Confusion Lurks in Even Basic Symbols
A previous blog discussed confusing points relating to order of operations. Unfortunately, confusion can arise even when using just one operation, namely division.
The first symbol students learn for division is ÷, but sometime in late elementary school or middle school, the ÷ is generally replaced by the /. In theory, both symbols indicate division, but there is a difference in practice, especially as the symbols are used in computer applications. The ÷ is not generally found on keyboards, but it is extended ASCII character 246. On Windows-based systems, you can produce this symbol in many programs by holding down the Alt key while typing 246 on the numeric keypad.
Most computer programming languages use the / to indicate division, and computer software often does not even recognize the ordinary meaning of the ÷ symbol. For example, some search engines will evaluate 1/2 / 1/2 as an arithmetic expression but treat 1/2 ÷ 1/2 as an ordinary search. One well-known search engine takes an interesting approach: it treats 1/2 / 1/2 as the division problem 1 divided by 2 then 1 then 2 and produces the answer 0.25. However, the same search engine interprets 1/2 ÷ 1/2 as division of the fraction one-half by itself and produces the answer 1. Both interpretations are defensible, but the different results mean that / and ÷ are not the same thing.
These observations are not intended as criticism of any particular software program; the unavoidable fact is that our system of mathematical notation contains oddities that lead to ambiguity. Smart, well-intended computer programmers make reasonable decisions, and reasonable minds can differ. Perhaps one day our system of mathematical notation will evolve to eliminate such basic ambiguity, but until then we should use parentheses, horizontal fraction bars, and other aids to clarify meaning where necessary and avoid nasty bugs and errors that can arise from misunderstandings.