Let’s suppose you’re in an exam, and you
need to calculate 1 over 2 root 3. So you’re used to putting this into your
trusty calculator as 1 divided by 2 root 3, like that. Hit equals, and you would get
the correct answer of 0.29. But let’s suppose on this particular day,
your calculator has run out of batteries, and you borrow a friend’s calculator. So
you type it in, as you normally would, 1 divided by 2 root 3, hit enter, and you get
0.87, which is incorrect if that’s what you’re trying to calculate. So what’s happening
here? Well this calculator is interpreting
this as half of root 3 whereas the HP interprets it as it’s written there. So
to find out why this happens let’s have a look at the manuals. Now when we write
something like 2 root 3 without any kind of multiplication symbol in between the
2 and the root 3 that’s called “multiplication by juxtaposition” or “implied
multiplication”. So on the TI’s manual under order of operations it has the
implied multiplication at the same level as division, whereas for the HP it has a
couple of extra levels. It has implied multiplication there, and also here, whereas
division is all the way down here. So this method, where implied multiplication
has the same priority as division, I’m going to be calling that PEMDAS. And this
one where there’s this extra level, I’m gonna be calling that PEJMDAS,
so parentheses, exponents, multiplication by
juxtaposition, and then multiplication and division (so explicit multiplication
with the times symbol, that still has the same level as division) and then there’s
addition and subtraction. So those are the two main methods that calculators
use: either PEMDAS or PEJMDAS. So this disagreement between PEMDAS and
PEJMDAS is what causes the two calculators to interpret this
differently. It’s also what causes them to interpret the… one of the famous
Facebook questions differently. These things keep popping up; I think the
current iteration is this: 8 divided by 2(2+2). But when I
made my first video on PEMDAS about this question a lot of people were just
saying, you know, “Who cares? It’s just a silly Facebook question.” But I’m hoping
that by showing the different ways that the calculators interpret this, it
demonstrates that this is actually kind of a problem, because students could be
losing marks, and not just that but… I don’t know if you’ve ever been in a
situation where you’re trying to work through a textbook and your answer to a
question doesn’t match what’s in the back of the book and you keep checking
and rechecking it and rereading the chapter and that can just take hours out
of your life. So I think this is actually kind of a problem. So let’s have a little look
into the history of PEMDAS. Now I wasn’t able to find out who invented the
acronym itself but the idea that multiplications and divisions have the same
precedence, let’s have a look at that. So there’s this article by Sarah Sass which
goes over quite thoroughly the history of this. Now right into the 1920s
there wasn’t any sort of agreement about the order that they should be done in;
some authors were saying multiplication should be done first and others were
saying that they should be done in the order they occur. So there’s these three
books that are putting forth that idea, this PEMDAS idea. Now let’s have a look
at them. So First Year Algebra that’s saying “all
multiplications and divisions in their order from left to right” then there’s
First Course in Algebra which says something similar and Second Course in
Algebra – similar thing again. Now what’s really interesting though is that if you
actually read through these books, all three of them break that rule when it
comes to juxtaposition. For example in First Year Algebra you see expressions
like this being taken to mean this which is actually following PEJMDAS.
