For anyone like me who has math as their worst subject: PEMDAS.
PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations. PEMDAS stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.
So we gotta do it in the proper order. And remember, if the number is written like 2(3) then its multiplication, as if it was written 2 x 3 or 2 * 3.
So we read 8/2(2+2) and need to do the following;
Read the Parentheses of (2 + 2) and follow the order of operations within them, which gets us 4.
Then we do 2(4) which is the same as 2 x 4 which is 8
8 / 8 is 1.
The answer is 1. The old calculator is correct, the phone app which has ads backed into it for a thing that all computers were invented to do is inaccurate.
Correct! 2(2+2) is a single term - subject to The Distributive Law - and 2x(2+2) is 2 terms. Those who added a multiply sign there have effectively flipped the (2+2) from being in the denominator to being in the numerator, hence the wrong answer.
But it’s not called “implicit multiplication” - it’s Terms and/or The Distributive Law which applies (and they’re 2 separate rules, so you cannot lump them together as a single rule).
Not quite, pemdas can go either from the left or right (as long as you are consistent) and division is the same priority as multiplication because dividing by something is equal to multiplying by the inverse of that thing… same as subtraction being just addition but you flip the sign.
8×1/2=8/2
1-1=1+(-1)
The result is 16 if you rewrite the problem with this in mind: 8÷2(2+2)=8×(1/2)×(2+2)
I’ve always heard it that way too but I think it is for consistency with students, imo
Logically, if you are looking at division = multiplying by inverse and subtraction = adding the negative, you should be able to do it both ways. Addition and multiplication are both associative, so we can do 1+2+3 = (1+2)+3 = 1+(2+3) and get the same answer.
But subtraction and division are not associative. Any time you work on paper, 2 - 2 - 2 would equal -2. That is, (2-2)-2=0-2=-2. If you evaluate right to left, you get 2-2-2=2-(2-2)=2-0=2
Correct, subtraction and division are not associative. However, what is subtraction if not adding the opposite of a number? Or division if not multiplying the inverse? And addition and multiplication are associative.
2-2-2 can be written as 2 + (-2) + (-2) which would equal -2 no matter if you solve left to right, or right to left.
In your example with the formula from right to left, distributing the negative sign reveals that the base equation was changed, so it makes sense that you saw a different answer.
But you broke the rule of left-associativity there. You can go right to left provided you keep each number with the sign to it’s left (and you didn’t do that when you separated the first 2 in brackets from it’s minus sign).
I’ve never had anyone tell me operations with the same priority can be done either way, it’s always been left to right
It’s left to right within each operator. You can do multiplication first and division next, or the other way around, as long as you do each operator left to right. Having said that, you also can do the whole group of equal precedence operators left to right - because you’re still preserving left to right for each of the two operators - so you can do multiplication and division left to right at the same time, because they have equal precedence.
Having said that, it’s an actual rule for division, but optional for the rest. The actual rule is you have to preserve left-associativity - i.e. a number is associated with the sign to the left of it - and going left to right is an easy way to do that.
No, that’s wrong. 2(2+2) is a single term, and thus entirely in the denominator. When you separated the coefficient you flipped the (2+2) into the numerator, hence the wrong answer. You must never add multiplication signs where there are none.
That’s not the same as 8 / 2 (2 + 2). In the original question, 2(2+2) is a single term in the denominator, when you added the multiply you separated it and thus flipped the (2+2) to be in the numerator, hence the wrong answer.
