6÷2(1+2)
I wrote a (very long) blog post about those viral math problems and am looking for feedback, especially from people who are not convinced that the problem is ambiguous.
It's about a 30min read so thank you in advance if you really take the time to read it, but I think it's worth it if you joined such discussions in the past, but I'm probably biased because I wrote it :)
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Obviously more letters would make the mnemonic worse, not better. I was making a joke.
As for the brackets 'the rules around expanding brackets' are only meaningful in the assumed context of our order of operations. For example, if we instead all agreed that addition should be before multiplication, then a×(b+c) would "expand" to a×b+c, because the addition is before multiplication anyway and the brackets do nothing.
I was making a joke.
Fair enough, but my point still stands.
if we instead all agreed that addition should be before multiplication
...then you would STILL have to do multiplication first. You can't change Maths by simply agreeing to change it - that's like saying if we all agree that the Earth is flat then the Earth is flat. Similarly we can't agree that 1+1=3 now. Maths is used to model the real world - you can't "agree" to change physics. You can't add 1 thing to 1 other thing and have 3 things now, no matter how much you might want to "agree" that there is 3, there's only 2 things. Multiplying is a binary operation, and addition is unary, and you have to do binary operators before unary operators - that is a fact that no amount of "agreeing" can change. 2x3 is actually a contracted form of 2+2+2, which is why it has to be done before addition - you're in fact exposing the hidden additions before you do the additions.
the brackets do nothing
The brackets, by definition, say what to do first. Regardless of any other order of operations rules, you always do brackets first - that is in fact their sole job. They indicate any exceptions to the rules that would apply otherwise. They perform no other function. If you're going to no longer do brackets first then you would simply not use them at all anymore. And in fact we don't - when there are redundant brackets, like in (2)(1+2), we simply leave them out, leaving 2(1+2).
I believe you're conflating the rules of maths with the notation we use to represent mathematical concepts. We can choose whatever notation we like to mean anything we like. There is absolutely nothing stopping us from choosing to interpret a+b×c as (a+b)×c rather than a+(b×c). We don't even have to write it like that at all. We could write a,b,c×+. (And sometimes people do write it like that.) Notation is just a way to communicate. It represents the maths, but it is not itself the maths. Some notation is more convenient or more intuitive than others. × before + is a very convenient choice, because it easier to express mathematical truths clearly and concisely - but nevertheless, it is still just a choice.
I believe you’re conflating the rules of maths with the notation we use to represent mathematical concepts.
You think a Maths teacher doesn't know the difference?
There is absolutely nothing stopping us from choosing to interpret a+b×c as (a+b)×c
Yes there is - the underlying Maths. 2x3 is short for 2+2+2, which is therefore why you have to expand multiplications before doing additions. If you "chose" to interpret 2+3x4 (which we KNOW is equal to 14, because 3x4=3+3+3+3 by definition) as (2+3)x4, you would get 20, which is clearly wrong, since 20 isn't equal to 14.
We don’t even have to write it like that at all
No that's right, because it IS written differently in different languages, but regardless of how you write it, it doesn't change that 2+3x4=14 - the underlying Maths doesn't change regardless of how you decide to write it. Maths is literally universal.
× before + is a very convenient choice
It's not a choice, it's a consequence of the fact that x is shorthand for +. i.e. 2x3=2+2+2.
it is still just a choice
It is a consequence of the definitions of what each operator does. If x is a contraction of +, then we have to expand x before we do +. If it were the other way around then we'd have to do it the other way around. Anything which is a contraction of something else has to be expanded first.
Hey man, if you want to resort to some weird appeal to authority argument despite having clear examples against what you are saying - go for it. You can choose to die on that hill if you want to.
clear examples against what you are saying
Which are where, exactly? You haven't presented any. You haven't, for example, shown how one can make (2+3)x4=14.
re: appeal to authority
The examples I gave were that the expansion of brackets would be done differently if the order of operations was "PESADM"; and I also drew your attention to the fact that reverse polish notation exists, in which there are no brackets at all and the order of operation is entirely determined by the order that operators appear, with no hierarchy of operations. As for your appeal to authority, let me just say that your level of qualification on this topic is not above mine. It adds no weight whatsoever to your argument.
I just glanced at your post history to get a sense of why you were so engaged in this. I was a bit startled to see that you've been on a bit of a posting spree in this thread, which I point out to you is a 3 month old post on a 'memes' channel. I see you've taken issue with a lot of what people have said here. My suggestion to you now is that there probably won't be a lot of engagement in this thread from this point on. So perhaps you should just ponder what is said, and prepare yourself again for next time this comes up. Perhaps you can start by seeing if you can get a consensus amongst fellow experts in a maths channel or something, because at the moment it seems like you're on your own.
The examples I gave were that the expansion of brackets would be done differently if the order of operations was “PESADM”
Yep I read it, and no it wouldn't. Expanding Brackets - or in the case of this mnemonic Parentheses - is done as part of B/D (as the case may be). i.e. expanding brackets isn't "multiplication" (no multiplication sign), but solving brackets (there are brackets there), which always come first in all the mnemonics.
reverse polish notation exists
...but is not taught in high school.
your level of qualification on this topic is not above mine
Maybe not, but it means it's not an "appeal to authority" (as per screenshot). Maths teachers ARE an authority on Maths. The most common appeal to authority I see from people is claiming that someone (not them) is a University professor, and "they would know". No, they wouldn't - this topic isn't taught at university - it's taught in high school.
why you were so engaged in this.
I'm a teacher. You say you're on the same level as me - don't you like to teach people what's correct?
3 month old post
Which will show up in search results for all eternity (it's how I found it - I was looking for something else!).
probably won’t be a lot of engagement in this thread from this point on
Got another 12 responses after yours. But the point is I'm not even LOOKING for responses, just to correct misinformation. As a teacher (a Maths teacher?) have you not had people say to you "But Google says"? I certainly have. It's the bane of my professions.
it seems like you’re on your own
Did you read my thread? Maths textbooks, calculators, proofs, etc. Also, someone else said what you just did, asked a Maths teacher, and was told I was correct, then was man enough to go back and edit his posts and admit I was correct and specifically said "SmartmanApps is not on his own with this".