In First Course in Algebra you see a similar kind of thing, and in Second
Course in Algebra we’ve got this 2ab divided by 2a making b. Now if you
followed the rule that they had stated to the letter you would interpret this
as 2ab divided by 2, so you work out that that’s ab, and then you multiply by
a, so you’d get a squared b, and not b as they’ve written here. So why haven’t they
stated that juxtapositions should come before division when they set their
rules? Well one possibility is that they just didn’t realise that it was
contradictory, that what they were saying in their rule didn’t agree with what
they’re actually doing in their book. Another possibility is that they did
realise that it was a contradiction but they actually thought, you know, “everyone
already knows that juxtaposition comes before division;
it’s so well-established that we don’t even need to say it.” But some authors of
the time did notice the contradiction. There’s this article by N J Lennes who
is pointing out that the rules stated in these books is a contradiction with the
actual usage. So he was saying like if we actually follow that rule then you would
find that, for example, 9a squared divided by 3a would equal 3 a cubed if
you use the rule that’s written in the books. But no one would interpret it in
that manner. So the point I want to hammer home here is that juxtaposition
going before division was already well-established well before anyone came
up with this idea of PEMDAS. So at the time of writing of this, people were
still arguing whether division or multiplication should be done first,
but everyone agreed that multiplication by juxtaposition always
comes first before division, even the people who were saying that divisions
and multiplications should be applied in the order that they occur from left to
right. In the early days of scientific calculators they
didn’t actually support multiplication by juxtaposition at all; if you left out the
times symbol they’d just give you a syntax error. Now the earliest one that I
could find a manual for online that supports juxtaposition was the EL-512
which was released in 1984. This is from Sharp. And they say… they’ve got
“multiplication cleared of X instruction” or “times instruction”, so
meaning multiplication where the times symbol has been left out, that’s at a
higher priority level than normal multiplication and division. And then
here’s the manual for the TI-81 which was released around 1990 and you can see
they’ve put implied multiplication at a higher level… at that level and that
level… higher than regular multiplication and division. So both of these
calculators are following PEJMDAS. The early Casios followed PEJMDAS
as well. This is actually my old calculator that my parents bought for me
around the year 2000, and you can see it’s interpreting this as 2 times the 3
first, making 6, and then 6 divided by that, making 1. So that’s following
PEJMDAS as well. Then in the 90s and 2000’s, everything
changed. Texas Instruments came out with the
TI-83 in 1996 and that followed the PEMDAS rules, so they put the implied
multiplication down all the way back to the same level as division. And then
Casio came out with the FX-ES series in 2005 and that was similarly PEMDAS. Sharp
seems to have stuck with PEJMDAS the whole way through though; every calculator
that I found from Sharp was PEJMDAS. And then HP seems to be all over the
place; some of the HP calculators follow PEMDAS; some of them follow PEJMDAS,
and I can’t really see any sort of pattern to it. So why did this switch
happen? Well I’ve been corresponding with a representative from Casio who was
saying that basically they make their calculators according to what teachers
want, so they do various hearings with teachers and schools, and they were
saying that, you know, they started out with PEJMDAS but then they switched
to PEMDAS because teachers, and mostly teachers just for North America, wanted
them to switch to PEMDAS. But now actually after hearing from a wider
range of people they’ve switched back to PEJMDAS. So I found it quite interesting
that they said that it was just North American teachers who were pushing this
idea of PEMDAS. I’ve been also talking to David Linkletter who wrote an article
about PEMDAS about the same question of like “What is 6/2(1+2)?”
and he was saying something similar, that it was mostly the American teachers
who were saying “okay follow PEMDAS; the answer’s 9,” whereas teachers from the
rest of the world, teachers that he’s spoken to from Germany, China, and India
have said “No, the answer is 1.” So I don’t know why that is, why North American
teachers are saying PEMDAS whereas the rest of the world is saying PEJMDAS. And
just to be clear, it’s just the North American teachers who are in favour of
PEMDAS; it’s not mathematicians or scientists or engineers from North
America that are saying “Okay PEMDAS is the whole truth, it’s a strict rule and
you have to follow it, even with juxtaposition.” There was also this
statement from TI which was… I just thought this was interesting, because
they were saying how they made the switch but they were also saying that
the actual convention when you’re writing things down on paper is
PEJMDAS. So they were saying implied multiplication like should have a higher
priority because that’s how you would write things down and… but, you know, we’ve
switched and now it’s this different way that you wouldn’t normally write on
paper. Okay enough about history. What about calculators that are
still being made today? So I went by my local store to see what was available,
and whether they’re following PEMDAS or PEJMDAS these days.
