were able to talk about arbitrarily
high powers of the unknown *x,* and work out the basic algebra of polynomials
(without yet using modern symbolism). This included the ability to multiply,
divide, and find square roots of polynomials as well as a knowledge of the binomial
theorem. The Persian mathematician, astronomer, and poet Omar
Khayyam showed
how to express roots of cubic equations by line segments obtained by intersecting
conic sections, but he could not find a formula for the roots. A Latin translation
of Al-Khwarizmi's *Algebra* appeared in the 12th century. In the early
13th century, the great Italian mathematician Leonardo Fibonacci achieved a
close approximation to the solution of the cubic equation *x*3 + 2*x*2
+ *cx* = *d*. Because Fibonacci had traveled in Islamic lands, he
probably used an Arabic method of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano solved the general cubic equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree, and as a result, mathematicians for the next several centuries tried to find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician Évariste Galois proved that no such formula exists.

An important
development in algebra in the 16th century was the introduction of symbols for
the unknown and for algebraic powers and operations. As a result of this development,
Book III of *La géometrie* (1637), written by the French philosopher and
mathematician René
Descartes,
looks much like a modern algebra text. Descartes's most significant contribution
to mathematics, however, was his discovery of analytic geometry,
which reduces the solution of geometric problems to the solution of algebraic
ones. His geometry text also contained the essentials of a course on the theory
of equations, including his so-called *rule of signs* for counting the
number of what Descartes called the "true" (positive) and "false"
(negative) roots of an equation. Work continued through the 18th century on
the theory of equations, but not until 1799 was the proof published, by the
German mathematician Carl
Friedrich Gauss,
showing that every polynomial equation has at least one root in the complex
plane (*see* Number:
Complex Numbers).

By the
time of Gauss, algebra had entered its modern phase. Attention shifted from
solving polynomial equations to studying the structure of abstract mathematical
systems whose axioms
were based on the behavior of mathematical objects, such as complex numbers,
that mathematicians encountered when studying polynomial equations. Two examples
of such systems are groups (*see* Group)
and quaternions, which share some of the properties of number
systems but
also depart from them in important ways. Groups began as systems of permutations
and combinations
of roots of polynomials, but they became one of the chief unifying concepts
of 19th-century mathematics. Important contributions to their study were made
by the French mathematicians Galois and Augustin
Cauchy, the
British mathematician Arthur
Cayley, and
the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered
by British mathematician and astronomer William
Rowan Hamilton,
who extended the arithmetic of complex numbers to quaternions while complex
numbers are of the form *a* + *bi,* quaternions are of the form *a*
+ *bi* + *cj* + *dk.*

Immediately
after Hamilton's discovery, the German mathematician Hermann Grassmann began
investigating vectors.
Despite its abstract character, American physicist J.
W. Gibbs recognized
in vector algebra a system of great utility for physicists, just as Hamilton
had recognized the usefulness of quaternions. The widespread influence of this
abstract approach led George
Boole to write
*The Laws of Thought* (1854), an algebraic treatment of basic logic. Since
that time, modern algebra—also called abstract algebra—has continued
to develop. Important new results have been discovered, and the subject has
found applications in all branches of mathematics and in many of the sciences
as well.

**Symbols and Special
Terms **

The symbols of algebra include numbers, letters, and signs that indicate various arithmetic operations. Numbers are, of course, constants, but letters can represent either constants or variables. Letters that are used to represent constants are taken from the beginning of the alphabet; those used to represent variables are taken from the end of the alphabet.

