I'm not going to open a can of worms and say humans invented math, but we can and did invent order of operations. Unless the body that oversees these kinds of things changed it recently, the order traditionally has been: Parenthesis, exponents, multiplication, division, addition, subtraction.
Of course in any language clarity is important. When in doubt add a set of parentheses where you want them.
Now that all knowledge is on the web, I see that some 'authorities' list multiplication then division, and some list multiplication or division, whichever comes first left to right, so that lack of standard makes the problem a problem.
Also, apparently different calculators differ in the order of operations. In practical terms I would consider the way the C language (and its derivatives) performs its order of operations as the 'social standard'. I think that method is multiplication and division and modulo performed left to right. Also, Number (Apple's spreadsheet) and I presume Excel will yield 288 for the problem.
So I'm changing my vote to 288. This would dovetail nicely into a discussion on Platonic vs. Sophist view on life and order of operations. Clearly, when the C Programming Language and spreadsheets interpret multiplication-division from left to right, that is the right way to go.
As Pete likes to point out, the real problem with everything is just humans. We would be much better off without them.
Google search engine says 288 Mac Spotlight says 2 Excel says error problem with the equation, please go to help section to learn how to write equations
nooooo...(*) is making the equation slightly to not at all, less ambiguous...numbers and excel have to read my mind, whether I am using order of operation or multiplication by juxtaposition or whether order of operations includes multiplication by juxtaposition.
I (should) have experience with this problem. I once wrote a C code program that emulated the TI 84 calculator—at least all the functionality of the calculator that you would need for your first calculus course. As I recall I scanned the expression first and then performed the operations in order of preference. But I seem to recall that I may have determined order for similar operations by which occurred first (left to right). I can't recall whether multiplication and division were consider the same or not. It would be interesting to see the answer the TI calculator gives. My code should have given the same thing.
I also don't remember if a number next to a parenthesis implied multiplication, but I think I was smart enough to allow for that.
It seems as though the orders of operation always has multiplication and division on equal footing (right to left) but when implicit multiplication is given priority over explicit multiplication it changes.
Your emulation of the TI-84 plus sounds right. From the Texas Instrument website- "Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.
This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication is given the same priority."
No no no, It's always whats in the parenthesis first. If you had spoken this equation from left to right though by saying something like Forty Eight divided by two multiplied by 9 plus 3 thennnnnn you would get 288 but as it stands as an equation we do the parenthesis' first. Google calculator will almost always read left to right.
No, no, no. No one ever said you don't do the parentheses first. (Read the comments:) ) After you get the 12, the question is: Is Multiplication and Division of equal footing so you work left to right?
Excel, Numbers, and the C Language and its derivatives say, "yes"
11 comments:
I'm not going to open a can of worms and say humans invented math, but we can and did invent order of operations. Unless the body that oversees these kinds of things changed it recently, the order traditionally has been: Parenthesis, exponents, multiplication, division, addition, subtraction.
Of course in any language clarity is important. When in doubt add a set of parentheses where you want them.
By the way, can anyone create a voting mechanism?
Now that all knowledge is on the web, I see that some 'authorities' list multiplication then division, and some list multiplication or division, whichever comes first left to right, so that lack of standard makes the problem a problem.
Also, apparently different calculators differ in the order of operations. In practical terms I would consider the way the C language (and its derivatives) performs its order of operations as the 'social standard'. I think that method is multiplication and division and modulo performed left to right. Also, Number (Apple's spreadsheet) and I presume Excel will yield 288 for the problem.
So I'm changing my vote to 288. This would dovetail nicely into a discussion on Platonic vs. Sophist view on life and order of operations. Clearly, when the C Programming Language and spreadsheets interpret multiplication-division from left to right, that is the right way to go.
As Pete likes to point out, the real problem with everything is just humans. We would be much better off without them.
Google search engine says 288
Mac Spotlight says 2
Excel says error problem with the equation, please go to help section to learn how to write equations
The error for Excel may be that you need a * between the 2 and the parenthesis, at least that is what Numbers required to get rid of error message.
nooooo...(*) is making the equation slightly to not at all, less ambiguous...numbers and excel have to read my mind, whether I am using order of operation or multiplication by juxtaposition or whether order of operations includes multiplication by juxtaposition.
I (should) have experience with this problem. I once wrote a C code program that emulated the TI 84 calculator—at least all the functionality of the calculator that you would need for your first calculus course. As I recall I scanned the expression first and then performed the operations in order of preference. But I seem to recall that I may have determined order for similar operations by which occurred first (left to right). I can't recall whether multiplication and division were consider the same or not. It would be interesting to see the answer the TI calculator gives. My code should have given the same thing.
I also don't remember if a number next to a parenthesis implied multiplication, but I think I was smart enough to allow for that.
It seems as though the orders of operation always has multiplication and division on equal footing (right to left) but when implicit multiplication is given priority over explicit multiplication it changes.
Your emulation of the TI-84 plus sounds right. From the Texas Instrument website-
"Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.
This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication is given the same priority."
Nice work. What does all this show? Well, if nothing else, that the seemingly simplest things are often not so simple.
No no no, It's always whats in the parenthesis first. If you had spoken this equation from left to right though by saying something like Forty Eight divided by two multiplied by 9 plus 3 thennnnnn you would get 288 but as it stands as an equation we do the parenthesis' first. Google calculator will almost always read left to right.
No, no, no. No one ever said you don't do the parentheses first. (Read the comments:) ) After you get the 12, the question is:
Is Multiplication and Division of equal footing so you work left to right?
Excel, Numbers, and the C Language and its derivatives say, "yes"
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