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8÷2(2+2) comes out to 16, not 1.
Saw it posted on Instagram or Facebook or somewhere and all of the top comments were saying 1. Any comment saying 16 had tons of comments ironically telling that person to go back to first grade and calling them stupid.
Let's see.
8÷2×(2+2) = 8÷2×4
At this point, you solve it left to right because division and multiplication are on the same level. BODMAS and PEMDAS were created by teachers to make it easier to remember, but ultimately, they are on the same level, meaning you solve it left-to-right, so....
8÷2×4 = 4×4 = 16.
So yes, it does equal 16.
Depends on whether you're a computer or a mathematician.
2(2+2) is equivalent to 2 x (2+2), but they are not equal. Using parenthesis implicitly groups the 2(2+2) as part of the paretheses function. A computer will convert 2(4) to 2 x 4 and evaluate the expression left to right, but this is not what it written. We learned in elementary school in the 90s that if you had a fancy calculator with parentheses, you could fool it because it didn't know about implicit association. Your calculator doesn't know the difference between 2 x (2+2) and 2(2+2), but mathematicians do.
Of course, modern mathematicians work primarily in computers, where the legacy calculator functions have become standard and distinctions like this have become trivial.
It seems you are partly correct. You are correct in saying that this is how it used to be done (but that was 100 years ago, it seems) and you are correct that in modern times, this would be interpreted as I did it, above.
Link: https://mindyourdecisions.com/blog/2019/07/31/what-is-8-÷-22-2-the-correct-answer-explained/
I'm old but I'm not that old.
The author of that article makes the mistake of youth, that because things are different now that the change was sudden and universal. They can find evidence that things were different 100 years ago, but 50 years ago there were zero computers in classrooms, and 30 years ago a graphing calculator was considered advanced technology for an elementary age student. We were taught the old math because that is what our teachers were taught.
Early calculators couldn't (or didn't) parse edge cases, so they would get this equation wrong. Somewhere along the way, it was decided that it would be easier to change how the equation was interpreted rather than reprogram every calculator on earth, which is a rational decision I think. But that doesn't make the old way wrong, anymore than it makes cursive writing the wrong way to shape letters.
it was decided that it would be easier to change how the equation was interpreted
No, it wasn't. The claim that the rules were changed is a debunked myth.
No, that video is wrong. Not only that, if you check the letter he referenced Lennes' Letter, you'll find it doesn't support his assertion that the rules changed at all! And that's because they didn't change. Moral of the story Always check the references.
A computer will convert 2(4) to 2 x 4
Only if that's what the programmer has programmed it to do, which is unfortunately most programmers. The correct conversion is 2(4)=(2x4).
in the 90s that if you had a fancy calculator with parentheses, you could fool it because it didn’t know about implicit association. Your calculator doesn’t know the difference between 2 x (2+2) and 2(2+2), but mathematicians do
Actually it's only in the 90's that some calculators started getting it wrong - prior to that they all gave correct answers.
8÷2×(2+2)
But that's not the same thing as 8÷2(2+2). 2x(2+2) is 2 Terms, 2(2+2) is 1 Term. 8÷2×(2+2)=16 ((2+2) is in the numerator), 8÷2(2+2)=1 (2(2+2) is in the denominator)
Under pemdas divisor operators must literally be completed after multiplication. They are not of equal priority unless you restructure the problem to be of multiplication form, which requires making assumptions about the intent of the expression.
Okay, let me put it in other words: Pemdas and bodmas are bullshit. They are made up to help you memorise the order of operations. Multiplication and division are on the same level, so you do them linearly aka left to right.
Pemdas and bodmas are not bullshit, they are a standard to disambiguate expression communication. They are order of operations. Multiplication and division are not on the same level, they are distinct operations which form the identity when combined with a multiplication.
Similarly, log(x) and e^x are not the same operation, but form identity when composited.
Formulations of division in algebra allow it to be at the same priority as multiplication by restructuring it as multiplication, but that requires formulating the expression a particular way. The ÷ operator however is strictly division. That's its purpose. It's not a fantastic operator for common usage because of this.
There are valid orders of operations, such as depmas which I just made up which would make the above expression extremely ambiguous. Completely mathematically valid, order of ops is an established convention, not mathematical fact.
This comment is the epitome of being confidently wrong on the internet.
