If you haven't already, start with
Part (14÷7(5-3)+2)÷3, where I introduced the problem. Then, in
Part (14÷7(5-3)+2)÷3, I began discussing what's actually going on, and introduced a few other mathematical ideas. Now, in Part (14÷7(5-3)+2)÷3, I will answer the question posed at the end of the last part. When given 12+33+72, which addition do you start with: 12+33, or 33+72?
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| WHAT?! Addition is associative?! |
Well in an incredibly climactic turn of events that none of you were expecting, it turns out that it doesn't matter. When adding three numbers, you can add them in any order you want. But let's take an example which isn't so friendly. Pretend you have a function "avg" that takes the average of two numbers: avg(2, 4) = 3, avg(14, 17) = 15.5, and so on. And just like add(2, 3) can also be written as 2+3, let's say avg(2, 3) can be written as 2#3.
Now what's 1#5#21? Well if you do 1#5 first, you get 3, and 3#21 = 12. But if you do 5#21 first, you get 13, and 1#13 = 7! So you do it in a different order, and you get a completely different answer (note: if you decided to break the "only two at a time" rule and find the average of all three at once, you'd get 9, another completely different answer)
Now, if instead you did it with functions, you couldn't write something unclear like 1#5#21; you
have to choose which average is done first. Your options are avg(avg(1, 5), 21), and avg(1, avg(5, 21)). When you write it like this, there's no ambiguity, because it's really clear what has to be done first. And
this reflects the actual mathematical ideas, which include knowing which functions depend on the results of other ones. 1#5#21, on the other hand,
doesn't. This is what I meant when I said that the mathematical expressions we're used to don't actually reflect the essence of what they try to describe - function notation succeeds in that far better than regular notation does.
Now, it would technically be possible to rewrite all elementary textbooks in function form. But when you realize that something as simple as 2(5+3-4) would then be written as mult(2, sub(add(5, 3), 4)), it begins to get a little hard on the eyes trying to figure out what's inside of what. And so people use expressions like 2(5+3-4), because they're easier. Simply put, these expressions are a
shorthand. They're a way of abbreviating mathematical ideas to make them easier to read and write.
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I did a Google Image Search for "AH"
and this was one of the results. I don't
really know why. But mm, dumplings |
And whenever you abbreviate something, you lose information. For example, a couple weeks ago I had a Facebook conversation and someone mentioned seeing something on "AH." Unfortunately, I didn't know what that acronym stood for at the time. So I looked the acronym up, and found it could have meant anything from Art History to Adaptive Hypermedia to American Health to Apocalyptic Harbingers to Artificial Horizon to... Later he clarified that he was referring to "Alternate History." But since I didn't know the context, the shortened version was incredibly ambiguous. Making a shorter, easier-to-read version ended up taking away certainty.
And that's exactly what happens with things like 6÷2(1+2). It's a shortened version of the actual mathematical idea behind it, and because of that, it's lost some of its information.
Again, sometimes losing information isn't so bad. Like in the case of 2+5+4. In the end, it doesn't matter which numbers you add first, because you'll get the same answer. But in a case like 2×5+4, how do you know which to do first? Is it add(mult(2, 5), 4) = 14, or mult(2, add(5, 4)) = 18?
Two things were done to try to help the shorthand become slightly more accurate: brackets, and order of operations. Brackets are used to state clearly "everything in here needs to be considered together. You can't do anything to just part of this while leaving the rest behind." Technically, if you had enough brackets, the shorthand could be just as clear as the original function form. For example, add(1, add(add(2, 3), 4))) and (1+((2+3)+4)) mean exactly the same thing, order and all. So people could have just demanded that every operation symbol (+, -, ×, ÷) have its own pair of brackets, and there would be no trouble. But as in the case of (1+((2+3)+4)), this can look almost as messy as the function version, when it doesn't need to be; 1+2+3+4 works just as well, even if it's not clear what to do first, because it doesn't
matter what you do first. It's a shorter abbreviation that still works just as well, so why worry about all the extra brackets? But if you drop that requirement, then you need to know what to do when two different operations aren't separated by brackets, when the order
does matter - like in 2×5+4.
