There's quite a few calculators that get this wrong. In college, I found out that Casio calculators do things the right way, are affordable, and readily available. I stuck with it through the rest of my classes.
Casio does a wonderful job, and it's a shame they aren't more standard in American schooling. Texas Instruments costs more of the same jobs, and is mandatory for certain systems or tests. You need to pay like $40 for a calculator that hasn't changed much if at all from the 1990's.
Meanwhile I have a Casio fx-115ES Plus and it does everything that one did, plus some nice quality of life features, for less money.
$40??!! My ti that was required was like over $200!!
TI did the same thing Quark and Adobe did later on – got dominance in their markets, killed off their competition, and then sat back and rested on their laurels thinking they were untouchable
EDIT: although in part, we should thank TI for one thing – if they hadn’t monopolized the calculator market, Commodore would’ve gone into calculators instead of computers
Huge failure my ass. Come at me on munch man, Alpiner, or Tombstone City. Or coding vaguely racist things like Mr. Bojangles, one of the first codes in the early books.
Had one at home and used the hell out of it, don't get me wrong. Was my first computer. Played the Zork series on that thing. But, it had issues and wasn't a financial success.
It had fewer issues than almost anything I've owned since. I bet it would still work if I got the right adaptors. Wasn't a huge financial success though. They seemed content with early coding and games, and didn't move into word processing etc.
If you're lucky, you can find these TI calculators in thrift shops or other similar places. I've been lucky since I got both of my last 2 graphing calculators at a yard sale and thrift shop respectively, for maybe around $40-$50 for both.
The TI equivalent to the Casio fx-115ES Plus is the TI-36X Pro, and they both cost $20 at Walmart.
My Casio calculators get this wrong, even the newer ones. BTW the correct answer is 16, right?
16 is the right answer if you use PEMDAS only: (8 ÷ 2) × (2 + 2)
1 is the right answer if you use implicit/explicit with PEMDAS: 8 ÷ (2 × (2 + 2))
both are correct answers (as in if you don’t put in extra parentheses to reduce ambiguity, you should expect expect either answer)
this is also one of the reasons why postfix and prefix notations have an advantage over infix notation
postfix (HP, RPN, Forth): 2 2 + 8 2 ÷ × .
prefix (Lisp): (× (÷ 8 2) (+ 2 2))
prefix notation doesn't need parentheses either though, at least in this case. lisp uses them for readability and to get multiple arity operators. infix doesn't have any ambiguity either if you parenthesize all operations like that.
infix doesn’t have any ambiguity either if you parenthesize all operations like that
There isn't any ambiguity even if you don't.
16 is the right answer if you use PEMDAS only: (8 ÷ 2) × (2 + 2)
You added brackets and changed the answer. 2(2+2) is a single term, and if you break it up then you change the answer (because now the (2+2) is in the numerator instead of in the denominator).
1 is the right answer
The only right answer
both are correct answers
Nope, 1 is the only correct answer.
this is also one of the reasons why postfix and prefix notations have an advantage over infix notation
Except they don't. This isn't a notation problem, it's a people don't remember the rules of Maths problem.
No. Order of operations is left to right, not right to left. 1 is wrong.
Pemdas.
Multiplication comes before division.
1 is the correct answer.
That's wrong. Multiplication and division have equal precedence, same as addition and subtraction. You do them left to right. PEMDAS could be rewritten like PE(MD)(AS). After parentheses and exponents, it"s Multiplication and division together, then addition and subtraction together. They also teach BODMAS some places, which is "brackets, order, division and multiplication, addition and subtraction" Despite reversing the division and multiplication, it doesn't change the order of operations. They have the same priority, so they are just done left to right. PEMDAS and BODMAS are the different shorthand for the same order of operations.
They were right but for the wrong reason. Implied multiplication–that is, a(b) or ab–often comes before explicit multiplication and division. Apparently it's up to the person writing the equation, so the meme is intentionally and explicitly ambiguous
They're still wrong, in my humble opinion. I'm aware of this notion, and I've even had people share a snip from some book that states this as fact. However, this is not standardized and without the convention being widely understood and recognized as the standard in the world of mathematics (which generally doesn't use the symbol (÷) at all at post-algebra levels), there is no reason to treat it as such just because a few people assert it is should be.