For one misinterpretation? Are you sure about that?
There was 3 misinterpretations - see my reply to them.
confidently wrong on the internet
I made a hashtag for people #LoudlyNotUnderstandingThings :-)
They are order of operations
No, they're not.
Multiplication and division are not on the same level
Yes, they are.
they are distinct operations which form the identity when combined with a multiplication
In other words, they are the inverse operation of each other - welcome to why they have the same precedence.
order of ops is an established convention, not mathematical fact
It's a mathematical fact.
Under pemdas divisor operators must literally be completed after multiplication
Not literally. It's only a mnemonic, not the actual rules.
They are not of equal priority
Yes, they are. Binary operators have equal precedence, and unary operators have equal precedence.
No, 2+2 = 🐟 so it would be 8÷2🐟 and since 🐟 is no longer a number it becomes 4🐟. So the answer is 4 fishes.
since 🐟 is no longer a number
It's still a pronumeral though, equal to 4, so the answer is still 8÷8=1.
And both you and people arguing that it's 1 would be wrong.
This problem is stated ambiguously and implied multiplication sign between 2 and ( is often interpreted as having priority. This is all matter of convention.
I see what you're getting at but the issue isn't really the assumed multiplication symbol and it's priority. It's the fact that when there is implicit multiplication present in an algebraic expression, and really best practice for any math above algebra, you should never use the '÷' symbol. You need to represent the division as a numerator and denominator which gets rid of any ambiguity since the problem will explicitly show whether (2+2) is modifying the numerator or denominator. Honestly after 7th grade I can't say I ever saw a '÷' being used and I guess this is why.
That said, I'll die on a hill that this is 16.
There is another example where the pemdas is even better covered than a simple parenthetical multiplication, but the answer there is the same: It's the arbitrary syntax, not the math rules.
You guys are both correct. It's 16 and the problem is a syntax that implies a wrong order of operations. The syntax isn't wrong, either, just implicative in your example and seemingly arbitrary in the other example I wish I remembered.
Do you not understand that syntax is its own set of rules?
Do you not understand that syntax is its own set of rules?
Yes, the rules of Maths, as I was already saying. I'm a Maths teacher. I take it you didn't read the link then.
the assumed multiplication symbol and it’s priority.
Precedence is the term usually used for this (at least anywhere where computers have to parse expressions)
I’ll die on a hill that this is 16
both you and people arguing that it’s 1 would be wrong
No, they're correct Order of operations thread index
This problem is stated ambiguously and implied multiplication
It's not ambiguous, there's no such thing as implicit multiplication
This is all matter of
...following the rules of Maths.
A matter of convention: true
Unless you specify you aren't using pemdas, that's generally the assumed order of ops.
This is not one of the ambiguous ones, but it's certainly written to be. Multiplication does indeed have priority under pemdas.
A matter of convention: true
False. Actual rules of Maths
This is not one of the ambiguous ones
There aren't any ambiguous ones - #MathsIsNeverAmbiguous
Back in gradeschool I was always taught that in Pemdas, the parenthesis are assumed to be there in 8÷(2×(2+2)) where as 8÷2×(2+2) would be 16, 8÷2(2+2) is the above and equals 1.
Not quite. It's true you resolve what's inside the parentheses first, giving you. 8÷2(4) or 8÷2x4.
Now this is what gets most people. Even though Multiplication technically comes before Division the Acronym PEMDAS, that's really just to make it sound correct phonetically. Really they have equal priority in the order of operations and the appropriate way to resolve the problem is to work from left to right solving each multiplication or division sign as you encounter them. Giving you 16. Same for addition and subtraction.So basically the true order of operations is:
- Work left to right solving anything inside parentheses
- Work left to right solving any exponentials
- Work left to right solving any multiplication or division
- Work left to right solving any addition or subtraction
Source: Mechanical Engineering degree so an unfortunate amount of my life spent in math and physics classes.
Absolutely, its all seen as equal so it has to go left to right However as I said in the beginning the way I was taught atleast, is when you see 2(2+2) and not 2×(2+2) you assume that 2(2+2) actually means (2×(2+2 )) and so must do it together.
Ah sorry just realized what you were saying. I've never been taught that. Maybe it's just a difference in teaching styles, but it shouldn't be since it can actually change the outcome. The way I was always taught was if you see a number butted up against an expression in parentheses you assume there is a multiplication symbol there.