*Operations and the
Grouping of Symbols *

The grouping of algebraic symbols and the sequence of arithmetic operations rely on grouping symbols to ensure that the language of algebra is clearly read. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and horizontal bars (also called vinculums) that are used most often for division and roots, as in the following:

The
basic operational signs of algebra are familiar from arithmetic: addition (+),
subtraction (-), multiplication (×), and division (÷). Often, in the case of
multiplication, the "×" is omitted or replaced by a dot, as in *a*
· *b.* A group of consecutive symbols, such as *abc,* indicates the
product of *a, b,* and *c.* Division is commonly indicated by bars,
as in the preceding example. A virgule, or slash (/), is also used to separate
the numerator from the denominator of a fraction, but care must be taken to
group the terms appropriately. For example, *ax* + *b/c* - *dy*
indicates that *ax* and *dy* are separate terms, as is *b/c,*
whereas (*ax* + *b*)/(*c* - *dy*) correctly represents the
fraction

*Order of Operations
*

Multiplications are performed first, then divisions, followed by additions, and then subtractions. Grouping symbols indicate the order in which operations are to be performed—that is, carry out all operations within a group first, beginning with the innermost group. For example

*Special Definitions
*

Any
statement involving the equality relation (=) is called an *equation.*
An equation is called an *identity* if the equality is true for all values
of its variables; if the equation is true for some values of its variables and
false for others, the equation is *conditional.* A *term* is any algebraic
expression consisting only of products of constants and variables; 2*x,*
-*a,* 3*s*4*x,* and *x*2(2*zy*)3 are all examples of
terms. The numerical part of a term is called its *coefficient.* The coefficients
of each term above are, respectively, 2, -1, 3, and 8 (the last term may be
rewritten 8*x*2(*zy*)3).

An expression
containing one term is called a *monomial;* two terms, a *binomial;*
and three terms, a *trinomial.* A *polynomial* is any finite sum (or
difference) of terms. For example, a general polynomial of degree *n* might
be expressed as

In this
context, *degree* refers to the largest exponent of the variables in a
polynomial. For example, if the largest exponent of a variable is 3, as in *ax*3
+ *bx*2 + *cx,* the polynomial is said to be of degree 3. Similarly,
the expression *x*n + *x*n-1 + *x*n-2 is of degree *n.*

A *linear
equation* in one variable is a polynomial equation of degree one—that
is, of the form *ax* + *b* = 0. These are called linear equations
because they represent the equation of straight lines in analytic geometry.

A *quadratic
equation* in one variable is a polynomial equation of degree two—that
is, of the form *ax*2 + *bx* + *c* = 0.

A *prime
number* is any integer (whole number) that can be evenly divided only by
itself and by the number 1. Thus, 2, 3, 5, 7, 11, and 13 are all prime numbers.

*Powers*
of a number are formed by successively multiplying the number by itself. The
term *a* raised to the third power, for example, can be expressed as *a·a·a,*
or as *a*3.

The
*prime factors* of any number are those factors to which it can be reduced
such that the number is expressed only as the product of primes and their powers.
For example, the prime factors of 15 are 3 and 5. Similarly, because 60 = 22
× 3 × 5, the prime factors of 60 are 2, 3, and 5. *See* Factor.

**Operations with Polynomials
**

In operating
with polynomials, the assumption is that the usual laws of the arithmetic of
numbers hold. In arithmetic, the numbers used are the set of rational numbers
(*see* Number;
Number Theory).
Arithmetic alone cannot go beyond this, but algebra and geometry can include
both irrational numbers, such as the square root of 2, and complex numbers.
The set of all rational and irrational numbers taken together constitutes the
set of what are called real numbers.

*Laws of Addition
*

A1.
The sum of any two real numbers *a* and *b* is again a real number,
denoted *a* + *b.* The real numbers are closed under the operations
of addition, subtraction, multiplication, division, and the extraction of roots;
this means that applying any of these operations to real numbers yields a quantity
that also is a real number.

A2.
No matter how terms are grouped in carrying out additions, the sum will always
be the same: (*a* + *b*) + *c* = *a* + (*b* + *c*).
This is called the associative law of addition.

A3.
Given any real number *a,* there is a real number zero (0) called the additive
identity, such that *a* + 0 = 0 + *a* = *a.*

A4.
Given any real number *a,* there is a number (-*a*), called the additive
inverse of *a,* such that (*a*) + (-*a*) = 0.

A5.
No matter in what order addition is carried out, the sum will always be the
same: *a* + *b* = *b* + *a.* This is called the commutative
law of addition.

