Why Multiplication Has Higher Priority than Addition: A Pedagogical Remark

Abstract

Traditionally, multiplication has higher priority over addition; this means that there is no need to add parentheses if we want to perform multiplication first, and we need to explicitly add parentheses if we want addition to be performed first. Why not use an alternative arrangement, in which addition has higher priority? In this paper, we explain the traditional priority arrangement by showing that in the general case, the traditional arrangement allows us to use fewer parentheses than the alternative one. 1 Why Multiplication Has Higher Priority than Addition: Formulation of the Problem Multiplication has higher priority than addition: a reminder. In our usual arithmetic notations, multiplication has priority over addition. This means that if the arithmetic expression has no parentheses, e.g., has the type a+ b · c, then we: • first multiply b and c, and then • add a to the resulting product. If we want to perform addition first, then we have to add parentheses. For example, if we want to first add a and b, and then multiply the sum by c, then we have to write an expression (a+ b) · c. Why? A natural question is: why does multiplication have higher priority than addition? Why not consider addition a higher-priority operation, so that: