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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)
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.
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.
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.
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).
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.
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.
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.