It doesn't make sense at all to me that implied multiplication would be treated any differently, let alone at a higher priority, than explicit multiplication. They're both the same operation, just with different notations, the former of which we use as shorthand.
There are obviously examples that show the use of the division symbol without parentheses sometimes leads to misunderstandings like this. It's why that symbol is not used by real mathematicians at all. It is just abundantly more clear what you're saying if you use the fraction bar notation (the line with numerator on top and denominator on bottom). But the rules as actually written, when followed, only reach one conclusion for this problem and others like it. x÷y(z) is the SAME as x÷y*z. There's no mathematical or logical reason to treat it differently. If you meant for the implicit multiplication to have priority it should be in parentheses, x÷(y(z)), or written with the fraction bar notation.
Implicit multiplication being before regular multiplication/division is so we can write 2y/3x instead of (2y)/(3x). Without priority, 2y/3x becomes (2y÷3)•x.
Coefficients are widely used enough that mathematicians don't want to write parentheses around every single one. So implicit multiplication gets priority.
I think one could argue a coefficient on an unknown variable, like 2y, should take higher priority simply because it cannot be any further resolved or simplified. That is not the case with, say, 2(3+1). Although that does still leave you with potential ambiguity with division/multiplication, such has 1/7y. Is the coefficient 7, or is it 1/7? i.e. Is that 1/(7y)? Or (1/7)y? Either way, if that's not the the standard understood by everyone, then it is a non-standard, inconsistent rule. And as demonstrated, if you do use that rule, it needs to be more clearly defined. That is the source of this "ambiguity". If you don't include it, the order of operations rules, as written, are clear.
2y, should take higher priority simply because it cannot be any further resolved or simplified
Bingo!
That is not the case with, say, 2(3+1)
It's the same thing, where y=3+1.
1/7y. Is the coefficient 7, or is it 1/7? i.e. Is that 1/(7y)
Yes, it's 1/(7y) as per the definition of Terms.
Either way, if that’s not the the standard understood by everyone
It's the standard in literally every Maths textbook.
Why the hell are you commenting on my 4 month old comments trying to argue. Fuck off.
I agree it needs to be more clearly defined, but one of the reasons it wasn't clearly defined was because mathematicians thought it was so universal it didn't need defining, like how parentheses work to begin with.
Casio tried not doing umplicit multiplication after some american teachers complained, then went back to doing it after everyone else complained. Implicit multiplication is the standard.
I agree it needs to be more clearly defined
It's clearly defined in any Maths textbook you pick up.
No, what people are calling "implicit multiplication" is either The Distributive Law - which is the first step in solving Brackets - or Terms - and neither of these things is "multiplication". Multiplication literally refers to multiplication symbols only.
It’s why that symbol is not used by real mathematicians at all. It is just abundantly more clear what you’re saying if you use the fraction bar notation
The division symbol is used - it is not the same thing as a fraction bar.
x÷y(z) is the SAME as x÷y*z.
No, it's the same as x÷(y*z).
There’s no mathematical or logical reason to treat it differently
Terms, The Distributive Law, are why it's treated differently.
There's an argument to be made that implicit multiplication comes before division, resulting in the answer 1, but all multiplication? That's wrong, full-stop. You calculate (explicit) multiplication and division in one step, left to right. Reason being that division is technically just multiplying by the reciprocal.
a(b) is a×b. Step 2 could be rewritten as 8 / 2 × 4. Working left to right, step 3 becomes 4 × 4.
No, because implicit multiplication binds more tightly than explicit. a/b(c) becomes a/(b×(c))
Says who?
Most maths textbooks written by mathematicians.
I don't mean when they're explaining "here's how the order of operations works". I mean in the basic way that they write more advanced problems and the answers they give for them.