Any set of numbers obeying laws A1 through A4 is said to form a group. If the set also obeys A5, it is said to be an Abelian, or commutative, group.

*Laws of Multiplication
*

Laws similar to those for addition also apply to multiplication. Special attention should be given to the multiplicative identity and inverse, M3 and M4.

M1.
The product of any two real numbers *a* and *b* is again a real number,
denoted *a·b* or *ab.*

M2.
No matter how terms are grouped in carrying out multiplications, the product
will always be the same: (*ab*)*c* = *a*(*bc*). This is
called the associative law of multiplication.

M3.
Given any real number *a,* there is a number one (1) called the multiplicative
identity, such that *a*(1) = 1(*a*) = *a.*

M4.
Given any nonzero real number *a,* there is a number (*a*-1), or (1/*a*),
called the multiplicative inverse, such that *a*(*a*-1) = (*a*-1)*a*
= 1.

M5.
No matter in what order multiplication is carried out, the product will always
be the same: *ab* = *ba.* This is called the commutative law of multiplication.

Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero (because division by zero is inadmissible), forms such a commutative group under multiplication.

*Distributive Laws
*

Another important property of the set of real numbers links addition and multiplication in two distributive laws as follows:

D1.
*a*(*b* + *c*) = *ab* + *ac*

D2.
(*b* + *c*)*a* = *ba* + *ca*

Any
set of elements with an equality relation and for which two operations (such
as addition and multiplication) are defined, and which obeys all the laws for
addition A1 through A5, the laws for multiplication M1 through M5, and the distributive
laws D1 and D2, constitutes a *field.*

**Multiplication of
Polynomials **

The following is a simple example of the product of a binomial and a monomial:

This same principle—multiplying each term of the one polynomial by each term of the other—is directly extended to polynomials of any number of terms. For example, the product of a binomial and a trinomial is carried out as follows:

After such operations have been performed, all terms of the same degree should be combined whenever possible to simplify the entire expression:

**Factoring Polynomials
**

Given
a complicated algebraic expression, it is often useful to factor it into the
product of several simpler terms. For example, 2*x*3 + 8*x*2*y*
can be factored as 2*x*2(*x* + 4*y*). Determining the factors
of a given polynomial may be a simple matter of inspection or may require trial
and error. Not all polynomials, however, can be factored using real-number coefficients,
and these are called *prime polynomials.*

Some common factorizations are given in the following examples.

Grouping may often be useful in factoring; terms that are similar are grouped wherever possible, as in the following example:

**Highest Common Factors
**

Given
a polynomial, it is frequently important to isolate the *greatest common factor*
from each term of the polynomial. For example, in the expression 9*x*3
+ 18*x*2, the number 9 is a factor of both terms, as is *x*2. After
factoring, 9*x*2(*x* + 2) is obtained, and 9*x*2 is the greatest
common factor for all terms of the original polynomial (in this case a binomial).
Similarly, for the trinomial 6*a*2*x*3 + 9*abx* + 15*cx*2,
the number 3 is the largest numerical factor common to 6, 9, and 15, and *x*
is the largest variable factor common to all three terms. Thus, the greatest
common factor of the trinomial is 3*x.*

**Least Common Multiples
**

Finding least common multiples is useful in combining algebraic fractions. The procedure is analogous to that used to combine ordinary fractions in arithmetic. To combine two or more fractions, the denominators must be the same; the most direct way to produce common denominators is simply to multiply all the denominators together. For example

But
*bd* may not be the *least* common denominator. For example

But 18 is only one possible common denominator; the least common denominator is 6, and

