The (Not Really) ‘Impossible Order of Operations’ Problem
Tuesday 3 November 2020, 8.00pm HKT
WE’VE all seen this at some point in our lives:—
6 ÷ 2 ( 1 + 2) = ?
We all know it’s bait meant to start an online argument, especially to pit Americans with the rest of the world.
The ‘trick’ is that it’s banking on the average American’s poor numeracy and poor understanding of arithmetic and mathematical notation.
THE WRONG IMMEDIATE ACTION
The immediate action is to clarify whether (1+2) is attached to the whole “6 ÷ 2” or just the “2.” It’s also the wrong immediate action, by the way.
THE RIGHT IMMEDIATE ACTION
If you know what the notation in 6÷2(1+2) means, then you will know what to do — and ace it like a walking scientific calculator.
The proper immediate action is to stop using the arithmetic division symbol (÷) and put the whole thing into standard mathematical format using the fractional notation.
If you know your basic arithmetic and basic algebra, you need not even ask.
WHAT IT REALLY MEANS
In all of arithmetic and mathematics, the notation 6÷2(1+2) means this:—

which otherwise means

All scientific and algebraic calculators give the same answer — 9. So does Excel and Wolfram Alpha.




NOT THE MEANING YOU THINK IT MEANS
What 6÷2(1+2) doesn’t mean is this:—

So,

WHERE’S THE PROBLEM?
Americans (that is, USA people) read 6÷2(1+2) mostly in both linear and literary fashion — treating the thing like a sentence and the numbers like words — so that the (1+2) is attached to the deniminator 2 instead of the numerator 6 as per normal fractional multiplication.
PEMDAS is also problematic. BODMAS is taught as standard everywhere except in the USA. You’ll see why a bit later.
THREE WAYS TO SOLVE IT, BUSTER
- Either use distributive property
- Or solve in pairs
- Or use BODMAS
It doesn’t matter which one to use. The answer is always 9. Always.
The simplest and most error-free is with BODMAS. You can do it linearly or notationally.
Linear BODMAS
Solve the bracketed stuff first but don’t bracket the result after solving it, or you will end up with the wrong order of operations.
6÷2 (1+2)
= 6÷2 × 3
= 3 × 3
= 9
Notational BODMAS (a.k.a. solve in pairs)
Fix the 6÷2 into an actual fraction. Then solve the bracketed stuff and the fraction separately.
6÷2 (1+2)
= 6/2 × (1+2)
= 3 × 3
= 9
Using distributive property
Treat 6÷2 as a whole entity and multiply it separately against the 1 and 2 in (1+2).
[6÷2] (1+2)
First [6÷2] × (1) = 3 × (1) = 3
Then [6÷2] × (+2) = 3 × (+2) = +6
Finally 3 + 6 = 9
Protip:— Since we have the two blocks [6÷2] and (1+2), solve the division block first and multiply that result with the 1 and 2 separately in the other block.
[6÷2] (1+2)
= [3] (1+2)
= [3×1] + [3×2]
= 3 + 6
= 9
THE PROOF, ONCE AND FOR ALL
Let’s put to rest this type of retarded online bait with the proof, once and for all.
In mathematics, what we don’t know is called a variable. We call that x.
6÷2 (1+2) = x
Now the part that is confusing you is (1+2) — let’s get rid of it.
So do unto one side as done to the other:—

The (1+2) cancels out on one side. Fix the 6÷2 into an actual fraction, and we are left with:—

Solve all we can for now:—

Isolate x:—

Solve out the brackets:—

Therefore the answer is x = 9
I’m not kidding when I say nearly all junior high students around the world outside the USA can do the above.
PEMDAS IS DYSLEXIC

