Simple Math Problem

[markem]

PEMDAS and BODMAS are mnemonics, not mathematical rules. Language is fickle and you can't have two letters occupying the same position in a word, so some order must be chosen. This is the ambiguity of these two mnemonics.

It helps to know that any division can be rewritten as a multiplication without loss of meaning and multiplication can be rewritten as division. The same is true for with regards to addition and subtraction. Remember too that multiplication is a shorthand for addition and division for subtraction.

So all this can be reduced to PEMA or PEDS / PODS. PEMA is the dominant form, probably because multiplication and addition are commutative, associative and multiplication has the distributed rule. PEMA stands for parentheses, exponentiation, multiplication, addition. Note that PEMA has no ambiguity since there is only one operation at each evaluation step (or level, if you know about the precedence pyramid: https://www.teacherspayteachers.com/...yramid-2358878). (Note that the difference between PEMDAS and BODMAS is really just selecting which item at a specific level is listed first, and thus ambiguous).

If we use multiplication for division and addition for subtraction, the solution is unambiguous (aka deterministic). Equivalent maths should give the same answer.

8 / 2(2+2) can be rewritten as
8 / 2 * (2+2) can be rewritten as
8 * (1/2) * (2+2) can be rewritten as
8 * (1/2) * 4 which reduces to
16

How the order of the operations in the final reduction step do not matter because multiplication is both commutative and associative:

(8 * (1/2)) * 4 == 8 * ((1/2) * 4) == 4 * (1/2) * 8

Note that I use (1/2) to indicate one-half, which could just as easily be replaced by 0.5, but that may complicate understanding for some.

The "DM" and "MD" of the mnemonics do not dictate a required ordering, they are just a handy memory device. There are not 6 cases to remember, just 4, since two cases can be recast in terms of other cases.

Here are some useful links:
http://mathforum.org/library/drmath/view/57199.html
http://thomas.tuerke.net/on/tech/?thread=1521859580

Since MD are at the same level of precedence, some method must be found to denote which order the operations will occur, or non-determinism results. Having a mathematical system that yields different results because of ambiguity is not good and would reduce the value of much of mathematics.

The "consensus" of about the last 100 years (yes, resolving this is a newish problem) is that operations at the same level of precedence are unambiguous when evaluated "left-to-right". Note that this is not a rule of mathematics, but a guide for resolving ambiguity.

Precedence plus the left-to-right guide removes the ambiguity. PEMA has no need for the left-to-right guide so is much simpler to get right.

The issue of PEMDAS and BODMAS (why no PEDMAS?) comes about through misunderstanding of the meaning of the mnemonics, not because math is non-deterministic. The underlying rule is PEMA. Any equation resolved with either of the mnemonics must be equivalent to PEMA or the answer is incorrect by mathematical rule, which cannot be changed. Note that you can reduce PEMA to PEA without introducing non-determinism.

So the real problem is NOT that the equation is ambiguous or opaque. It clearly is not under PEMA. The problem is a misunderstanding about the mnemonics being a mathematical rule rather than just a memory device.

[stearns]

I was following up until we got to the clarifying post above

[Weelok]

16. Here is the physics link for math orders.

https://en.m.wikipedia.org/wiki/Order_of_operations

Since the division is not a slash, think fractions, then division and multiplication are equal and evaluated left to right.

PEMDAS is flawed and incorrect.

[Weelok]

Originally Posted by markem

If we use multiplication for division and addition for subtraction, the solution is unambiguous (aka deterministic). Equivalent maths should give the same answer.

8 / 2(2+2) can be rewritten as
8 / 2 * (2+2) can be rewritten as
8 * (1/2) * (2+2) can be rewritten as
8 * (1/2) * 4 which reduces to
16

2) to indicate one-half, which could just as easily be replaced by 0.5, but that may complicate understanding for some.

.

Wow, so well done. Perfect.

[icehog3]

My brain hurts.

[longknocker]

Everything In My Old Body Hurts, Now, & I Still Don't Know The Answer!

[longknocker]

Just Looked Up Adam's Original Post On The Internet, And The Bottom Line Of The Article Gives The Correct Answer As "16".

[icehog3]

Originally Posted by longknocker

Just Looked Up Adam's Original Post On The Internet, And The Bottom Line Of The Article Gives The Correct Answer As "16".

Yeah, and they used to say the Earth was flat.

[Weelok]

Originally Posted by icehog3

Yeah, and they used to say the Earth was flat.

Wait, you mean the Earth isn’t flat?

I once convinced a Sophomore university French class that the planets in our solar system didn’t rotate around the sun which is why it’s so easy to plot a course for a satellite to take pictures of the planet.

[icehog3]

Originally Posted by Weelok

Wait, you mean the Earth isn’t flat?

I once convinced a Sophomore university French class that the planets in our solar system didn’t rotate around the sun which is why it’s so easy to plot a course for a satellite to take pictures of the planet.

I put instant coffee in a microwave oven and almost went back in time.

[markem]

[Brlesq]

Old math tells me its 1. But new math with spiky haired teenagers?

[mk05]

Microsoft Excel says 16, so it is 16. I do agree that the above 2 pages of disagreements, misunderstandings, etc could have been resolved with a clarifying bracket/parenthesis.

I didn't realize that this was an instagram question. Apparently, this has been so widely debated that Popular Science wrote about this today. The spokesman for the American Mathematical Society commented on this:

"According to order of operations, you solve whatever is in the parentheses first. That gives you 4. Then, in PEMDAS, multiplication and division take equal precedence, so you’d do the first that occurs from left to right. So you’d do 8 divided by 2 first, which is 4. Thus, it’s 16 according to classic order of operations.

