1.3.1 Introduction

What exactly are terms?

Terms are arithmetic expressions that are combinations of numbers, variables, brackets, and appropriate arithmetic operations.

Terms can be interpreted in two ways:

As functional expressions: If each variable contained in the term is substituted with a specific number, the term can be evaluated to a certain value. For example, $x + x - 1$ is a term; once $x = 2$ is inserted one gets the value $3$ . The expression $2 x - 1$ is a term as well, this term can be transformed into $x + x - 1$ , and hence it evaluates to the same value if $x = 2$ is inserted. As a symbolic expression, $x + x - 1$ is different from $2 x - 1$ , but as functional expressions they are both the same (equivalent): No matter which value is inserted for $x$ , both terms are always evaluated to the same value. A term can also be a value on its own, if no variables occur in it. For example, $3 \cdot (2 + 4)$ is a term with the value $18$ .
As evaluation rules: A term can be interpreted as a type of instruction how to calculate a new value from given values (inserted into the variables). For example, the term $x^{2} - 1$ can be read as "square the value of $x$ and subtract one from the result". This is different from the term $(x + 1) (x - 1)$ , even if both terms have the same value. The second term describes the evaluation as "add one to $x$ and multiply the result with the value, resulting if $x$ is subtracted by one". The two terms are mathematically equal. One writes $x^{2} - 1 = (x - 1) (x + 1)$ , but they represent two different ways for calculating the value. Depending on the problem setting, one of the two terms may be more convenient for solving the problem.