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.

Above: Result from Excel
Above: Result from Symbolab
Above: Calculator result
Above: Result from 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

  1. Either use distributive property
  2. Or solve in pairs
  3. 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

Wrong turn

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

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.

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