Last week, a seemingly simple math problem caused deep divisions among DailyMail.com readers, sparking an argument so furious it resembled a furious political debate.
A British mathematician and TV presenter, Carol Vorderman, was furious enough to take to Twitter to call it an ‘attempt to change the truth/rules of algebra’ and demand an apology from all math teachers.
The story’s most popular comment said that the answer is “Absolutely 1 and I will fight anyone who says otherwise.”
Dailymail.com spoke to math experts to understand how algebra rage could become a topic so polarizing that people were willing to fight over it. Most importantly, we wanted to know who is right once and for all.
Unfortunately, you may not be satisfied with the answer, which is that there is no “objectively” correct solution. That’s because while the principles of mathematics are not open to interpretation, the symbols invented to describe it are.
‘It is a very artfully conceived problem. It’s like it was designed to maximize confusion,” said Steven Strogatz, a math professor at Cornell University who has written about this type of equation. for. “You have to admire the cunning of the little devil who made them,” he added.
The problem that occurred last week caused a lot of division
Last week’s problem was posed as above.
Many people were convinced that the only truthful answer is 1. However, some computers and calculators gave an answer of 9.
“It’s not cut and dried like that.” [one] thinks. I may have thought so at first… but when you really dive in, a lot happens,” Strogatz said.
Much simplified 6 ÷ 2(1 + 2) to 6 ÷ 2(3).
There was no doubt that the first operation to be performed was the addition of 1 and 2
But the order in which operations should be applied from here is not objectively. Therefore, everything came down to which part of the equation you did first: expanding the hook or doing the division.
PEMDAS (or BIDMAS/BODMAS as it is often taught in the UK) is the simplistic formula taught to primary school students.
Strict and pedantic application of PEMDAS says that calculations should be made from left to right, because division and multiplication have the same priority. That makes the answer 9.
Phrasing the original problem in this less elegant way and pedantically applying PEMDAS yields an answer of 9
However, many readers adopted a more sophisticated convention that dictates that “implicit” multiplication takes precedence over “explicit” multiplication or division.
This means that when multiplication is implied by a parenthesis – in this case 2(3) – it is processed before multiplication or division – which is explicitly indicated by an ‘x’ or ‘÷’ symbol.
“The whole question boils down to this: Is implicit multiplication given higher priority than explicit division? With the symbol 6 divided by 2, there is an explicit division symbol, but there is only an implied multiplication symbol,” Strogatz said.
This approach is clearly indicated when the problem is expressed in a form that excludes the division symbol and uses fractions instead:
Expressing the original problem in this notation makes solution 1 seem more intuitive. The division symbol (÷) is very rarely used in high school and beyond math
Last week’s issue provoked a furious reaction from many, including British TV personality Carol Vorderman (pictured), who said the answer was clearly 1.
Steven Strogatz, a professor of applied mathematics at Cornell University, described the class of problems as “artfully conceived”
What makes the original problem especially confusing is that it combines basic notation (the division symbol) with implied multiplication (the parentheses).
Further evidence that the interpretation of the original problem is not black and white lies in the fact that calculators and other calculators handle it differently.
Wolfram Alpha, one of the most established semantic search engines, as well as Google, give the answer as 9. They are not wrong, as they also specify how they chose to interpret the confusing division symbol.
Strogatz pointed to another programming language, Julia, which is capable of applying the second approach that takes implicit multiplication into account.
High school math students may recall that electronic calculators would accurately solve most problems, but that for PEMDAS-like questions, they would often apply operations in a sequence that would produce answers that teachers consider incorrect.
In fact, the Physical Review Letters, one of the most recognized academic journals in the field of physics, provides editorial guidance on how to avoid exactly this kind of dilemma.
Strogatz pointed to the need to clear up any misunderstanding as evidence that there is no one universally accepted convention.
“People need to be told,” he said.
Presh Talwalkar, who created the YouTube video that spawned last week’s puzzle, agreed that the solution isn’t black and white.
“We can’t forget that this question went viral years ago because two calculators gave different answers. If you group, the answer is 1. If you use PEMDAS, the answer is 9,” he said.
“In general, mathematical notation can have multiple meanings,” he said. So I think it’s important to know the context of the question. I studied math and I can honestly say we’ve never argued over ambiguous expressions. Most textbooks are precise in defining a convention or using brackets carefully.’
Strogatz added that while the controversy surrounding the problem is divisive, ultimately it’s not a matter of math.
‘It’s quite interesting, more sociologically than mathematically or pedagogically. There’s a lot of different things at play here,” he said.