This video, and the prequel to it linked in the description, go into some detail showing who uses what convention and why.
Interestingly I’ve wondered if this is regional, as a fellow Aussie I learned the same as you but it seems in other places they learn the other way
FWIW I went to school in Asia, using an internationally-focused curriculum, rather than going through the Australian curriculum here in Aus.
The video I linked includes some discussion with a calculator manufacturer who apparently is under the impression that teachers in North America are asking for strict BIDMAS, so the calculator manufacturer actually switched their calculators to doing that. Until they then got blowback from the rest of the world's teachers, so they switched back to BIDMAS with juxtaposition being prioritised over division. The video also presents the case that outside of teachers—among actual maths and physics academics—prioritising juxtaposition is always preferred, even in North America.
I'm an Australian teacher who has also taught the U.K. curriculum (so I have textbooks from both countries) and, based on these comments you mention, have also Googled some U.S. textbooks, and I've yet to see any Maths textbooks that teach it "the other way". I have a very strong suspicion that it's just a lot of people in the U.S. claiming they were taught that way, but not actually being true. I had someone from Europe claim the way we (and the U.K.) teach it wasn't taught there (from memory it was Lithuania, but I'm not sure now), so I just Googled the curriculum for their country and found that indeed it is taught the same way there as here. i.e. people will just make up things in order not to admit they were wrong about something (or that their memory of it is faulty).
Huh, I'll be darned. I'm not as much of a math nerd as I thought
That's exactly where the calculators in the op differ. For more examples, Casio calculators do implicit multiplication first, while ti's treat it the same as explicit multiplication and division. I think that the latter is more predictable personally, but really you just need to know your calculator.
Casio calculators do implicit multiplication first
Actually they follow the actual rules of Maths - Terms and The Distributive Law.
Right answer, wrong words. The actual rules are Terms and The Distributive Law.
There's quite a few calculators that get this wrong. In college, I found out that Casio calculators do things the right way, are affordable, and readily available. I stuck with it through the rest of my classes.
Casio does a wonderful job, and it's a shame they aren't more standard in American schooling. Texas Instruments costs more of the same jobs, and is mandatory for certain systems or tests. You need to pay like $40 for a calculator that hasn't changed much if at all from the 1990's.
Meanwhile I have a Casio fx-115ES Plus and it does everything that one did, plus some nice quality of life features, for less money.
$40??!! My ti that was required was like over $200!!
TI did the same thing Quark and Adobe did later on – got dominance in their markets, killed off their competition, and then sat back and rested on their laurels thinking they were untouchable
EDIT: although in part, we should thank TI for one thing – if they hadn’t monopolized the calculator market, Commodore would’ve gone into calculators instead of computers
https://en.wikipedia.org/wiki/TI-99/4A
It was a huge failure, but they tried.
Huge failure my ass. Come at me on munch man, Alpiner, or Tombstone City. Or coding vaguely racist things like Mr. Bojangles, one of the first codes in the early books.
Had one at home and used the hell out of it, don't get me wrong. Was my first computer. Played the Zork series on that thing. But, it had issues and wasn't a financial success.
It had fewer issues than almost anything I've owned since. I bet it would still work if I got the right adaptors. Wasn't a huge financial success though. They seemed content with early coding and games, and didn't move into word processing etc.
If you're lucky, you can find these TI calculators in thrift shops or other similar places. I've been lucky since I got both of my last 2 graphing calculators at a yard sale and thrift shop respectively, for maybe around $40-$50 for both.
The TI equivalent to the Casio fx-115ES Plus is the TI-36X Pro, and they both cost $20 at Walmart.
My Casio calculators get this wrong, even the newer ones. BTW the correct answer is 16, right?
(8 ÷ 2) × (2 + 2)8 ÷ (2 × (2 + 2))2 2 + 8 2 ÷ × .(× (÷ 8 2) (+ 2 2))prefix notation doesn't need parentheses either though, at least in this case. lisp uses them for readability and to get multiple arity operators. infix doesn't have any ambiguity either if you parenthesize all operations like that.
