For about a decade now, mathematicians and mathematics educators have been weighing in on a particular debate rooted in school mathematics that shows no signs of abating.<\/p>\n
The debate, covered by Slate, Popular Mechanics, The New York Times and many other outlets, is focused on an equation that went so \u201cviral\u201d that it, eventually, was lumped with other phenomena that have \u201cbroken\u201d or \u201cdivided\u201d the internet.<\/p>\n
On the off chance you\u2019ve yet to weigh in, now would be a good time to see where you stand. Please answer the following:<\/p>\n
8\u00f72(2+2)=?<\/p>\n
If you\u2019re like most, your answer was 16 and are flabbergasted someone else can find a different answer. Unless, that is, you\u2019re like most others and your answer was 1 and you\u2019re equally confused about seeing it another way. Fear not, in what follows, we will explain the definitive answer to this question \u2014 and why the manner in which the equation is written should be banned.<\/p>\n
Our interest was piqued because we have conducted research on conventions about following the order of operations \u2014 a sequence of steps taken when faced with a math equation \u2014 and were a bit befuddled with what all the fuss was about.<\/p>\n
Clearly, the answer is\u2026<\/p>\n
Two viable answers to one math problem? Well, if there\u2019s one thing we all remember from math class: that can\u2019t be right!<\/p>\n
Many themes emerged from the plethora of articles explaining how and why this \u201cequation\u201d broke the internet. Entering the expression on calculators, some of which are programmed to respect a particular order of operations, was much discussed.<\/p>\n
Others, hedging a bit, suggest both answers are correct (which is ridiculous).<\/p>\n
The most dominant theme simply focused on implementation of the order of operations according to different acronyms. Some commentators said people\u2019s misunderstandings were attributed to incorrect interpretation of the memorized acronym taught in different countries to remember the order of operations like PEMDAS, sometimes used in the United States: PEMDAS refers to applying parentheses, exponents, multiplication, division, addition and subtraction.<\/p>\n
A person following this order would have 8\u00f72(2+2) become 8\u00f72(4) thanks to starting with parentheses. Then, 8\u00f72(4) becomes 8\u00f78 because there are no exponents, and \u201cM\u201d stands for multiplication, so they multiply 2 by 4. Lastly, according to the \u201cD\u201d for division, they get 8\u00f78=1.<\/p>\n
Were different ways of teaching the order of operations responsible for confusion? (Shutterstock)
\nBy contrast, Canadians may be taught to remember BEDMAS, which stands for applying brackets, exponents, division, multiplication, addition and subtraction. Someone following this order would have 8\u00f72(2+2) become 8\u00f72(4) thanks to starting with brackets (the same as parentheses). Then, 8\u00f72(4) becomes 4(4) because (there are no exponents) and \u201cD\u201d stands for division. Lastly, according to the \u201cM\u201d for multiplication, 4(4)=16.<\/p>\n
Do not omit multiplication symbol
\nFor us, the expression 8\u00f72(2+2) is syntactically wrong.<\/p>\n
Key to the debate, we contend, is that the multiplication symbol before the parentheses is omitted.<\/p>\n
Such an omission is a convention in algebra. For example, in algebra we write 2x or 3a which means 2 \u00d7 x or 3 \u00d7 a. When letters are used for variables or constants, the multiplication sign is omitted. Consider the famous equation e=mc2, which suggests the computation of energy as e=m\u00d7c2.<\/p>\n
The real reason, then, that 8\u00f72(2+2) broke the internet stems from the practice of omitting the multiplication symbol, which was inappropriately brought to arithmetic from algebra.<\/p>\n
Inappropriate priority
\nIn other words, had the expression been correctly \u201cspelled out\u201d that is, presented as \u201c8 \u00f7 2 \u00d7 (2 + 2) = ? \u201d, there would be no going viral, no duality, no broken internet, no heated debates. No fun!<\/p>\n
Had the problem been correctly presented as 8 \u00f7 2 \u00d7 (2 + 2) = ?, there would be no heated debate. (Egan J. Chernoff), Author provided
\nUltimately, omission of the multiplication symbol invites inappropriate priority to multiplication. All commentators agreed that adding the terms in the brackets or parentheses was the appropriate first step. But confusion arose given the proximity of 2 to (4) relative to 8 in 8\u00f72(4).<\/p>\n
We want it known that writing 2(4) to refer to multiplication is inappropriate, but we get that it\u2019s done all the time and everywhere.<\/p>\n
Nice symbol for multiplication
\nThere is a very nice symbol for multiplication, so let\u2019s use it: 2 \u00d7 4. Should you not be a fan, there are other symbols, such as 2\u20224. Use either, at your pleasure, but do not omit.<\/p>\n
As such, for the record, the debate over one versus 16 is now over! The answer is 16. Case closed. Also, there should have never really been a debate in the first place.<\/p>\n
Egan J Chernoff, Professor of Mathematics Education, University of Saskatchewan and Rina Zazkis, Professor, Faculty of Education, Simon Fraser University<\/p>\n
This article is republished from The Conversation under a Creative Commons license. Read the original article.<\/p>\n","protected":false},"excerpt":{"rendered":"
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