So this question got a lot of attention
on the internet, and if you apply the
order of operations that you would have
learned in primary school, which is
usually called something like PEMDAS or
BODMAS or BIDMAS or BEDMAS
depending on what country you're in, you
would get the answer of nine. But here's
the thing...
PEMDAS is a lie. PEMDAS is an
oversimplification that we teach to
children to make things easy. It's like
saying that negative numbers don't have
square roots - we just want to get on with
the lesson. But PEMDAS isn't the way that
mathematicians, physicists, or engineers
actually evaluate mathematical
expressions. I'll talk about the real
rule in a minute but first let's see how
PEMDAS applies to our question. So what
you need to understand about the order
of operations as they teach it in
primary school is that multiplication
and division have equal precedence. So
I'm just going to write PEMDAS like that
to show that it's just four things:
parentheses, exponents, multiplication and
division in whatever order they occur
from left to right and then addition and
subtraction in whatever order they occur.
So with this question we start with
what's inside the parentheses: the one
plus two, that makes three.
And then there's no exponents; we move on
to multiplication and division in the
order they occur, so we've got a division
there and a multiplication there. This
one's to the left so that one happens
first.
So you do the 6 divided by 2; that makes 3.
And then that's multiplied by the 3. So
now we do that and that makes 9. So let's
see how the order of operations applies
to something like a/bc. So again we've
got a division and a multiplication. We
start on the left, so a divided by b is
the first thing you do. And then you
multiply the result of that by c so this,
according to PEMDAS, is the same as a
over b times c. But is that how
mathematicians, scientists, and engineers
would actually interpret this expression?
So to test this out I just googled
"mathematics textbook PDF" and this is the
first one that I clicked on - this is
abstract algebra by Robert B Ash, and you
can see on the second page of the PDF - I
just went straight to the answers to
exercises because that's where I was
expecting to find expressions like this.
So we've got this. What they've done to
get that is multiplied the m/r by n/s.
So that would make mn over rs.
And then they've written it like that
though, whereas the order of operations
would tell you that that means mn/r -
the division comes first - and then you
would multiply by the s and that of
course makes mns/r but
they're using it to mean that so they're
obviously not using the order of
operations as it's taught in primary
school.
What about physics? This is from the
Feynman lectures on physics. We've got a
deviation of one on (2 root N) and then
they've written that again down here,
in-line, as 1/2 root N. Now, order of
operations would say that that means
half root N but they're using it to
mean one on (2 root N) so they're doing
the multiplication first before the
division.
What about engineering? I've just googled
"engineering textbook PDF" and this was
the first one that I clicked on. This is
on page 15. I'll put a link to it
in the description. But what they've done
is they've solved these equations for W
and got PVMg on (RT) and then
they've written that like this with a
slash RT and PEMDAS would say that that
means PVMg divided by R, and then you
would multiply by T and that is not the
same thing as that so they're not using
PEMDAS. So the textbook authors aren't
following PEMDAS but what do the
students think? Well Oliver Knill of
Harvard asked his calculus students this
question: what is 2x/3y - 1 if x=9 and y=2?
Well according to PEMDAS you would say 2
times 9 divided by 3 times 2 minus 1.
2 times 9 is 18, then you divide by 3 that
makes 6, you multiply by 2, that makes 12,
and then you subtract 1; that's 11. And
this is what this fifth-grade teacher
was saying. They asked their students this
question and it caused a bit of a stir
because the parents and students didn't
agree.
So anyway Oliver Knill asks his calculus
class this question and 58 out of 60 of
them said no it's 2 times 9 over (3 times 2)
and then you subtract 1 so that's 18 over
6 which is 3 and then minus 1 makes
2. 58 out of 60 said that. The two that
didn't interestingly didn't say 11 they
did 2 times 9 over 3 times 2 and put the
minus 1 in the denominator as well, so
absolutely nobody would interpret the
question like that. It's easy to see why
this became a rule. If you want to write
something like 1/(2x) in one line
you would have to use brackets which is
annoying so mathematicians tend to be
quite lazy and they just write 1/2x
when they mean 1/(2x). Now the
order of operations that you would learn
in school says that this is the same as
half of x but there's a much easier way
of writing this: you could just say x/2.
So people have developed
this rule that 1/2x means 1/(2x)
and if you want to write (1/2)x you just
write x/2. So it's an unofficial
rule that's used by almost everyone in
maths, science, and engineering but has it
ever been an official rule? Well indeed
it has! This is from a style guide put
out by the American Mathematical Society
and here they clearly say that they use
the rule that multiplication indicated
by juxtaposition is carried out before
division. And here they're just saying
that if you write something like 1 over
(2 pi i) they're going to change it to 1/2pi i
and that these mean the same thing.
Likewise, the American Physical Society in
their style guide say in mathematical
formulas this is the accepted order of
operations: raising to a power, then
multiplication, and then division, and
then addition and subtraction have equal
precedence. So how does this rule apply
to our question? Well you still start by
evaluating what's in the parentheses -
that makes three.
But then multiplication by juxtaposition
comes before division so we've got six
divided by six which makes one. So
although PEMDAS says that the answer to
this is nine, no one actually uses
PEMDAS after primary school. Let me know
in the comments if your teacher ever
told you that PEMDAS isn't the whole
truth, and if you'd like to see more
videos like this then subscribe because
there's lots more coming up about the
how and why of mathematics. :)