In algebra,
the problem of finding least common multiples of denominators is similar. Given
several algebraic expressions, the least common multiple is the expression of
lowest degree and least coefficient that can be divided evenly by each of the
expressions. Thus, to find a common multiple of the terms 2*x*2*y,*
30*x*2*y*2, and 9*ay*3, all three expressions could simply be
multiplied together, and it would be easy to show that (2*x*2*y*)(30*x*2*y*2)(9*ay*3)
is evenly divisible by each of the three terms; however, this would not be the
*least* common multiple. To determine which is the least, each of the terms
is reduced to its prime factors. For the numerical coefficients 2, 30, and 9,
the prime factors are 2, 2·3·5, and 3·3, respectively; the least common multiple
for the numerical coefficients must therefore be 2·3·3·5, or 90. Similarly,
because the constant *a* appears only once, it too must be a factor. Of
the variables, *x*2 and *y*3 are required, so that the least common
multiple of the three terms is 90*ax*2*y*3. Each term will evenly
divide this expression.

**Solution of Equations
**

Given
an equation, algebra proceeds to supply solutions based on the general idea
of the identity *a* = *a.* As long as the same arithmetic or algebraic
procedure is applied simultaneously to both sides of the equation, the equality
remains unaffected. The basic strategy is to isolate the unknown term on one
side of the equation and the solution on the other. For example, to solve the
linear equation in one unknown

the
variable terms are isolated on one side and the constant terms on the other.
The term 3*x* can be removed from the right side by subtracting; 3*x*
must then be subtracted from the left side as well:

The number 6 is then subtracted from both sides:

To isolate
*x* on the left side, both sides of the equation are divided by 2:

and
the solution then follows directly: *x* = 3. This can easily be verified
by substituting the solution value *x* = 3 back into the original equation:

*Solution of Quadratic
Equations *

Given any quadratic equation of the general form

a number of approaches are possible depending on the specific nature of the equation in question. If the equation can be factored, then the solution is straightforward. For instance:

First the equation is put into the standard form

which can be factored as follows:

This
condition can be met, however, only when the individual factors are zero—that
is, when *x* = 5 and *x* = -2. That these are the solutions to the
equation may again be verified by substitution.

**Method of Completing
the Square **

If, on inspection, no obvious means of factoring the equation directly can be found, an alternative might exist. For example, in the equation

the
expression 4*x*2 + 12*x* could be factored as a perfect square if
it were 4*x*2 + 12*x* + 9, which equals (2*x* + 3)2. This can
easily be achieved by adding 9 to the left side of the equation. The same amount
must then, of course, be added to the right side as well:

This can be reduced to

or

and

(because
º has two values). The first equation leads to the solution *x* = 1 (because
2*x* + 3 = 4, 2*x* = 1 [subtracting 3 from both sides], and *x*
= 1 [dividing both sides by 2]). The second equation leads to the solution *x*
= -7/2, or *x* = -31. Both solutions can be verified, as before, by substituting
the values in question back into the original equation.

*The Quadratic Formula
*

Any quadratic equation of the form

can
be solved using the quadratic formula. In all cases the two solutions of *x*
are given by the formula:

For example, to find the roots of

the equation is first put into the standard form

As a
result, *a* = 1, *b* = -4, and *c* = 3. These terms are then
substituted into the quadratic formula

*Solution of Two Simultaneous
Equations *

Algebra frequently has to solve not just a single equation but several at the same time. The problem is to find the set of all solutions that simultaneously satisfies all equations. The equations to be solved are called simultaneous equations, and specific algebraic techniques can be used to solve them. For example, given the two linear equations in two unknowns

a simple
solution exists: The variable *y* in equation (2) is isolated (*y*
= 5 - 2*x*), and then this value of *y* is substituted into equation
(1):

This
reduces the problem to one involving the single linear unknown *x,* and
it follows that

or

so that

When this value is substituted into either equation (1) or (2), it follows that

A faster method of solving simultaneous equations is obtained by observing that, if both sides of equation (2) are multiplied by 4, then

If equation
(1) is subtracted from equation (2), then 5*x* = 10, or *x* = 2. This
procedure leads to another development in mathematics, matrices, which help
to produce solutions for any set of linear equations in any number of unknowns
(*see* Matrix
Theory and Linear Algebra).
*See also* Mathematical
Symbols.

Contributed by:

Joseph Warren Dauben

J. Lennart Berggren