Introduction to Algebra

Intro to 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 3² + 4² = 5²). But algebra can make a purely general statement that fulfills the conditions of the theorem: a² + b² = c². 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 a² (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+3b-5
b. 3x²-2+4xy
c. (3x+4)/(9-2x)
d. (3x+2)(x-3)

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)