So you were taught that 2(2+2) == (2(2+2))
I was taught 2(2+2)==2*(2+2)Interesting difference though because again, assuming invisible parentheses can really change up how a problem is done.
Edit: looks like theshatterstone54's comment assumed a multiplication symbol as well.
if you see a number butted up against an expression in parentheses you assume there is a multiplication symbol there
No, it means it's a Term (product). If a=2 and b=3, then axb=2x3, but ab=6.
I was taught 2(2+2)==2*(2+2)
2(2+2)==(2*(2+2)). More precisely, The Distributive Law says that 2(2+2)=(2x2+2x2).
That's basically what I was taught, too.
Edit to add: Ha, I just realized how similar our usernames are. Neat! :)
Basically the normal arithmetic operators are all left-associative which means if you have more than one you solve them left to right.
It’s true you resolve what’s inside the parentheses first, giving you. 8÷2(4) or 8÷2x4.
Not "inside parenthesis" (Primary School, when there's no coefficient), "solve parentheses" (High School, The Distributive Law). Also 8÷2(4)=8÷(2x4) - prematurely removing brackets is how a lot of people end up with the wrong answer (you can't remove brackets unless there is only 1 term left inside).
Yes, it's The Distributive Law.
Great explainer on the subject: https://youtu.be/lLCDca6dYpA?si=gUJlQJgfDxi-n_Y6
And a follow up on how calculators actually implement this inconsistently: https://youtu.be/4x-BcYCiKCk?si=g5pqwXvBqSS8Q5fX
Here is an alternative Piped link(s):
https://piped.video/lLCDca6dYpA?si=gUJlQJgfDxi-n_Y6
https://piped.video/4x-BcYCiKCk?si=g5pqwXvBqSS8Q5fX
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I'm open-source; check me out at GitHub.
Both of those Youtubes debunked in this thread.
2(4) is not exactly same as 2x4.
2(4) is not exactly same as 2x4
Correct! It's exactly the same as (2x4).
No. No. You choose to be ignorant.
Ummm, I was agreeing with you??
Anyways, I'm a Maths teacher who has taught this topic many times - what would I know?
Math should be taught with postfix notation and this wouldn't be an issue. It turns your expression into this.
8 2 ÷ 2 2 + ×It already isn't an issue if people just follow all the rules of Maths.
Is that
8——— 2(2+2)
Or
8 — (2+2) 2
The division symbol is ambiguous so it could mean either
8÷2(2+2) comes out to 16, not 1
No, it's 1, and only 1. Order of operations thread index
P.S. this is Year 7 Maths, not Year 1.
Under normal interpretations of pemdas this is simply wrong, but it's ok. Left to right only applies very last, meaning the divisor operator must literally come after 2(4).
This isn't really one of the ambiguous ones but it's fair to consider it unclear.
Pemdas puts division and multiplication on the same level, so 34/22 is 12 not 3. Implicit multiplication is also multiplication. It's a question of convention, but by default, it's 16.
Implicit multiplication is also multiplication
There's no such thing as implicit multiplication. The answer is 1.
I don't know what you're on about with your distributive law thing. That just states that
a*(b + c) = a*b + a*c, and has literally no relation to notation.And "math is never ambiguous" is a very bold claim, and certainly doesn't hold for mathematical notation. For some simple exanples, see here: https://math.stackexchange.com/questions/1024280/most-ambiguous-and-inconsistent-phrases-and-notations-in-maths#1024302
That just states that a*(b + c) = ab + ac
No, The Distributive Law states that a(b+c)=(ab+ac), and that you must expand before you simplify.
For some simple exanples,
Examples by people who simply don't remember all the rules of Maths. Did you read the answers?
Please learn some math before making more blatantly incorrect statements. Quoting yourself as a source is... an interesting thing to do.
https://en.m.wikipedia.org/wiki/Distributive_property
I did read the answers, try doing that yourself.
Please learn some math
I'm a Maths teacher - how about you?
Quoting yourself as a source
I wasn't. I quoted Maths textbooks, and if you read further you'll find I also quoted historical Maths documents, as well as showed some proofs.