There isn't any ambiguity even if you don't.
You added brackets and changed the answer. 2(2+2) is a single term, and if you break it up then you change the answer (because now the (2+2) is in the numerator instead of in the denominator).
The only right answer
Nope, 1 is the only correct answer.
Except they don't. This isn't a notation problem, it's a people don't remember the rules of Maths problem.
Yes
8 / 2 (2+2)
8 / 2 (4)
4 (4)
16
No
8 / 2 (2+2)
8 / 2 (4)
8 / 8
1
No. Order of operations is left to right, not right to left. 1 is wrong.
Pemdas.
Multiplication comes before division.
1 is the correct answer.
That's wrong. Multiplication and division have equal precedence, same as addition and subtraction. You do them left to right. PEMDAS could be rewritten like PE(MD)(AS). After parentheses and exponents, it"s Multiplication and division together, then addition and subtraction together. They also teach BODMAS some places, which is "brackets, order, division and multiplication, addition and subtraction" Despite reversing the division and multiplication, it doesn't change the order of operations. They have the same priority, so they are just done left to right. PEMDAS and BODMAS are the different shorthand for the same order of operations.
They were right but for the wrong reason. Implied multiplication–that is, a(b) or ab–often comes before explicit multiplication and division. Apparently it's up to the person writing the equation, so the meme is intentionally and explicitly ambiguous
They're still wrong, in my humble opinion. I'm aware of this notion, and I've even had people share a snip from some book that states this as fact. However, this is not standardized and without the convention being widely understood and recognized as the standard in the world of mathematics (which generally doesn't use the symbol (÷) at all at post-algebra levels), there is no reason to treat it as such just because a few people assert it is should be.
It doesn't make sense at all to me that implied multiplication would be treated any differently, let alone at a higher priority, than explicit multiplication. They're both the same operation, just with different notations, the former of which we use as shorthand.
There are obviously examples that show the use of the division symbol without parentheses sometimes leads to misunderstandings like this. It's why that symbol is not used by real mathematicians at all. It is just abundantly more clear what you're saying if you use the fraction bar notation (the line with numerator on top and denominator on bottom). But the rules as actually written, when followed, only reach one conclusion for this problem and others like it. x÷y(z) is the SAME as x÷y*z. There's no mathematical or logical reason to treat it differently. If you meant for the implicit multiplication to have priority it should be in parentheses, x÷(y(z)), or written with the fraction bar notation.
Implicit multiplication being before regular multiplication/division is so we can write 2y/3x instead of (2y)/(3x). Without priority, 2y/3x becomes (2y÷3)•x.
Coefficients are widely used enough that mathematicians don't want to write parentheses around every single one. So implicit multiplication gets priority.
I think one could argue a coefficient on an unknown variable, like 2y, should take higher priority simply because it cannot be any further resolved or simplified. That is not the case with, say, 2(3+1). Although that does still leave you with potential ambiguity with division/multiplication, such has 1/7y. Is the coefficient 7, or is it 1/7? i.e. Is that 1/(7y)? Or (1/7)y? Either way, if that's not the the standard understood by everyone, then it is a non-standard, inconsistent rule. And as demonstrated, if you do use that rule, it needs to be more clearly defined. That is the source of this "ambiguity". If you don't include it, the order of operations rules, as written, are clear.
Bingo!
It's the same thing, where y=3+1.
Yes, it's 1/(7y) as per the definition of Terms.
It's the standard in literally every Maths textbook.
Why the hell are you commenting on my 4 month old comments trying to argue. Fuck off.
I agree it needs to be more clearly defined, but one of the reasons it wasn't clearly defined was because mathematicians thought it was so universal it didn't need defining, like how parentheses work to begin with.
Casio tried not doing umplicit multiplication after some american teachers complained, then went back to doing it after everyone else complained. Implicit multiplication is the standard.
It's clearly defined in any Maths textbook you pick up.
A Maths textbook.
It's standard in every Maths textbook.
The "few people" are Maths teachers and Maths textbook authors.
