#### Chapter 1 Elementary Arithmetic

Section 1.3 Transformation of terms

# 1.3.1 Introduction

What exactly are terms?
##### Info 1.3.1

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 $2x-1$ 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, $3·\left(2+4\right)$ 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 $\left(x+1\right)\left(x-1\right)$, 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 ${x}^{2}-1=\left(x-1\right)\left(x+1\right)$, 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.