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Even More on Order of Operations

Date: 10/26/2017 at 16:22:13
From: Steven
Subject: Solving 8/4(3-1). PEMDAS vs Distributive

I'm curious to know what the answer is for this:

   8/4(3 - 1)

Following strictly PEMDAS, the answer is 4:


However, if you follow the distributive property, you get 1:

   8/((4*3) - (4*1))
   8/(12 - 4)

Which one would be correct and why?
Both are valid, so I'm conflicted as to what would be the correct answer.

It should be right or wrong, not two different answers being right.

Date: 10/26/2017 at 17:01:15
From: Doctor Peterson
Subject: Re: Solving 8/4(3-1). PEMDAS vs Distributive

Hi, Steven.

The problem is not a conflict between PEMDAS and distribution; it is that
strict interpretation of PEMDAS conflicts with one's natural impression of
the meaning of the expression, so that you unknowingly apply an
alternative interpretation when you think you are just applying the
distributive property.

When you distributed, you ASSUMED that it was the 4, not 8/4, that was
multiplying the (3 - 1). In doing so, you were bypassing the rules and
just doing what felt right. If you followed the rules AND distributed, you
would get this:

   (((8/4)*3) - ((8/4)*1))
   ((2*3) - (2*1))
   6 - 2

Because it is so natural to see the multiplication as tightly attached to
the parenthesis (especially when the division is indicated by an obelus
rather than a slash/virgule), some authorities state an extra rule: when a
multiplication is indicated only by juxtaposition (no symbol), it is done
before divisions.

What many people don't realize is that the "rules" we teach are only an
attempt at DESCRIBING what mathematicians did for a long time without
explicitly stating what rules they were following. They do not PRESCRIBE
what inherently must be done, a priori. In just the same way, English
grammar came long after English itself, and has sometimes been taught in a
way that is inconsistent with actual practice, in an attempt to make the
language seem perfectly rational. See this page:

  History of the Order of Operations 

In my opinion, the rules as usually taught are not the best possible
description of how expressions are evaluated in practice. (This is
supported by a recent correspondent who found articles from the early
twentieth century arguing that the rules newly being taught in schools
misrepresented what mathematicians actually did back then.)
Unfortunately, for decades schools have taught PEMDAS as if it must be
taken literally, so that one must do all multiplications and divisions
from left to right, even when it is entirely unnatural to do so. The
better textbooks have avoided such tricky expressions; but others actually
drill students in these awkward cases, as if it were important.

The added rule about juxtaposition is an attempt to correct this. But the
reality is that, since teachers give two different sets of rules that
produce different values for expressions like yours, all we can really say
is that such an expression SHOULD NEVER BE WRITTEN! In the absence of
agreement on the meaning of such an expression, we have to consider it
ambiguous unless you first state what rules you are following.

And in fact, such expressions are rarely written in practice; beyond the
elementary grades, we typically represent most divisions with a horizontal
fraction bar. That, I think, is why this "error" has never been corrected:
it doesn't matter much.

We get questions about this quite often, and they routinely raise other
issues. Here is a typical recent reply:

This question, and others like it, have been going around on social media
for years, always causing this sort of conflict (perhaps even
intentionally). We've had several questions about it just this month.

It's totally unnecessary.

On the whole, this is what you will get if you follow the "rules" as
usually taught:

   6 / 2 (1 + 2)
   6 / 2 *  3
     3   *  3

Those who say you should distribute first are putting the cart before the
horse: you can't apply tricks to evaluate an expression before you first
know what it MEANS, but they are thinking that the distributive property
affects the meaning. (In fact, the distributive property is a waste of
time here, because it makes you do two multiplications where only one is

The meaning is determined by the order of operations. Is the
multiplication supposed to be done before or after the division?

The trouble is that sources differ on the "rules." The basic "order of
operations" as commonly taught says only that multiplication and division
are to be done (in the order they appear, left to right) before addition
and subtraction, as above.

But some authors add an "extra" rule that when multiplication is indicated
by mere juxtaposition -- without any explicit symbol, as in "2x" or 
"2(3)" -- it is done first, taking it as a unit:

   6 / 2 (1 + 2)
   6 / 2 (  3  )
   6 /     6
This is what your "opponents" are doing, though without a proper reason;
it is also popular among kids who just do what feels right to them. But,
as I said, there are also textbook authors and others who explicitly teach

In books and handwritten math beyond the elementary level, we hardly ever
use the horizontal division symbol, but use fraction bars instead, which
leaves no ambiguity. As a result, the math community has never had a need
to make a choice on this situation! It's essentially been left undefined,
and it is textbook authors who came up with explicit "rules" to describe
what is really just a language that developed organically, based not on
carefully stated rules but on tacit agreement.

So which is the "right" way to read such an expression depends on what
rules are in force in a particular community (math class, journal, 
etc.) -- and what was intended by the writer.

As a result, in problems such as this, the error is being made primarily
not by those who give "wrong" answers, but by those who post the problem
in the first place (or pass it on). Anyone who really wants to do math
correctly will want to communicate clearly about it, and will avoid
anything ambiguous or uncertain. They should either fully parenthesize, or
use the horizontal fraction bar, which makes the order clear:

      6             6
   --------   or   ---(2 + 1)
   2(2 + 1)         2

Here are some examples of past answers to similar questions:

  Order of Operations Dispute 

  More on Order of Operations 

Even some calculators differ on this:

  Implied Multiplication and TI Calculators 

Here is an example of someone who teaches the "extra" rule and believes 
that it is generally accepted: 

- Doctor Peterson, The Math Forum at NCTM
Associated Topics:
Elementary Addition
Elementary Division
Elementary Multiplication
Elementary Number Sense/About Numbers
Elementary Subtraction

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