This post is a continuation of “What is a variable? Part I: Symbols.”
We can now precisely define the terms algebraic expression (both rational and polynomials), and numerical expressions in terms of constants (numbers), variables, operators, and functions.
Definition. An algebraic expression is either
- a numerical symbol or a variable symbol or,
- the result of placing previously generated algebraic expressions into the two blanks of one of the four operators ((__+__), (__−__), (__×__), (__÷__)) or into the base blank of an exponentiation with an exponent that is a rational number.
Expressions are recursively built from already known expressions. Here are the rules: start with all constant symbols (e.g. 3) and variables (e.g.. x), which are expressions by the first part of the definition. Next, insert constant or variable symbols into the blanks of operator symbols to create new expressions, e.g. (3×x). This new object is also an expression, which can then be inserted into further operator symbols. To keep track of the order in which we are creating an expression, we place parenthesis around expressions that are not just constant or variable symbols when we insert the into new operator blanks: ((3×x)×y).
For example, the expression ((3×x)×y)+5 is recursively built by noting:
- 3, x, y, and 5 are all expressions.
- (3×x) is an expression by placing 3 and x in the blanks of a multiplication symbol.
- ((3×x)×y) is an expression by placing (3×x) and y into the blanks of another multiplication symbol, using parenthesis () to keep track of the order of we used to create the expression.
- (((3×x)×y)+5) is an expression by placing ((3×x)×y) and 5 into the blanks of an addition symbol.
In practice, of course, teachers and students often skip the building-up process and just write down the expression ((((3×x)×y)+5)), using parentheses to indicate the order in which the expression was created. Unfortunately, the seemingly squirrelly topic of “order of operations” actually hides this absolutely clear building-up process from students, often causing confusion for students. Textbook School Mathematics exacerbates this problem with its emphases of “order of operations” as a set of rules to be memorize. On the other hand, parentheses can quickly become quite annoying for expressions with lots of symbols, like
so any reduction in the use of parentheses afforded by “order of operations” and other tricks (like using 3x instead of 3×x) is welcome relief.
In school, there is a curriculum sequence building up to the full definition of algebraic expression. Initially (grade 7 and below in the CCSS), students are basically working with algebraic expressions that are built out of the four operators, possibly with some simple exponentials (square, cube, etc.) included. In grade 8, students learn the rules of exponentials and increase their knowledge of algebraic expressions to include any integer exponents. Also in grade 8, students begin to study simple fractional exponents (using the square root and cubed root symbols). In Algebra I and more extensively in Algebra II, students are introduced to rational expressions (algebraic expressions generated only by the four operators, or equivalently, the four operators together with exponentiation by an integer power). Algebra I students study an important subset of the rational expressions—polynomials (expressions generated by numbers, variables, addition and multiplication). Two years later, in Algebra II, students work with the full definition of algebraic expression when they study rational expressions as well as expressions involving nth roots.
As the term expression grows in meaning, so does the term, numerical expression. In fact, it usually precedes algebraic expression. Numerical expression is introduced and/or the idea is used from 1st grade onward:
Definition. A numerical expression is an algebraic expression that contains only numerical symbols (no variable symbols) that evaluates to a single number.
- “3” when evaluated is the number three.
- “3×(4+17)” when evaluated is the number sixty-three.
- “3+” is not a numerical expression, nor is “3÷0”.
- In grade 11, “4 sin(30)” is a numerical expression.
A numerical expression is a way to represent or “express” a number. Algebraic expressions containing variables are incomplete thoughts in the sense that they are waiting for numbers to assigned to the variables (after which they become numerical expressions). For example, x+3 is not a number until we substitute a number, say 5, in for x. The result then denotes a number, in this case 8.
It is easy to tell when two numerical expressions are equivalent—just evaluate both of them to see if they are the same number. Two expressions are equivalent if, whenever the same number is assigned to the corresponding variable in each expression for all variables, then the resulting numerical expressions are equivalent.
We investigate equality of two expressions in Part III.
CHANNEL: Engineering School Mathematics