Yeah, I know, but it’ll show up in search results for all eternity, and it’s full of disinformation. As a Maths teacher/tutor who is sick of hearing “But Google/Wolfram/TI says…”, I’m doing my best to try and get said people to fix their damn calculators (and also make people aware that those calculators are wrong. (sigh) I miss the old days when all calculators gave the right answer - what happened??). If you feel the same way then feel free to share my links - that’s what they’re there for. :-)
And right at the moment I’m feeling too tired to do anything which requires me to think about it, so just doing what I can do on automatic. ;-)
There’s no multiplication in this question - multiplication refers literally to multiplication signs - only division and brackets, and addition within the brackets. So you have to use The Distributive Law to solve the brackets, then do the division, giving you 1.
not to be That Guy, but the phone is actually correct… multiplication and division have the same precedence, so 8 / 2 * 4 should give the same result as 8 * 4 / 2, ie 16
The problem with this is that the division symbol is not an accurate representation of the intended meaning. Division is usually written in fractions which has an implied set of parenthesis, and is the same priority as multiplication. This is because dividing by a number is the same as multiplying by the inverse, same as subtracting is adding the negative of a number.
8/2(2+2) could be rewritten as 8×1/2×(2+2) or (8×(2+2))/2 which both resolve into 16.
Fractions are explicitly Terms. Terms are separated by operators (such as division) and joined by grouping symbols (such as a fraction bar), so 1÷2 is 2 terms, but ½ is 1 term.
8/2(2+2) could be rewritten as 8×1/2×(2+2)
No, it can’t. 2(2+2) is 1 term, in the denominator. When you added the multiply you broke it into 2 terms, and sent the (2+2) into the numerator, thus leading to a different answer. 8/2(2+2)=1.
The latter is correct, Multiplication/Division, and Addition/Subtraction each evaluate left to right (when not made unambiguous by Parentheses). I.e., 6÷2×3 = 9, not 1. That said, writing the expression in a way that leaves ambiguity is bad practice. Always use parentheses to group operations when ambiguity might arise.
Ignore the idiots telling you you’re wrong. Everyone with a degree in math, science or engineering makes a distinction between implicit and explicit multiplication and gives implicit multiplication priority.
Turns out I’m wrong, but I haven’t been told how or why. I’m willing to learn if people actually tell me
Well, I don’t know what you said originally, so I don’t know what it is you were told was wrong - 1 or 16? 😂 The correct answer is 1.
Anyhow, I have an order of operations thread which covers literally everything there is to know about it (including covering all the common mistakes and false claims made by some). It includes textbook references, historical Maths documents, worked examples, proofs, memes, the works! I’m a high school Maths teacher/tutor - I’ve taught this topic many times.
PEMDAS is what they teach school children to help them learn. Like most such rules, it’s oversimplified and doesn’t quite work all the time.
The proper way is to write a fraction top and bottom. Thus, 8(2+2) / 2 = 16, while 8 / 2(2+2) = 1. The implicit multiplication simply does not work if they’re split across the fraction.
For anyone like me who has math as their worst subject: PEMDAS.
So we gotta do it in the proper order. And remember, if the number is written like
2(3)then its multiplication, as if it was written2 x 3or2 * 3.So we read
8/2(2+2)and need to do the following;(2 + 2)and follow the order of operations within them, which gets us 4.2(4)which is the same as2 x 4which is88 / 8is1.The answer is 1. The old calculator is correct, the phone app which has ads backed into it for a thing that all computers were invented to do is inaccurate.
Well that’s just wrong… Multiplication and division have equal priorities so they are done from left to right. So: 8 / 2 * (2 + 2)=8 / 2 * 4=4 * 4=16
This but unironically
Implicit multiplication takes priority over explicit multiplication or division. 2(2+2) is not the same thing as 2*(2+2).
Correct! 2(2+2) is a single term - subject to The Distributive Law - and 2x(2+2) is 2 terms. Those who added a multiply sign there have effectively flipped the (2+2) from being in the denominator to being in the numerator, hence the wrong answer.
But it’s not called “implicit multiplication” - it’s Terms and/or The Distributive Law which applies (and they’re 2 separate rules, so you cannot lump them together as a single rule).
Not quite, pemdas can go either from the left or right (as long as you are consistent) and division is the same priority as multiplication because dividing by something is equal to multiplying by the inverse of that thing… same as subtraction being just addition but you flip the sign.