This is where the order of operations comes from. It's a convention for reading a shorthand notation, so we can all agree that it refers to the same idea. People decided that certain functions were "more important" than others, and gave them different priorities. There are actually some good reasons behind the order*, but it turns out that things would work just as well if the order of operations was completely flipped! You'd just need to learn how to read and write in the new system.
*One example of a good reason is the distributive property, which describes how multiplication and addition relate to each other: a×(b+c) = a×b+a×c. If addition had priority, this would have to be written a×b+c = (a×b)+(a×c) - you'd need an extra pair of brackets.
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Excuse me waiter, but I believe
I ordered a sled. |
This is what I mean when I say that why it bugs me that people get so uptight about the order of operations. A lot of them seem to think that they're arguing over unchangeable facts. But no, it's way less significant than that. Some people might think that there's a cultural element involved, but that there's a single correct standard accepted today (Even though words can change over time, "I through the ball over their" is just wrong). But no, it's less than that. Some people might think that there are two acceptable versions, but that either one should be seen as, in a way, "as good as possible" ("favourite" is British English, "favorite" is American English - and there's no better way to express the same meaning, so these words are as good as it gets). But I'd still say it's even less than that; I'd say people are arguing over a
shorthand notation which doesn't completely describe what's going on. It's like they're arguing over whether "sld" means "sold," "sled," "solid," or "salad." If there's any argument, it's not a problem with English, and it's not a problem with whoever disagrees with your interpretation; it's a problem who came up with the abbreviation. It's up to him to explain why he thought "sld" worked better than writing the whole thing out, or choosing another abbreviation that's more clear.
So what do I say to whoever first wrote 6÷2(1+2)?
Explain why you chose that abbreviation. I can understand why you wouldn't write it out fully (as either mult(div(6, 2), add(1, 2)) or div(6, mult(2, add(1, 2)))), but if you're going to use an abbreviation, why didn't you use an abbreviation that causes less argument, like (6÷2)(1+2) or 6÷(2(1+2))? Or better yet, get rid of that division sign altogether and use fraction notation instead - it makes very clear what gets divided by what. You could even use an entirely different system of shorthand, like
Polish Notation (where the choices would be × ÷ 6 2 + 1 2 or ÷ 6 × 2 + 1 2) or
Reverse Polish Notation (6 2 ÷ 1 2 + × or 6 2 1 2 + × ÷), which in many ways are better than the system we usually use* because no brackets or order of operations are necessary to be perfectly clear about what's going on. So with so many great, unambiguous options, why oh why did you choose 6÷2(1+2)?
*which, in case you're wondering, is called Infix Notation
Of course, the person who came up with that expression probably had a very good reason: to mess with people. And for that, I applaud him, because he has succeeded immensely.
Yes, it is important to come up with a standard convention that we can all agree on, but the reason for this is so that we can spend more time on the math and less time decoding stuff. So when faced with something ambiguous, especially when it
doesn't have to be, it's entirely against the point to spend so much time arguing about how to interpret it. Until we see a mathematical expression that is inherently ambiguous - that is, there's no better way to write it - the proper response should be to teach people how to express their mathematical ideas in a non-ambiguous way. Just as how a good English teacher shouldn't be teaching grammar in order to show students how to write really complex, convoluted sentences that are still "correct," but rather teaching students
how to write clearly.
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| Jingle Bells, Batman Spells |
In
Part (14÷7(5-3)+2)÷3, I asked why so many people who would claim to hate math would get so riled up over a discussion like what the answer to 6÷2(1+2) is. And I guess I answered my own question. The reason they can get so involved in the topic, even though they hate math, is because they're not doing math at all. You want to do math?* Then tell me what you're actually trying to say, and then we can use that to have an epic discussion about commutativity and associativity and inverses and divisibility and factorization and modular arithmetic and all sorts of other cool stuff. But don't expect me to argue about what you're trying to say.
*Most people: PLEASE NO
That about sums it up for now. If you have any additions, questions, things I left out, things you think I got dead-wrong, or correct answers to 6÷2(1+2), please let me know!
