bcplimpton Loc: Southern New Jersey

Unfortunately according to correct mathematical notation only the 2 is under the /. If an expression is to be the divisor it must be placed under a horizontal line which acts as both the division operator and parentheses.

bcplimpton Loc: Southern New Jersey

From the article I quoted earlier:
Here is one common situation in which parentheses are assumed to be present but don't usually appear explicitly.

Example 3: Compute

fraction expression

In this situation the fraction line (also called a vinculum) indicates that the entire numerator and entire denominator must each be computed separately before the division indicated by the fraction line is carried out.

If you compute this on a calculator you must explicitly include parentheses around each of the numerator and denominator. You would enter: (2*4-3)/(√(9)+5). Note carefully the parentheses.

Dannj

For what it’s worth, I called in the experts: my 5 grandchildren currently in grades 6 thru 10. Their classroom experience in this type of math is more recent than mine and they all agree: 16.

Longshadow Loc: Audubon, PA, United States

jbk224 Loc: Long Island, NY

In general I am with you regarding your explanations. However, I don't think you go far enough with the parentheses and operations.
Everyone here agrees that operations within parentheses go first. The error is thinking that once you complete the operation within the 'p'; you continue with what appeared to be the original intent. i.e. 2(2) = 4. This is technically the same as (2)(2)=4. Also the same as 2*2=4. Yet when written as 2(2+2), the operation ends when you add what's in the P. The parentheses disappear. The expression 2(2+2) is 24. The expression 2*(2+2)=2*4=8. There is a way to get to the same answer with using the P properly. (2)(2+2)=(2)4=8. Or, 2((2+2))=2(4)=8. In both of these examples the primary operation is completed first (2+2). This leaves another operation to complete designated by either the '2' or '4' in parentheses. Complex formulas follow this requirement when requiring operations on different combinations of formulas and numbers.
None of the spreadsheet applications accept operations of multiplication, division, and addition without putting in the 'operator'--x,/,+. The assumption that this operator exists only works when the P are properly used.
Following this with 8/2(2+2)
The divisor is 2(2+2). The first operation to be done is (2+2)=4. Once this is completed, there is no further operation to be completed. The assumption is that we are to multiply 2*(2+2). This is how the spreadsheets will show this. The '*' is 2nd operation to be followed. No different than 2/(2+2), or 2/(2*2). In all of these cases there is another operation specified once the primary is completed within the P.
Try this....put these 'expressions' in order...
1(5+4), 2(5-2), 1(4*2), (3-1)1, 2(4/2), 2(2-2)
Now, put these expressions in order...
1((5+4)),2((5-2)), 1((4*2)), ((3-1))1, 2((4/2)), 2((2-2))
These are different. You can replace the 2nd set of P with * for the same result.
So, even though we assume an intent; it is the operation that must be properly specified.
8/2(2+2)= 8/24=1/3.
8/2*(2+2) = 8/2*4=8/8=1
8/2((2+2)) = 8/2(4)=8/1
Last...
There is no math that first has one divide 8/2=4 then multiply by (2+2) to equal 16.
The divisor must be fully 'expressed' before the division takes place.
This is not new math or old math. It is just math.

Longshadow Loc: Audubon, PA, United States

Wow, look what I tripped over trying to find a citation as to why "The expression 2(2+2) is 24." could ever be 24.
In my world "2(" is an implied multiplication "2*(", the way we used to hand write it before computers needed the "*".

https://mindyourdecisions.com/blog/2019/07/31/what-is-8-%C3%B7-22-2-the-correct-answer-explained/

John7199 Loc: Eastern Mass.

8

Curtis_Lowe Loc: Georgia

one is the only one

bcplimpton Loc: Southern New Jersey

What the error is that once all the operations within the parentheses are completed we must go back to the beginning and work from left to right then the division by 2 comes before the multiplication by 4 yielding 16.
You show "8/2*(2+2) = 8/2*4=8/8=1" which shows that you worked from right to left instead of left to right. Most math teachers never mention the "from left to right" in their teaching of the order of operations.

BTW, your 8/2(2+2)=8/24=1/3 ignoring the implied multiplication operator (that only those of us that learned math before calculators would see) doesn't violate any of the order of operations rules.

I think we can all agree that the original was poorly represented especially to those of us in the calculator and computer age.

Longshadow Loc: Audubon, PA, United States

Oh-oh, I understood it.....

Dleib38 Loc: Lakewood, CA

1

srscary Loc: Cary, NC

1.

Chembro2

MR BOB, THAT IS INCORRECT. IF YOU APPLY PEMDAS CORRECTLY THE ANSWER IS '16' NOT '1'. PEMDAS IS AN ACRONYM FOR PARENTHESES, EXPONENTS, MULTIPLY, DIVIDE, ADD, SUBTRACT AND IS THE ORDER OF OPERATION TO SOLVE A COMPLEX MATH PROBLEM. HENCE 2+2= 4: 8/2=4; THEREFORE 4X4='16'. I UNDERSTAND YOUR POINT THAT IF YOU (MIS)INTERPRET THE PROBLEM AS A SINGLE LINEAR LINE YOU GET '1' BUT THIS IS NOT PEMDAS MAKING THE ANSWER '1' ; NO, THIS IS A FORMAT INABILITY TO EXPRESS NUMERICAL PROBLEMS IN A THREE LINE FORMAT [ONE LINE OVER A SECOND LINE OVER A THIRD LINE] SUCH AS HOW FRACTIONS ARE NORMALLY WRITTEN. SO, MY POINT IS, WHEN YOU ELIMINATE FORMAT PROBLEM [WHICH IS JUST TRICKERY] AND INTERPRET IT CORRECTLY AS, IS AND HAS BEEN, STANDARD PRACTICE FOR CENTURIES THE CORRECT ANSWER IS '16'. REMEMBER MATH IS AN EXACT SCIENCE; THERE CAN ONLY BE ONE CORRECT ANSWER NO MATTER WHAT LIMITATIONS A CALCULATOR HAS.

papakatz45 Loc: South Florida-West Palm Beach

1

SteveR Loc: Michigan

Einstein will be totally misunderstood by this new generation.