I didn't say the distributive property, I said The Distributive Law. The Distributive Law isn't ax(b+c)=ab+ac (2 terms), it's a(b+c)=(ab+ac) (1 term), but inaccuracies are to be expected, given that's a wikipedia article and not a Maths textbook.
I did read the answers, try doing that yourself
I see people explaining how it's not ambiguous. Other people continuing to insist it is ambiguous doesn't mean it is.
About the ambiguity: If I write
f^{-1}(x), without context, you have literally no way of knowing whether I am talking about a multiplicative or a functional inverse, which means that it is ambiguous. It's correct notation in both cases, used since forever, but you need to explicitly disambiguate if you want to use it.I hope this helps you more than the stackexchange post?
If I write f^{-1}(x), without context, you have literally no way of knowing whether I am talking about a multiplicative or a functional inverse, which means that it is ambiguous
The inverse of the function is f(x)^-1. i.e. the negative exponent applies to the whole function, not just the x (since f(x) is a single term).
If you read the wikipedia article, you would find it also stating the distributive law, literally in the first sentence, which is just that the distributive property holds for elemental algebra. This is something you learn in elementary school, I don't think you'd need any qualification besides that, but be assured that I am sufficiently qualified :)
By the way, Wikipedia is not intrinsically less accurate than maths textbooks. Wikipedia has mistakes, sure, but I've found enough mistakes (and had them corrected for further editions) in textbooks. Your textbooks are correct, but you are misunderstanding them. As previously mentioned, the distributive law is about an algebraic substitution, not a notational convention. Whether you write it as
a(b+c) = ab + acor asa*(b+c) = a*b + a*cis insubstantial.If you read the wikipedia article
...which isn't a Maths textbook!
also stating the distributive law, literally in the first sentence
Except what it states is the Distributive property, not The Distributive Law. If I call a Koala a Koala Bear, that doesn't mean it's a bear - it just means I used the wrong name. And again, not a Maths textbook - whoever wrote that demonstrably doesn't know the difference between the property and the law.
This is something you learn in elementary school
No it isn't. This is a year 7 topic. In Primary School they are only given bracketed terms without a coefficient (thus don't need to know The Distributive Law).
be assured that I am sufficiently qualified
No, I'm not assured of that when you're quoting wikipedia instead of Maths textbooks, and don't know the difference between The Distributive Property and The Distributive Law, nor know which grade this is taught to.
Wikipedia is not intrinsically less accurate than maths textbooks
BWAHAHAHAHA! You know how many wrong things I've seen in there? And I'm not even talking about Maths! Ever heard of edit wars? Whatever ends up on the page is whatever the admin believes. Wikipedia is "like an encyclopedia" in the same way that Madonna is like a virgin.
but you are misunderstanding them
And yet you have failed to point out how/why/where. In all of your comments here, you haven't even addressed The Distributive Law at all.
Whether you write it as a(b+c) = ab + ac or as a*(b+c) = ab + ac is insubstantial
And neither of those examples is about The Distributive Law - they are both to do with The Distributive Property (and you wrote the first one wrong anyway - it's a(b+c)=(ab+ac). Premature removal of brackets is how many people end up with the wrong answer).
Let me quote from the article:
"In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
x*(y+z) = x*y + x*zis always true in elementary algebra."This is the first sentence of the article, which clearly states that the distributive property is a generalization of the distributive law, which is then stated.
Make sure you can comprehend that before reading on.
To make your misunderstanding clear: You seem to be under the impression that the distributive law and distributive property are completely different statements, where the only difference in reality is that the distributive property is a property that some fields (or other structures with a pair of operations) may have, and the distributive law is the statement that common algebraic structures like the integers and the reals adhere to the distributive property.
I don't know which school you went to or teach at, but this certainly is not 7th year material.
Incorrect, pemdas puts multiplication before division.
I always thought pemdas was more like P/E/MD/AS with MD and AS occurring left to right
This is how I was taught, but also people don't really use the ÷ symbol in algebra beyond like 6th grade
people don’t really use the ÷ symbol in algebra beyond like 6th grade
Yes they do, just pick up a high school Maths textbook (in a country which uses obelus rather than colon).
And "Multiplication" refers literally to multiplication signs, of which there are none in this question.
This isn’t really one of the ambiguous ones but it’s fair to consider it unclear.
#MathsIsNeverAmbiguous if you follow all the rules of Maths (there's a lot of people here who aren't).