Last week, a seemingly simple math problem caused a deep division among DailyMail.com readers, sparking an argument so furious it resembled an angry political debate.
A British mathematician and TV presenter, Carol Vorderman, was furious enough to take to Twitter to call it an “attempt to bend the truth/rules of algebra” and demand an apology from all math teachers.
The most popular comment ever stated that the answer is “Absolutely definitely 1 and I will fight anyone who says otherwise.”
Dailymail.com spoke to math experts to understand how the rage of algebra could become such a polarizing topic that people were willing to fight over it. Most importantly, we wanted to find out once and for all who is right.
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 subject to interpretation, the symbols invented to describe them are.
It is a cleverly devised problem. It’s like it’s designed to maximize confusion,” said Steven Strogatz, a mathematics professor at Cornell University who has written about this type of equation. before. “You have to admire the cunning of whatever imp created them,” she added.
The issue raised last week was very divisive.
Last week’s problem was posed as above.
Many people were convinced that the only true answer is 1. However, some computers and calculators produced an answer of 9.
‘It’s not as simple as (one) thinks. I might have thought about it at first…but when you really dive in, there’s a lot going on,” Strogatz said.
Many simplified 6 ÷ 2(1 + 2) to 6 ÷ 2(3).
It was indisputable that the first operation to be treated was the sum of 1 and 2
But the order in which the operations ought be applied from here is not aim. So it came down to which part of the equation you did first: expand the parentheses or do the division.
PEMDAS (or BIDMAS/BODMAS, as it is usually taught in the UK) is the simplistic formula given to primary school students.
The strict and pedantic application of PEMDAS says that calculations should be done from left to right because division and multiplication have the same priority. That makes the answer 9.
Expressing the original problem in this less elegant way and pedantically applying PEMDAS yields an answer of 9
However, many readers applied a more sophisticated convention which dictates that ‘implicit’ multiplication takes precedence over ‘explicit’ multiplication or division.
This means that when multiplication is implicit in a bracket, in this case 2(3), it is processed before multiplication or division, which is explicitly signaled by an ‘x’ or ‘÷’ symbol.
‘The whole problem boils down to whether implicit multiplication has higher priority than explicit division. With the symbol 6 divided by 2, there is an explicit division symbol, but there is only an implicit multiplication symbol,” Strogatz said.
This approach is clearly indicated when the problem is expressed in a way 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 used very rarely in high school math and beyond
Last week’s issue sparked 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 devised.”
What makes the original problem especially confusing is that it combines basic notation (the division symbol) with implicit multiplication (the brackets).
Further evidence that the original problem interpretation is not black and white lies in the fact that calculators and other computational ‘answer engines’ treat 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 because they also specify how they chose to interpret the confusing division symbol.
PEMDAS (or BIDMAS)
The acronym describes the order in which operations must be performed:
P – parentheses
E – Exponents
M – Multiplication
D – Division
A – Addition
S – Subtraction
Strogatz pointed to another programming language, Julia, which is capable of applying the second approach that considers implicit multiplication.
High school math students may remember that electronic calculators would solve most problems accurately, but for PEMDAS-type questions, they would often apply operations in an order that would produce answers that teachers would consider incorrect.
In fact, Physical Review Letters, one of the most renowned academic journals in physics, provides editorial guidance for avoiding dilemmas exactly like this.
Strogatz pointed to that need to clear up any misunderstanding as proof that there is no universally accepted convention.
“People need to be told,” he said.
Presh Talwalkar, who created the YouTube video from which last week’s puzzle emerged, agreed that the solution isn’t black and white.
“We have to remember 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,’ she 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 completed a degree in mathematics and can honestly say that we never had any discussions about ambiguous expressions. Most textbooks are precise in the definition of a convention or in the careful use of parentheses.’
Strogatz added that while the controversy surrounding the problem is divisive, it is ultimately not a matter of mathematics.
“It is something interesting, more sociologically than mathematically or pedagogically. There are a lot of different things that play here,’ she said.