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If you went to elementary school in the United States (or much of the rest of the world)
you almost certainly learned about something boringly called the "order of operations"
- a set of rules for whether or not you should do multiplication before addition or addition
before subtraction to get the right answer on your math test.
Except, you don't always get the right answer, or even, one answer - I mean, is 8-2+1 equal
to 5 or 7? and is 6/3/3 equal to two thirds or six? - the problem is, focusing on the
order of operations can lead to ambiguity and obscures the real, underlying, and often
beautiful mathematics.
A mathematician will tell you that 8-2+1 is really 8+(-2)+1, which is unambiguously equal
to 7, even though the so-called "order of operations" standard in the US tells you the
answer is 5. If you want five for your answer, then you really need some parentheses!
But why is this ambiguity even possible? Because fundamentally, all of these operations are
simply different procedures that start with two numbers and combine them in some way to
give you a single number. Each operation takes two numbers as input - two, and no more. If
you want to be entirely unambiguous (and pedantic), every single pair of numbers in an algebraic
expression should be inside parentheses, and then there's no need to know ANY order of
operations - just evaluate the innermost parentheses first, and work outwards, collapsing them
down pairwise like a championships bracket.
But it turns out that's not the only way - it's possible to CHANGE the order in which you
do the operations, to rearrange the parentheses, as long as you know what the underlying mathematics
For example, if you want to add (3+4) and then multiply the result by 5, you can either
do the addition first and get 7 times 5 = 35, or you can do the multiplication first as
long as you know that multiplication "distributes" across all the terms in the parentheses...
that is, (5*3) plus (5*4) = 15 plus 20 = 35 - the same answer! And how do we know multiplication
distributes? One way is to draw rectangles... but I've done that before.
The same rearranging can be done for exponentiation and multiplication: (3*2) all squared (or
6^2=36) is the same as 3 squared times 2 squared - 36; It even works for addition and subtraction:
5 minus (1+2) is (5 minus 1) minus 2.
So, the true order of operations is this: use parentheses, and learn what exponentiation,
multiplication, addition and the rest are REALLY doing - then you can proceed however
you want.
That's not to say that we don't have a conventional order of operations in mathematics... I mean,
deciding that we evaluate multiplication before addition allows us to get rid of lots and
lots of redundant parentheses, and noticing that (1+2)+3 = 1+ (2+3) and 2*(3*4)=(2*3)*4
removes a ton more... but the point is that those parentheses are still there, still implied
- just like how 3 minus 4 is secretly implying 3+ negative 4 and 3 divided by 4 is really
3 times one fourth. But any time the result might be ambiguous, you really need to use
parentheses. Then you can proceed in whatever order you want.
The order of operations learned in school, however, is different - it's a mechanical
set of instructions that dictates just one of many ways you can do algebra - it locks
you into a single path through the beautiful mathematical landscape, which, while necessary
for a computer whose goal is merely to give you the right answer, doesn't really give
any insight onto WHAT IT IS that you're doing when you do algebra.
So, while, the order of operations isn't technically wrong, because most of the time it'll give
you the right answer, it's morally wrong, because it turns humans into robots.