This archive report was first published on 5 August 2019.
Mathematicians have long been drilled in the order of operations, which dictates the order in which mathematical operations are carried out. However, a recent debate has highlighted the complexity of this convention, particularly when it comes to implicit multiplication.
At the heart of the debate is the equation 8 ÷ 2(2+2). Some argue that the answer is 16, while others claim it's 1. The issue lies in the interpretation of the implicit multiplication in the expression 2(2+2).
According to the more sophisticated convention used in algebra, implicit multiplication is given higher priority than explicit multiplication or division. Under this convention, the expression 2(2+2) should be evaluated first, yielding 8÷8 = 1.
However, this convention is not universally adopted. Calculators built into Google and WolframAlpha use the more elementary convention, which makes no distinction between implicit and explicit multiplication. As a result, they interpret 8÷2(2+2) as 8÷2×(2+2) = 8÷2×(4), and arrive at an answer of 16.
Other commenters have objected to the original question itself, arguing that it is poorly posed and could have been made clearer with additional parentheses. However, this misses the point: the question was not meant to ask anything clearly, and its obscurity seems almost intentional.
Steven Strogatz, a professor of mathematics at Cornell, notes that the equation uses parentheses in a 'jujitsu manner' to exacerbate the murkiness. He argues that the viewer is left wondering whether to use the sophisticated convention for implicit multiplication from algebra class or to fall back on the elementary PEMDAS convention from middle school.
Ultimately, there is no clear-cut answer to this question. Some spreadsheets and software systems flatly refuse to answer the question, and most mathematicians agree that it's best to ask a clearer question if you want a clearer answer.
