jbk224 wrote:
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.
In general I am with you regarding your explanatio... (
show quote)
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.