Elementary Algebra With Geometry

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. The major difference between algebra and arithmetic is the inclusion of variables. While in arithmetic only numbers and their arithmetical operations (such as , −, ×, ÷) occur, in algebra, one also uses symbols such as x and y, or a and b to denote variables.

Features of algebra

Variables

The purpose of using variables, symbols that denote numbers, is to allow the making of generalizations in mathematics. This is useful because:

• It allows arithmetical equations (and inequalities) to be stated as laws (such as a b = b a for all a and b), and thus is the first step to the systematic study of the properties of the real number system.
• It allows reference to numbers which are not known. In the context of a problem, a variable may represent a certain value which is not yet known, but which may be found through the formulation and manipulation of equations.
• It allows the exploration of mathematical relationships between quantities (such as "if you sell x tickets, then your profit will be 3x − 10 dollars").

Expressions

In elementary algebra, an expression may contain numbers, variables and arithmetical operations. These are conventionally written with 'higher-power' terms on the left (see polynomial); a few examples are:

x 3\,

y^{2} 2x - 3\,

z^{7} a(b x^{3}) 42/y - \pi.\,

In more advanced algebra, an expression may also include elementary functions.

Operations

Properties of operations

• The operation of addition...
• • has an inverse operation called subtraction: (a b) − b = a, which is the same as adding a negative number, a − b = a (−b);
  • The operation of multiplication...
  • means repeated addition: a × n = a a ... a (n number of times);
  • has an inverse operation called division that works for non-zero numbers: (ab)/b = a, which is the same as multiplying by a reciprocal, a/b = a(1/b);
  • distributes over addition: (a b)c = ac bc;
  • is abbreviated by juxtaposition: a × b ≡ ab;
  • The operation of exponentiation...
  • means repeated multiplication: a ⁿ = a × a ×...× a (n number of times);
  • has an inverse operation, called the logarithm: a ^{log _a b} = b = log _a a ^b ;
  • distributes over multiplication: (ab) ^c = a ^c b ^c ;
  • can be written in terms of n-th roots: a ^m/n ≡ ( ⁿ √ a ) ^m and thus even roots of negative numbers do not exist in the real number system. (See: complex number system)
  • has the property: a ^b a ^c = a ^{b c} ;
  • has the property: (a ^b ) ^c = a ^bc .
  • in general a ^b ≠ b ^a and (a ^b ) ^c ≠ a ^{(b c )} ;

Order of operations ...