So the Casios are doing PEJMDAS. The TI was the one that I showed at the start,
that one’s doing PEMDAS. This one wasn’t in stock and the manual I think doesn’t
actually mention in the priority levels implied multiplication so I think that
one might be PEMDAS but I’m not sure. All of the Sharps are
PEJMDAS. That one’s PEJMDAS. That one wasn’t available but from
reading its manual I think that it’s PEMDAS. And these are all PEJMDAS as
well; those are Sharps too. So most of them are following PEJMDAS these days
with the exception of the TI and probably these HP ones that weren’t even
in stock so probably aren’t sold as often as the others. Now I just want to
show you what Casio is doing because I think this is really a good solution to
this problem. So this is a modern Casio fx-100AU that I bought a few months
ago. So if you type in 6/2(1+2), let’s see what
happens when I hit equals. It actually inserts brackets there, so instead of
just evaluating it some way and not letting you know, it actually tells you
by putting the brackets in how it’s interpreting that expression. So I think
that is a really good solution. Like unlike the HP and the TI and the Sharps
which, you know, don’t really explicitly tell you whether it’s using PEMDAS or
PEJMDAS, the Casios will put in these brackets to let you know. As for online
calculators, google calc follows PEMDAS, but Wolfram Alpha is really weird
because ok here it’s following PEMDAS giving an answer 9; if you put it as 6
divided by 2x it’s still following PEMDAS, but if you put in 6 divided by xy, it
will interpret that as like PEJMDAS; it interprets that as 6
divided by x and then divided by y as well and gives the answer of 1. So I’m
not sure what kind of priority levels they’re working with there but they’re
a bit inconsistent. So I think we need to come to a conclusion about which one of
these is correct. Do we interpret 1 divided by 2 root 3 as half of root 3 or
do we interpret it the way that the majority of mathematicians, scientists and
engineers would interpret it? So I’m firmly in favour of the PEJMDAS. Now
in my first video that I made on this I was arguing based on usage, like this is
how mathematicians interpret it, therefore this is the rule. Now a lot of
people correctly pointed out that if there is a strict rule of PEMDAS, then,
you know, mathematicians just shouldn’t be lazy; they should use brackets or
something. Now I think that this comes from a misconception that mathematicians
have gotten together and decided “Ok PEMDAS is the rule; this is a strict rule;
we should all follow this.” That has not happened.
PEMDAS was something that some educator wrote down as… you know, either as an
oversimplification or because they didn’t understand the actual rules of
mathematics. Mathematicians have not decided in any sort of way that
multiplication and division should always have the same precedence even if
it’s by juxtaposition. In fact if you want to see some official statements
from mathematicians or scientists then we could look at the American
Mathematical Society style guide which said “the rule that multiplication
indicated by juxtaposition is carried out before division.” It actually doesn’t say
that anymore; this is a past style guide, but they do still use that rule. For
example here is a recent article where they’re intending this to be
following PEJMDAS. We could look at the American Physical Society style
guide where they put multiplication before division. They actually put it for
you know explicit multiplication as well but I I’m guessing that’s just because
you hardly ever would use explicit multiplication in a physics journal, it’s
always by juxtaposition. We could also look at the American Institute of
Physics style guide where they’ve said that 1/3x always means 1/(3x). So they’re saying PEJMDAS
is the case. Now you might still disagree with me, which is fine; you might
think that PEMDAS is the way to go. But i think that we can all agree that
students need to be aware that some calculators use PEMDAS and some use
PEJMDAS and that most mathematicians and scientists will be following PEJMDAS in
their publications. So what I would do is probably introduce this idea around the
time that you learn about juxtaposition in algebra, so when you’re beginning
algebra you learn that for example 2a means 2 times a, so maybe when
students are learning that we can just say, “Oh by the way, multiplication by
juxtaposition – some people consider it to have priority over division.” So I don’t
think that that would add too much difficulty to algebra class or to
textbooks, and it would clear up a lot of confusion for a lot of people. So if you
know any teachers or textbook writers that would benefit from seeing this
video then send it along to them.

The Problem with PEMDAS: Why Calculators Disagree
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