But the way it’s written, it’s ambiguous. In math, a lot of times there are ambiguities. Mathematicians try to make rules as precise as possible."

A professor from a university wrote that the intent was to be a convention, a question delivered in such a way to obfuscate the reader by design. Specifically:

"Of course this isn't math. This is convention. We have conventions on how to write these things just like we have conventions on how to spell stuff. But still, there are different conventions. Some people spell it as ‘gray’ and others as ‘grey.’ We still understand what's going on. For me, I would write this more explicitly so that there is no confusion. Like this: 8/(2*(2+2)), if that's what you are trying to do. That way no one will get it wrong."

[Weelok]

Originally Posted by mk05

Microsoft Excel says 16, so it is 16. I do agree that the above 2 pages of disagreements, misunderstandings, etc could have been resolved with a clarifying bracket/parenthesis.

I didn't realize that this was an instagram question. Apparently, this has been so widely debated that Popular Science wrote about this today. The spokesman for the American Mathematical Society commented on this:

A professor from a university wrote that the intent was to be a convention, a question delivered in such a way to obfuscate the reader by design. Specifically:

Hah, yes, it was written to be an ambiguous trick but totally cool it’s sparked a national conversation on math!!!!! Much more constructive arguing over an intellectual question then on Bachelorette show drama imho

[Brlesq]

Well now I know why I answered "1". From a news article on this today:

"The confusion has to do with the difference between modern and historic interpretations of the order of operations.
The correct answer today is 16. An answer of 1 would have been correct 100 years ago." <--- THAT explains it!

[icehog3]

Originally Posted by Brlesq

Well now I know why I answered "1". From a news article on this today:

"The confusion has to do with the difference between modern and historic interpretations of the order of operations.
The correct answer today is 16. An answer of 1 would have been correct 100 years ago." <--- THAT explains it!

I wanna say it was correct 40 years ago as well.

[shilala]

Best. Post. Ever.
I just went around about this with my kid who is a physics major. She insisted it was 1.
I win. Although she still doesn’t agree.
I’m pretty sure that the fact that I make money with math and she takes that money to argue with me about math equals I’m always right. 😁

Originally Posted by markem

PEMDAS and BODMAS are mnemonics, not mathematical rules. Language is fickle and you can't have two letters occupying the same position in a word, so some order must be chosen. This is the ambiguity of these two mnemonics.

It helps to know that any division can be rewritten as a multiplication without loss of meaning and multiplication can be rewritten as division. The same is true for with regards to addition and subtraction. Remember too that multiplication is a shorthand for addition and division for subtraction.

So all this can be reduced to PEMA or PEDS / PODS. PEMA is the dominant form, probably because multiplication and addition are commutative, associative and multiplication has the distributed rule. PEMA stands for parentheses, exponentiation, multiplication, addition. Note that PEMA has no ambiguity since there is only one operation at each evaluation step (or level, if you know about the precedence pyramid: https://www.teacherspayteachers.com/...yramid-2358878). (Note that the difference between PEMDAS and BODMAS is really just selecting which item at a specific level is listed first, and thus ambiguous).

If we use multiplication for division and addition for subtraction, the solution is unambiguous (aka deterministic). Equivalent maths should give the same answer.

8 / 2(2+2) can be rewritten as
8 / 2 * (2+2) can be rewritten as
8 * (1/2) * (2+2) can be rewritten as
8 * (1/2) * 4 which reduces to
16

How the order of the operations in the final reduction step do not matter because multiplication is both commutative and associative:

(8 * (1/2)) * 4 == 8 * ((1/2) * 4) == 4 * (1/2) * 8

Note that I use (1/2) to indicate one-half, which could just as easily be replaced by 0.5, but that may complicate understanding for some.

The "DM" and "MD" of the mnemonics do not dictate a required ordering, they are just a handy memory device. There are not 6 cases to remember, just 4, since two cases can be recast in terms of other cases.

Here are some useful links:
http://mathforum.org/library/drmath/view/57199.html
http://thomas.tuerke.net/on/tech/?thread=1521859580

Since MD are at the same level of precedence, some method must be found to denote which order the operations will occur, or non-determinism results. Having a mathematical system that yields different results because of ambiguity is not good and would reduce the value of much of mathematics.

The "consensus" of about the last 100 years (yes, resolving this is a newish problem) is that operations at the same level of precedence are unambiguous when evaluated "left-to-right". Note that this is not a rule of mathematics, but a guide for resolving ambiguity.

Precedence plus the left-to-right guide removes the ambiguity. PEMA has no need for the left-to-right guide so is much simpler to get right.

The issue of PEMDAS and BODMAS (why no PEDMAS?) comes about through misunderstanding of the meaning of the mnemonics, not because math is non-deterministic. The underlying rule is PEMA. Any equation resolved with either of the mnemonics must be equivalent to PEMA or the answer is incorrect by mathematical rule, which cannot be changed. Note that you can reduce PEMA to PEA without introducing non-determinism.

So the real problem is NOT that the equation is ambiguous or opaque. It clearly is not under PEMA. The problem is a misunderstanding about the mnemonics being a mathematical rule rather than just a memory device.

[longknocker]

Originally Posted by shilala

Best. Post. Ever.
I just went around about this with my kid who is a physics major. She insisted it was 1.
I win. Although she still doesn’t agree.
I’m pretty sure that the fact that I make money with math and she takes that money to argue with me about math equals I’m always right. 😁

If Scott Says The Answer Is 16, It's 16!

[shilala]

16, *****es!!!

[icehog3]

Originally Posted by shilala

16, *****es!!!

Your favorite number, the age of most of your hose monkeys.