This equation 6÷2(1+2) = ? in fact is a good test that PEMDAS doesn’t work.
6÷2 (1+2)
Step One — solve the brackets
6÷2 (1+2)
= 6÷2(3)
Step Two — multiply cuz PEMDAS
6÷2(3)
= 6÷6
Step Three — divide cuz PEMDAS
6/6
= 1
Result: Wrong!
Step Two is where the average American is fallen on. Once you’ve solved the bracketed stuff, don’t keep the result bracketed. The teachers mean well to keep the brackets to aid visualisation, but that just wreaks mathematical understanding.
GUESS WHO DIDN’T LEARN HOW TO READ
But even if Step Two didn’t have an internal result in brackets, PEMDAS is so flawed that it just defies belief.
Division has priority over multiplication. End of discussion.
Does that compute, or do I have to draw you a schematic?
Guess what, when I was a kid and grew up for a while in the USA, the “America, Fuck Yeah!” memory aid taught to us was THIS:—
- PEDMAS — not PEMeffingDAS
It had always been PEDMAS in America all the way back to the 1940s, and then before that, BODMAS or BIDMAS.
Are there no Americans left who remembers this stuff?
Until (obviously) some dyslexic cnut misread and misremembered PEDMAS with the “D” and “M” switched around. And the rest is history, as they say.
In places like Japan, China, Korea and India traditionally well known for their people’s numeracy, pupils are taught the actual order of operations without any recourse to using aids like BODMAS (or their unique language versions of it). The average liberal arts student from those countries can generally outdo the average North American STEM student. Think about that for a second. I kid you not, peep’l.
When your basic aid for arithmetic has “M” before “D” then it really isn’t an aid — it’s an abortion.

