Algebraic expressions

An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7−’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it's undefined (in which case you can work that out).

An algebraic expression is any grammatically sensible expression made up of any or all of the following:

The arithmetic operations are not limited to those that I've discussed in this course; whenever you learn a new operation, you get more algebraic expressions to work with!

Types of expressions

A constant expression is an algebraic expression with no variables in it; it's the same thing as an arithmetic expression. A literal expression is an algebraic expression with no constants or operations in it; it simply consists of a single variable. Most algebraic expressions, of course, are more complicated than these.

In this course, we'll mostly be dealing with linear expressions. A linear expression is an algebraic expression made up only from any or all of these:

Note that the optional bits are really included in the previous items: taking the opposite is equivalent to multiplying by the constant −1, then subtraction is simply adding the opposite, and division by a nonzero constant is simply multiplication by some other constant (the reciprocal).

Typical linear expressions are things like x + 2, 2t + 4, and 5x − y. But notice some things that you can't do in a linear expression: you can't divide by a non-constant expression, or use exponentiation, or take absolute values. So for example, 1/x, y2, and |a| are not linear expressions.

A monomial is an algebraic expression made up only from any or all of these:

Again, the optional bits are really included in the previous items. For example, raising to a constant whole number is either the constant 1 (if the exponent is 0), trivial (if the exponent is 1), or made up of multiplication (if the exponent is more than 1).

Typical monomials are things like x2, 2xy, and −xy3/5. But again some things you can't do in a monomial: you can't add or subtract, or divide by a non-constant expression, or raise to a negative exponent, and you still can't take absolute values. So for example, x + 5, 1/x, y−2, and |a| are not monomials.

A polynomial is an algebraic expression made up only from any or all of these:

Basically, anything you can do in a linear expression or a monomial is fair game in a polynomial. Again, the optional bits are really included in the previous items.

Typical polynomials include the linear expressions and monomials above, as well as things like x2 + 5 and 2xy2 + 3x. But there are still some things you can't do in a polynomial: you still can't divide by a non-constant expression, raise to a negative exponent, or take absolute values. So for example, 1/x, y−2, and |a| are still not polynomials.

Finally, a rational expression is an algebraic expression made up only from any or all of these:

The name ‘rational expression’ comes from the fact that these are the same operations that you use in forming a rational number out of 0 and 1. Again, the optional bits are really included in the previous items; for example, division is the same as multiplication by a reciprocal, and raising to a negative integers as exponent is built out of the constant 1 and division.

Typical rational expressions include all the examples above, as well as things like 1/x, y−2, and even (2x2 + 5)/(4xy − 7z). But there are still a few things that you can't do in a rational expression; most of these are operations that you won't study until Intermediate Algebra, but you still can't take absolute values. So for example, |a| is still not a rational expression.

We really won't use rational expression in this course, except as occasional examples. We'll use polynomials a bit, but mostly this course is about linear expressions. However, we'll also study expressions that are almost linear but include absolute values; I don't know a special name for those.

Evaluating expressions

If you have an expression with a variable in it, and you later learn what the value of that variable is, then you can figure out what the value of the expression is. For example, if you have the expression x + 5 and later learn that x is 3, then you can change x + 5 to 3 + 5, which works out to 8. This is called evaluating an expression; you can also say that you substitute 3 for x.

The instruction to substitute 3 for x is often written as ‘x = 3’, because it's based on the idea that the variable x is equal to the number 3. But a more precise notation is ‘x := 3’, which indicates not only that x is equal to 3 but also that you want to use that fact in a certain way. The book uses the simpler notation (without the colon) and it's all right with me if you do that too, but I'm going to use the colon.

Notice that sometimes evaluating an expression shows it to be undefined. For example, if you have 5/t and you substitute t := 0, then you get 5/0, which is undefined. That's OK; just say that it's undefined, that's the answer.

An expression can have more than one variable in it; to evaluate the expression, you must substitute something for each variable. For example, in the expression x + y, if you substitute x := 3, you get 3 + y, which is still not evaluated. But if you substitute x := 3 and y := 4, then you get 3 + 4, which evaluates to 7. You can indicate both these substitutions at once as (x := 3, y := 4) or as (x, y) := (3, 4).

Different variables can be substituted with the same value or with different values, but a single variable can only be substituted with one value in any given situation. For example, in the expression x + x2, you can substitute x := 3 and get 3 + 32 (which evaluates to 12), or you can substitute x := 4 and get 4 + 42 (which evaluates to 20), but it makes no sense to use 3 + 42 (where the first x is 3 and the second is 4). The value of x may vary from situation to situation, but it can only be one thing at a time!

Often you want to put in extra parentheses when you substitute. For example, if you have 2 · x and you substitute x := 4, then you get 2 · 4, which is fine (and works out to 8). But if you abbreviate 2 · x as 2x (as we usually do in Algebra) and again substitute x := 4, you don't want to say simply 24! Instead, you must either put the dot back and use 2 · 4 again, or else put in some parentheses and use 2(4), which means the same thing. As another example, in the expression a2, if you substitute a := 3, then you get 32, which is fine (and works out to 9). But if you substitute a := −3 instead, then you don't want to say −32, because this means the opposite of 32, which is not want you want! Instead, you should say (−3)2, which means −3 times −3, which is what you want. That is, the correct answer is 9, not −9. If in doubt, use parentheses (or another grouping symbol) when substituting.

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