A couple of days ago, I got a text from my boss. Usually, a text from the editor means breaking news is afoot - something big is happening and you need to be on it, now. Drop what you're doing; whatever is on the other end of this message needs immediate attention. Stat. Pronto. Straight away.
The text was a photograph of a mathematics problem: 8 divided by 2(2+2) = ....
A brief, fleeting, existentially horrifying second of thinking "Oh, god, they've sent me the launch codes", quickly and mercifully gave way to a milder form of panic as the slow recall to school maths classes came lurching out of a dark and repressed recess of my brain.
A call followed to explain: The problem had been posed at a primary school trivia competition a few nights back. It was presented in a line - no fractions - and to get the points you had to solve it.
Now class, before we get too far ahead, who would like to take a moment to solve it? Bonus points if you show your working (topics@newcastleherald.com.au for proof of solving).
I'll wait right here ....
Ok, are you back? Great. Let's see how we all did, starting with the school tutor and former teacher who was in the trivia comp at the time, and who posts our first recorded answer as 16.
I'm assuming at least a few of you are feeling pretty chuffed right now, but before we start handing out the ribbons, we need to address the whole controversy that caused this problem to be texted to my boss (and then to me) in the first place.
The quizmaster says the answer is 1.
And now, I reckon about half of you are throwing the paper across the room and the other half are madly Googling for that one Brooklyn 99 "Vindication" gif to flash about on your victory lap.
When I, a less-than-impressive maths student, subsequently arrived at both answers, I decided to call in help.
Things, meanwhile, were starting to get serious. HSC marks were being quoted as credentials in the Journo group chat as we all had a crack at the problem. Wait staff had reportedly been roped in at one stage. ChatGPT was consulted. My girlfriend and her mum had both arrived at different answers, and had called in friends to back them up, including at least one former maths teacher (answer: 1) and a civil engineer (answer: pending).
Reputations were on the line. I had to call in the big guns. And that's how I ended up on the phone to Emeritus Professor of Mathematics at the Univeristy of Newcastle, Dr George Willis.
The problem, Dr Willis explained, is that the equation as it was stated - in a single line - is ambiguous. To use an English analogy, it's like putting a comma in the wrong place in a sentence and changing the meaning of the whole thing. Depending on the order that you solved it, you can either come to 16 or 1 as the answer - and only one of those is slightly more correct than the other.
Dr Willis suggested that writing the question as a fraction - 8 / 2(2+2) - would be clearer, and would suggest that the correct answer was, in fact, 1. Incorporating more brackets could help as well, but too many and you go full circle and end up back at 16:
8 / (2(2+2)) = 8 / (2x2 + 2x2) = 8 / 8 = 1
But: (8/2)(2+2) = 4(4) = 16
If I really wanted to resolve this argument once and for all, Dr Willis said (and I began to feel that old school days dread of knowing homework was forthcoming), I should go away and learn about Reverse Polish Notation, which removes the brackets altogether and uses a "postfix" logical expression that is so unambiguous it's typically used to code computers.
You see, the maths problems you and I are probably most accustomed to seeing are written in an "infix" style, which means the symbols are placed in between the numbers (eg. 2 + 2 = 4) and, in more complicated equations, you have to know which order to approach them.
A postfix or Reverse Polish Notation places the symbols at the end of the expression (eg. 22+ = 4) and is meant to remove some of the doubt about the order of operations by - in the logical sense - listing the numbers and then telling you what to do with them ie. "2, 2, add them together".
For more complex equations that have multiple operations, Reverse Polish Notation uses a logical method of stacking numbers and symbols to resolve the equation in order.
Our same trivia problem written out in Reverse Polish Notation reads as: 8222+x/ (where the slash indicates a division symbol). And it's solved by descending the "stack" of numbers, first by adding the two 2s, then multiplying the result (4) with the remaining 2, then dividing that total (8) by the final number 8. And the answer is 1.
(It was about this point that I needed a Bex and a lie down.)
But you're probably wondering where all this leaves us, and whether our tutor friend was robbed on trivia night.
Well, Dr Willis says the question as stated in the quiz is unclear and poorly articulated. It's true that mathematics is an axiomatic language - things are typically either one thing or another, and the arts kids can argue over the grey areas - but in this case at least, "you shouldn't feel bad about getting it wrong," he says.
If the question was more clearly articulated, the answer would be more obvious, but as-written 16 might not be entirely correct, but it's somewhat logical.
So, now I'm off to find the federal watchdog for trivia night quizzes. I have a report to file and an injustice to expose.
WHAT IS THE ORDER OF OPERATIONS IN MATHEMATICS?
Solving mathematics problems follows a convention called the order of operations, or the order in which you should approach any maths problem.
One abbreviation is PEMDAS. Another is BODMAS. But they both follow the same rule:
- Brackets or parentheses come first
- Then solve any exponents (numbers to a power)
- Then multiplication and/or division
- And finally addition and/or subtraction.
BRIEF TOPICS OF NOTE
SPECIAL DELIVERY:
There's service and then there's *service* - and when this epic photo of a Lynch's Hub barista passing an order onto a floating tug in the Harbour surfaced this week, I think it's probably safe to say just about every bartender and waiter in the city did a double take. Service to your tug?! Outstanding!
THINGS TO DO:
For one weekend only, Newcastle Museum will be hosting one of Australia's best model railways in the city that inspired it.
Ross Balderson and a small group of fellow enthusiasts have worked together to create a stunning representation of the Newcastle Railway station precinct in 1899. The exhibition opens on July 7.
GOOD NEWS:
A Stockholm building company has set out to build an entire Swedish suburb from timber to prove the versatility of the material.
The project spans 25 blocks, with 30 sustainable mass timber buildings to add 7000 new offices and 2000 new homes. Construction is slated to begin in 2025 with first builds finished by 2027.
