Algebraic Expressions

The essence of algebra is the use of what are called algebraic expressions. Algebraic expressions are very similar to arithmetic expressions except that there may be symbols in place of some, or all, of the numbers. In High School algebra the symbols used are almost always lower case Roman letters: a, b, c, ..., x, y, z (we will print these symbols in an italic (i.e., slanted) font and in blue to make it easier to distinguish the symbols from the other text). In mathematics beyond the High School level it is common to use many other alphabets and fonts. Here are some examples of algebraic expressions

x+3

x - y

3(x + 17)

4xy + 7x -2y

ax²+bx + c

As you can see from these examples, algebraic expressions look very similar to general arithmetic expressions, the only difference is that some numbers may have been replaced by letters.

In an algebraic expression such as 4xy + 7x -2y the numbers, 4, 7, and -2 are called coefficients and the symbols, x and y are called variables.

Given an algebraic expression, e, we write Var(e) for the set of all variables occuring in e.

Given a set, V, of variables we write Exp(V) for the set of all expressions e such that var(e) V. Note, for some applications it might be wise to extend this notation so that it also indicated what number system was being used.

Sometimes it is also useful to use letters to represent the coefficients. Thus in an algebraic expression such as ax²+bx + c the letters a, b and c are coefficients. When letters are used for both coefficients and variables it may be hard to tell which is which. Indeed the distinction is not a sharp one and reflects intended use rather than a fundamental distinction. The intuitive idea is that a coefficient represents a constant (a number that will not change "during the solution of a problem") while a variable represents a "varying value". This intuition should become clearer over time.

The general convention is to use letters from the front of the alphabet for coefficients and letters from the end of the alphabet for variables. Furthermore, by convention, the coefficients are written to the left of the variables, that is, we write ax rather than xa even though xa = ax in all the number systems with which we deal in high school algebra.

Assignments and the evaluation of algebraic expressions

Given any algebraic expression we can make it into an arithmetic expression by assigning a number to each variable in the expressing and then transforming the algebraic expression into an arithmetic expression by replacing each variable by its assigned number For example, given the algebraic expression, 4xy + 7x -2y , then taking

if we assign x the value 1 and assign y the value 1, we get the arithmetic expression 411 + 71 - 21 = 9

if we assign x the value 2 and assign y the value 3, we get the arithmetic expression 423 + 72 - 23 = 32

if we assign x the value 1/2 and assign y the value 2/3, we get the arithmetic expression 41/22/3 + 71/2 - 22/3 = 7/2
etc.

Mathematically speaking, an assignment for an algebraic expression, e, is a function, a:VN, from any set, V, containing all the variables in the algebraic expression e, to the set, N, of numbers in the underlying number system. Given any set of variables, V, then every function f:VN, is an assignment for any expression all of whose variables are in V. The process of applying an assignment, a:VN, to the variables in an expression, eExp(V) to get an arithmetic expression which is then evaluated to give an element of N, gives us a "rule" which defines an new function a^# :Exp(V)N.

As these example should suggest: whatever number system we want we have that for every choice of values for x and y we get an arithmetic expression, Thus we can think of an algebraic expression as an abbreviation for an infinite number of arithmetic expressions.