© The Naked Listener’s Weblog, 03 Nov 2020. (B20.1103) All images by the author.
L’article crée le dimanche 31 octobre 2020. Touts images par l’auteur.
Mediaeval weasels and modern wonks
Tuesday 15 September 2020, 8.00pm HKT
BIOL(LOL)OGY
HOW different is a person from mediaeval times from a regular modern person?
What are the physical and intellectual differences? Would a normal modern person be considered ‘smart’ then?
You tell me. But here’s what I can tell you.
—♦—
THE OVERBITE
Post-mediaeval vs. mediaeval skulls
(via Beauty in the Bones)
The conventional wisdom
The first physical difference that comes immediately to mind is the dental “overbite” (the upper teeth overlapping the lower teeth), which most modern-day people have but most mediaeval people didn’t.
I can’t remember the reasons for this, but my schooldays taught it came only around 200 or 250 years ago (at least for Europeans).
Other than that, there’s no straightforward answer to the physique and physiology of mediaeval vs. modern-day people. (See later below.)
“I’m made of wax, Larry. What are you made of?” — President Teddy Roosevelt in “Night At The Museum” (2006)
—♦—
SMARTNESS
No comparison in ‘smartness’
I think it just stands to common sense that the modern-day person has a higher intellectual operating demand than the mediaeval person.
We moderns simply have lots more happening around us.
I once read that the average person today gets more facts about the world around us in one day than an average 19th-century person got in an entire lifetime.
If that’s a span of just 100 to 200 years, the difference is just phenomenal between the mediaevals and us.
We all dance to the tune of our own times
(via Middle Ages: scenes from medieval life)
Is the modern person smarter?
That’s exactly the problem with a word like “smarter.”
“Intelligent” isn’t the same as “smart.”
A mediaeval person is bound to be more adroit and resourceful (smarter) than we could be in his mediaeval environment. He knew how to ‘stretch’ his penny, groat and shilling. He would know where and how to get food and blankets, and how to sleep in the rough almost as a matter of reflex. For sure he’d know how to keep food in an edible state without the fridge.
By contrast, we moderns would be blinking furiously and scratching our heads at the currency, which was non-decimal in most places then. Where’s the blimmin’ restaurant?
Conversely, the mediaeval person is going to be completely screwed (and screwed up) in our modern world. He’d be totally retarded with half the stuff of life that we do almost by intuition or reflex (e.g. looking both ways before crossing the road, dealing with stairs — both suggested by my late grandpa).
—♦—
PHYSIQUE & PHYSICALNESS
Some of the stuff we have about mediaeval times are ‘received conventions’ and may not be robustly objective facts.
As far as I understand from my schooldays and general reading, we take the general line from anthropologists that people in the pre-Industrial Age eras were shorter and more lightweight in build than the post-Industrial people.
The main reasons given are lower availability of food, nutrition and healthcare, and higher incidence of pollution and disease due to increasing population levels. But that’s common sense, really.
Adam and Eve by Jan van Eyck (1432)
(via The Lamb Of God)
Same enough
I don’t think the differences are huge between the mediaevals and the moderns.
We could reasonably assume the normal mediaeval physique to be something like in “Adam and Eve” (1432) by Jan van Eyck and part of the Ghent Altarpiece. The naked figures (top row, left and right above) were drawn in realistic fashion, not idealised.
The pair of them looked just like any European couple today.
Lean, mean and weaselly
(via Engineering the Medieval Achievement)
We could look at portraits, the armour and the fighting manuals from the 15th century. They all showed quite lean bodies — muscular, but lean, more like Lance Armstrong rather than Conan the Barbarian (or Marvin the Martian!).
Looking at the works of various mediaeval artists (like Dürer), mediaeval people in general stature and build looked more like weasels rather than bears or hogs.
Back to the future, please!
(via Marvin the Martian)
What goes up … comes down too
Yet anthropologists also tell us height and weight AREN’T precise trend indicators — because living conditions and nutrition varied over the centuries, even just in Europe:—
In 10th-century England, the average person wasn’t much shorter than an average Englishman today — shorter by only 6 inches (15 cm) at most. A 6-foot, 10-stone Englishman from today (1.83 m, 140 lbs/63 kg) wouldn’t turn heads if he time-travelled back to the 10th century.
Shorter still by 14th century due to increased pollution, disease and competition owing to increased population levels.
Taller between 14th and 18th centuries due to improvements in agriculture and general living conditions.
Shorter again after the early 19th century because the Industrial Age caused people to overwork and live in cramped, unhygienic conditions.
Taller again by the 20th century due to better healthcare, nutrition and general sanitation.
They’re all just 5 foot 7!
(via American Civil War Quotes)
To give an idea of how quickly height and weight can vary:—
In the 156 years between 1850 and 2006, the average American male became 2 inches taller (5–6 cm) and 45 pounds heavier (ca. 20 kg):—
1850:— Average 5 feet 7½ inches and 146 lbs (1.71 m, 66 kg) (according to Union Army records)
2006:— Average 5 feet 9½ inches and 191 lbs (1.77 m, 86.6 kg)
(via The New York Times, 2006 news item on a study on body size and health over the last 150 years in the USA)
That roughly comes to a yearly increase of 0.016 inch (0.385 mm) and 4.62 ounces (130.97 g) for the last 156 years for the average American male.
—♦—
GEOGRAPHY & GENETICS
European genes mirror European geography
(via National Geographic)
Where you live is what you are
When considering the Middle Ages, we shouldn’t just look at the timeframe.
The geographical range is important — the range of physical environments even in Europe is huge.
A 10th-century Dane and Italian clearly wouldn’t resemble at all. A 13th-century Englishman wouldn’t resemble a Greek.