There's no such thing as implicit multiplication
No, what people are calling "implicit multiplication" is either The Distributive Law - which is the first step in solving Brackets - or Terms - and neither of these things is "multiplication". Multiplication literally refers to multiplication symbols only.
The division symbol is used - it is not the same thing as a fraction bar.
No, it's the same as x÷(y*z).
Terms, The Distributive Law, are why it's treated differently.
As are you. There is no such thing as implicit multiplication - the actual relevant rules are Terms and The Distributive Law.
#MathsIsNeverAmbiguous #DontForgetDistribution
There's an argument to be made that implicit multiplication comes before division, resulting in the answer 1, but all multiplication? That's wrong, full-stop. You calculate (explicit) multiplication and division in one step, left to right. Reason being that division is technically just multiplying by the reciprocal.
There's no such thing as implicit multiplication.
1 is the correct answer, but it's because Brackets comes before Division - there is no Multiplication in this problem.
Order of operations is BEDMAS, THEN left to right within each operator.
1 is the only correct answer.
a(b) is a×b. Step 2 could be rewritten as 8 / 2 × 4. Working left to right, step 3 becomes 4 × 4.
No, because implicit multiplication binds more tightly than explicit. a/b(c) becomes a/(b×(c))
Says who?
Most maths textbooks written by mathematicians.
I don't mean when they're explaining "here's how the order of operations works". I mean in the basic way that they write more advanced problems and the answers they give for them.
This video, and the prequel to it linked in the description, go into some detail showing who uses what convention and why.
Interestingly I’ve wondered if this is regional, as a fellow Aussie I learned the same as you but it seems in other places they learn the other way
FWIW I went to school in Asia, using an internationally-focused curriculum, rather than going through the Australian curriculum here in Aus.
The video I linked includes some discussion with a calculator manufacturer who apparently is under the impression that teachers in North America are asking for strict BIDMAS, so the calculator manufacturer actually switched their calculators to doing that. Until they then got blowback from the rest of the world's teachers, so they switched back to BIDMAS with juxtaposition being prioritised over division. The video also presents the case that outside of teachers—among actual maths and physics academics—prioritising juxtaposition is always preferred, even in North America.
I'm an Australian teacher who has also taught the U.K. curriculum (so I have textbooks from both countries) and, based on these comments you mention, have also Googled some U.S. textbooks, and I've yet to see any Maths textbooks that teach it "the other way". I have a very strong suspicion that it's just a lot of people in the U.S. claiming they were taught that way, but not actually being true. I had someone from Europe claim the way we (and the U.K.) teach it wasn't taught there (from memory it was Lithuania, but I'm not sure now), so I just Googled the curriculum for their country and found that indeed it is taught the same way there as here. i.e. people will just make up things in order not to admit they were wrong about something (or that their memory of it is faulty).
Huh, I'll be darned. I'm not as much of a math nerd as I thought
Here is an alternative Piped link(s):
This video
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I'm open-source; check me out at GitHub.
Ignore the video - she completely ignored Maths textbooks (and yes, you're right, the rules are in Maths textbooks - quoted multiple times here).
The rules of Maths
That's exactly where the calculators in the op differ. For more examples, Casio calculators do implicit multiplication first, while ti's treat it the same as explicit multiplication and division. I think that the latter is more predictable personally, but really you just need to know your calculator.
Actually they follow the actual rules of Maths - Terms and The Distributive Law.
Right answer, wrong words. The actual rules are Terms and The Distributive Law.
a(b) is (a×b) - you can't remove brackets unless there is only 1 term left inside.
8/(2x4)=8/8=1
No.
8 / 2 (4) 8/(2x4) 8/8 1
Depends on the system you use. Most common system worldwide and in the academic circles (the oldest of the two) has 1 as the answer.
There are no other systems - only people who are following the actual rules of Maths and those who aren't. And yes, 1 is the correct answer
No, the correct answer is 1.
Ditto for Sharp. It's really only Texas Instruments that is the ongoing exception to the rule.
Sharp as well.