8×1/2=8/2 1-1=1+(-1)
The result is 16 if you rewrite the problem with this in mind: 8÷2(2+2)=8×(1/2)×(2+2)
I’ve never had anyone tell me operations with the same priority can be done either way, it’s always been left to right.
I’ve always heard it that way too but I think it is for consistency with students, imo Logically, if you are looking at division = multiplying by inverse and subtraction = adding the negative, you should be able to do it both ways. Addition and multiplication are both associative, so we can do 1+2+3 = (1+2)+3 = 1+(2+3) and get the same answer.
But subtraction and division are not associative. Any time you work on paper, 2 - 2 - 2 would equal -2. That is, (2-2)-2=0-2=-2. If you evaluate right to left, you get 2-2-2=2-(2-2)=2-0=2
Correct, subtraction and division are not associative. However, what is subtraction if not adding the opposite of a number? Or division if not multiplying the inverse? And addition and multiplication are associative.
2-2-2 can be written as 2 + (-2) + (-2) which would equal -2 no matter if you solve left to right, or right to left.
In your example with the formula from right to left, distributing the negative sign reveals that the base equation was changed, so it makes sense that you saw a different answer.
2 - (2 - 2) = 2 + ((-2) + 2) = 2
But you broke the rule of left-associativity there. You can go right to left provided you keep each number with the sign to it’s left (and you didn’t do that when you separated the first 2 in brackets from it’s minus sign).
It’s left to right within each operator. You can do multiplication first and division next, or the other way around, as long as you do each operator left to right. Having said that, you also can do the whole group of equal precedence operators left to right - because you’re still preserving left to right for each of the two operators - so you can do multiplication and division left to right at the same time, because they have equal precedence.
Having said that, it’s an actual rule for division, but optional for the rest. The actual rule is you have to preserve left-associativity - i.e. a number is associated with the sign to the left of it - and going left to right is an easy way to do that.
No, that’s wrong. 2(2+2) is a single term, and thus entirely in the denominator. When you separated the coefficient you flipped the (2+2) into the numerator, hence the wrong answer. You must never add multiplication signs where there are none.
That’s not the same as 8 / 2 (2 + 2). In the original question, 2(2+2) is a single term in the denominator, when you added the multiply you separated it and thus flipped the (2+2) to be in the numerator, hence the wrong answer.
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Me searching the place for my old Ti83 after stumbling on this thread.
My partner: why are you even looking for this thing??
Me: Somebody on the Internet is WRONG.
My partner: Good enough, I’ll help!
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I know, it’s good to have someone in your life with priorities.
I happened to have my ti-83 plus next to me. I got 16!
YES finally, some sense in here. 🤣
I ended up downloading a Ti-83 emulator because mine has been devoured by the magical house gremlins.
Except TI calculators also give the wrong answer when a bracketed term has a coefficient.
Then I admit I’m wrong, I just don’t get how PEMDAS/I got it wrong.
Short story, PEMDAS kinda sucks, because it has 4 levels of priority, but 6 letters:
M and D are the same level, priority is from left to right.
A and S are the same level, priority is from left to right.
Just tested this on a Ti-83 too.
8/2(2+2) = 16
My one big criticism of the Ti-83/84 is implicit multiplication. Ti says 1/2x is 0.5x when I needed the reciprocal of 2x.
Yeah, that’s exactly the problem with TI calculators - ignoring the rules of Maths.
My calculator gets 1. Weird.
That is the correct answer.
I thank you for proving me right, but this a 4 month old thread. Not complaining, just giving you a heads up.
Yeah, I know, but it’ll show up in search results for all eternity, and it’s full of disinformation. As a Maths teacher/tutor who is sick of hearing “But Google/Wolfram/TI says…”, I’m doing my best to try and get said people to fix their damn calculators (and also make people aware that those calculators are wrong. (sigh) I miss the old days when all calculators gave the right answer - what happened??). If you feel the same way then feel free to share my links - that’s what they’re there for. :-)
And right at the moment I’m feeling too tired to do anything which requires me to think about it, so just doing what I can do on automatic. ;-)
And also if I hadn’t been replying to this thread then you wouldn’t have got the proof that you were right. :-)
I love you brought the TI truth lul
TI calculators disobey The Distributive Law and give the wrong answer when a bracketed term has a coefficient.
