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·(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 x2 -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 x2 -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.