Hot climes tend to produce leaner, slenderer, flatter bodies to dissipate heat. Cold climes lead to bulkier, rounder bodies to generate and retain more heat.
The human body adapts to the natural environment, resulting in a variety of builds and statures. That’s the generality, yet a useful one.
As for genetics, we could presume when a mixture of body types migrate and interbreed, then over time we should see one consistent body type emerging. I don’t know enough about this, but the stuff I’ve read over the years say this situation hardly ever occurred.
—♦—
NUTRITION & GENETICS
(via Nutrition and the Epigenome)
Nutrition across generations
While body size is based on genetic factors and nutrition generally, I understand that a bigger influence is childhood nutrition.
What further complicates the picture is the nutrition of both the mother and the grandmother — so we’re talking about generational nutrition here.
The reason I gathered from general biology is that the developing ova and embryos are rather vulnerable to even slight nutritional imbalances.
This makes sense just looking at people around me — women whose nutritional level were lower before being mothers tend to bear smaller kids, who then grow up to be smaller-sized people.
(via Buying Your Food In Bulk Is A Shocking Waste Of Money)
Time and money on food and the home
Food and nutritional availability have vastly increased in just the last 100 years. Today, we can buy an entire day’s nutrition for one hour’s minimum wage (roughly speaking).
Not so just a century ago.
At around the First World War (1914–18), the average person in the UK spent half the income just on food alone. Even in the mid-1970s, I remember I spent a third of my weekly pay just on groceries. Then it started going down towards the 1980s and the ’90s — but started flying up since 2005.
—♦—
FORAGING & CHORES
Time to forage for food takes it toll too.
I know from my grandparents (who were extremely well-off) in their 20s didn’t have to spend two to four hours daily traipsing all over the place just to get the groceries that most other normal 20-somethings had to.
There was simply no “one-stop shop” for groceries then like we have today.
(via Dreamstime)
Then there’s the matter of housework.
Around the First World War, housewives were spending an unbelievable 40 or even 50 hours a week just doing the housework.
That defies understanding. Today our average workweek is 45 hours — and we’re already beat from THAT.
Women at that time did 40 hours at home PLUS another 20, 30 or even 40 hours at work to make ends meet. That wrecks big time.
So if you have to spend literally hours working and doing chores at home AND literally hours just getting the actual food, your overall expenditure of energy on a daily basis is bound to affect your overall stature and health.
No wonder people died just a year after retirement.
—♦—
QUESTION TIME
‘Housework’ for a modern person is much different from a medieval person, or even someone from the 19th century. [I’m] from an American city where few people have to daily scrub the hearth, or bring in heavy coal or wood to feed a fire, or dig privies, or even black and clean one’s boots from horse manure and more. — Elizabeth P.
The Naked Listener writes:—
Of course. The difference is staggering in just a space of 100 years, from the First World War to now.
Is modern person smarter? Hardly. Erich Fromm once told that average mediaeval person was much more resistant to tricks and manipulations. Maybe that’s why there was so much bloody struggle — people were much more independent and much less obedient to central rule. Remember Robin Hood challenging city authority — impossible thing today. — Dmitriy S.
Yes, what you say is true enough too. But Erich Fromm was taken out of context, and Robin Hood wasn’t an actual historical person.
Fun supplementary fact:— The overbite apparently comes from our move towards forks for bringing food to our mouths. Using the top row of teeth to pull food from a fork or spoon has had an effect on our dental structure. Apparently, people from chopstick-using cultures have an equivalent underbite that stems from their ancestors essentially dropping food into their mouths. — Stacey O.
Yes, that’s one of several reasons I’ve been hearing and reading for a long time (well, since the 1980s, if memory serves). I’m not hip to the matter so I just run with whatever is the prevailing academic or scientific position on the matter.
You know, when you don’t know, you don’t have to run with anything, except “I don’t know.” — Geoffrey R. D.-T.
As in not running your mouth off too, I suppose…
Come on, it’s not that “I don’t know.” It’s “what I know” is “what the prevailing academic or scientific position on the matter is.”
I literally don’t know for myself if the Earth is round or flat, but I do know science has it that it’s round or spherical. In the same measure, what makes those other people so sure the overbite model is wrong? It’s a two-way street.
It’s the same as asking, do you have a brain? Have you seen it? How do you know you have one? It isn’t that you “don’t know.” It is that what you do know from general knowledge that almost without exception everyone has a brain.
Hey! My [nationality] wife has the overbite. So maybe something’s wrong with these theories. — Geoffrey R. D.-T.
Some people have them, some don’t. Dietary and other habits determine the presence or not. Nearly all ancient Egyptians had no overbite, but nearly all Egyptians today have it.
*shrug* It’s one of several explanatory models to account for certain features observed.
Really enjoyed the thoughts and work that went into this presentation. Some of it was “Duh!” in the sense that “Yeah, makes sense” — just never considered the changes that have occurred even in my lifetime of about 70 years. […] Vegetable gardens were a necessity even back in the 1950s: only one income for expenses, then someone had to make up the extra income needed to make it go. But they did it (because they HAD to)! — Paul H.
That is very true.
© The Naked Listener’s Weblog, 15 Sept 2020. (B17094)
Images originally via Quora image search.
Originale depuis 11 avril 2016 et l’article crée le 18 juillet 2017.