Intro to Algebra Click
here for a history of Algebra
The language of Algebra
Summary and Review:
Algebra is a 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 such mathematical relations 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 produce specific instances of these relations
(for example, 3, 4, and 5, where 3^{2} + 4^{2} = 5^{2}). 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 3^{2};
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 employs arithmetic operations to
establish procedures for manipulating 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 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.
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 may represent either constants or variables. Letters 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.
Signs of Aggregation and Operation
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:
2+4x
x
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
ax + b
c  dy
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:
Carry out all operations within a group first, beginning with the innermost group.
Expressions:
An expression is made up of Variables,
Numbers, and Operations.
Examples:
a. 2x+3b5
b. 3x^{2}2+4xy
c. (3x+4)/(92x)
d. (3x+2)(x3)
Equivalent Expressions:
Expressions that represent the same information.
Examples:
a. x+x is equivalent
to 2x
b. x(x+2)(x+1) + x(x+2)(x+1) is equivalent to 2x(x+2)(x+1)
Real Numbers:
Numbers that are not imaginary (such as 1, 73, 5, 49/12)
Natural Numbers
1,2,3,4
Whole Numbers
0,1,2,3
Integers
3,2,1,0,1,2,3
Prime Number
A number divisible only by one and itself
Rational Numbers
Numbers that are in a ratio. ( a/b , 2/3 , 2/1)
Rational numbers can be represented as terminating or
nonterminating
Terminating 4/5 = .8 

Nonterminating 2/3 = .33333333 etc. 
Irrational
Numbers
are nonterminating and nonrepeating decimals