Uh… no the 1 is wrong? Division and multiplication have the same precedence, so the correct order is to evaluate from left to right, resulting in 16.
The real correct order is to use brackets to remove ambiguity.
Exactly, these types of problems are designed to make people confused and discuss PEMDAS and drive social media engagement.
There’s no multiplication in this question - multiplication refers literally to multiplication signs - only division and brackets, and addition within the brackets. So you have to use The Distributive Law to solve the brackets, then do the division, giving you 1.
not to be That Guy, but the phone is actually correct… multiplication and division have the same precedence, so
8 / 2 * 4should give the same result as8 * 4 / 2, ie 16No, it’s actually wrong.
It’s 8/(2x4). You can’t remove brackets unless there is only 1 term left inside.
The problem with this is that the division symbol is not an accurate representation of the intended meaning. Division is usually written in fractions which has an implied set of parenthesis, and is the same priority as multiplication. This is because dividing by a number is the same as multiplying by the inverse, same as subtracting is adding the negative of a number.
8/2(2+2) could be rewritten as 8×1/2×(2+2) or (8×(2+2))/2 which both resolve into 16.
You left out the way it can be rewritten which most mathematicians would actually use, which is 8/(2(2+2)), which resolves to 1.
Division and fractions aren’t the same thing.
Fractions are explicitly Terms. Terms are separated by operators (such as division) and joined by grouping symbols (such as a fraction bar), so 1÷2 is 2 terms, but ½ is 1 term.
No, it can’t. 2(2+2) is 1 term, in the denominator. When you added the multiply you broke it into 2 terms, and sent the (2+2) into the numerator, thus leading to a different answer. 8/2(2+2)=1.
P E M D A S
vs
P E M/D A/S
The latter is correct, Multiplication/Division, and Addition/Subtraction each evaluate left to right (when not made unambiguous by Parentheses). I.e., 6÷2×3 = 9, not 1. That said, writing the expression in a way that leaves ambiguity is bad practice. Always use parentheses to group operations when ambiguity might arise.
PEMDAS evaluated from left to right. If you followed that you’d get 16. 1 is ignoring left to right.
1 isn’t ignoring anything. 16 can be arrived at by ignoring any one of multiple order of operations rules.
You are correct. This is the right sequence of operations done here.
Ignore the idiots telling you you’re wrong. Everyone with a degree in math, science or engineering makes a distinction between implicit and explicit multiplication and gives implicit multiplication priority.
Well, I don’t know what you said originally, so I don’t know what it is you were told was wrong - 1 or 16? 😂 The correct answer is 1.
Anyhow, I have an order of operations thread which covers literally everything there is to know about it (including covering all the common mistakes and false claims made by some). It includes textbook references, historical Maths documents, worked examples, proofs, memes, the works! I’m a high school Maths teacher/tutor - I’ve taught this topic many times.
You’re a lifesaver, thank you so much. I actually didn’t know about PEMDAS, I was never taught it before…
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Ohh, I see. Thank you for clarifying, Math has always been the bane of my existence. x_x
PEMDAS is what they teach school children to help them learn. Like most such rules, it’s oversimplified and doesn’t quite work all the time.
The proper way is to write a fraction top and bottom. Thus, 8(2+2) / 2 = 16, while 8 / 2(2+2) = 1. The implicit multiplication simply does not work if they’re split across the fraction.
PEMDAS sucks as a mnemonic, we can agree on this.
Try 8/2(2+2) on an actual calculator that’s used by scientists.
Something like a Ti-83 would show 16, not 1.
Replace (2+2) with x:
8 / 2x = 4 / x = 4 / (2+2) = 1
Implicit multiplication takes priority over division.
My Casio fx-991 gives 1. If I put in an explicit multiplication, then I get 16.