#### Chapter 1 Elementary Arithmetic

**Section 1.3 Transformation of terms**

# 1.3.1 Introduction

What exactly are terms?

Terms can be interpreted in two ways:

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 $2x-1$ 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 $2x-1$, 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, $3\xb7(2+4)$ 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.