What's the answer to this very basic maths problem?

h4rm0ny · 15 Oct 2017 at 22:27

dowie said:
I did read that post you quoted thanks. Perhaps you can explain how it has changed over time? At what point in time did it change?

I suspect you're just making things up based on some Facebook chat/confusion over a silly question that has been posted in order to cause confusion but am willing to be corrected.

It's news to me that there is a new "accepted" way to do things because I'm pretty sure most people still do it the way "old people" do. And that it is in fact still taught that way. The problem is that they write it out as if it's a sequence of steps. Like using a basic calculator. Normally it would be written like this:

And that's how a scientific calculator would treat it.

Simon Parker said:
the calculator has precedence doesn't it ?

Scientific / proper calculators do. By default. Very basic calculators don't because each operator you type is a separate sum. You can tell because it gives you each answer as you go along. Whereas in actual maths, it's not a series of steps but an equation that has to all exist because you can begin solving it. Basic calculators have no concept of waiting for everything to be entered before working it out. Mathematicians, however, don't base how maths works around a cheap calculator, however.

StuBob · 15 Oct 2017 at 22:32

I end up with -9???

Akagi · 15 Oct 2017 at 23:23

Vonhelmet said:
A friend of mine consistently posts ridiculous answers on every stupid maths thing that turns up on Facebook.

Hi Pete.

ubersonic · 15 Oct 2017 at 23:43

simpletom said:
When did this ever equal 1?

The accepted method used to be that using the divide symbol meant that you divided by the entire product of the right equation.

Put simply the current accepted method goes like this:
6÷2(1+2)
= 6÷2(3)
= 6÷2×3
= 3×3
= 9

However the previous convention was to do it like this:
6÷2(1+2)
= 6÷2(3)
= 6÷(2(3))
= 6÷6
= 1

Hence it being an example of how the changing usage of mathematics conventions over time has changed the answer deemed "correct" by the masses. The reason it causes arguments is because many older people who were taught the previous method and even older scientific calculators will still give the now depreciated answer.

simpletom · 15 Oct 2017 at 23:51

ubersonic said:
The accepted method used to be that using the divide symbol meant that you divided by the entire product of the right equation.

However the previous convention was to do it like this:
6÷2(1+2)
= 6÷2(3)
= 6÷(2(3))
= 6÷6
= 1

Hence it being an example of how the changing usage of mathematics conventions over time has changed the answer deemed "correct" by the masses. The reason it causes arguments is because many older people who were taught the previous method and even older scientific calculators will still give the now depreciated answer.

Ummm... when did this change occur? Was it a sudden decision by... everybody on Earth?

Vonhelmet · 16 Oct 2017 at 00:10

Acme said:
Hi Pete.

muon · 16 Oct 2017 at 00:22

ubersonic said:
The accepted method used to be that using the divide symbol meant that you divided by the entire product of the right equation.

Put simply the current accepted method goes like this:
6÷2(1+2)
= 6÷2(3)
= 6÷2×3
= 3×3
= 9

However the previous convention was to do it like this:
6÷2(1+2)
= 6÷2(3)
= 6÷(2(3))
= 6÷6
= 1

Hence it being an example of how the changing usage of mathematics conventions over time has changed the answer deemed "correct" by the masses. The reason it causes arguments is because many older people who were taught the previous method and even older scientific calculators will still give the now depreciated answer.

The issue is that there is no order for the division and multiplication.

But where it is badly written I will always go left to right just as a computer program would.

There is no other convention. If what you said was true, it could create massive nested formulae just by using a divisor.

dowie · 16 Oct 2017 at 07:20

h4rm0ny said:
The problem is that they write it out as if it's a sequence of steps. Like using a basic calculator. Normally it would be written like this:

And that's how a scientific calculator would treat it.

that isn't strictly correct, it depends on whether multiplication by juxtaposition is treated as a higher precedence than division or not (which is what I referred to in the post on the previous page), some calculators would treat it like that but I suspect that most wouldn't

I think most calculators would give multiplication by juxtaposition the same precedence as multiplication using the * symbol and therefore not treat it like that

most calculators (AFAIK) would treat 6÷2(3) in the same way as 6÷2*(3) and give the answer = 9

I've dug out two Casio calculators and as it happens one actually does give multiplication by juxtaposition higher precedence:

I think this is the main source of confusion in the question

ubersonic said:
The accepted method used to be that using the divide symbol meant that you divided by the entire product of the right equation.

This on the other hand I'm skeptical of, while I don't doubt that the ÷ symbol can cause confusion (just as the / symbol can) I'm not sure that there was ever a convention that meant either operator applied to everything to the right of it? Most confusion AFAIK mostly stems from the use of multiplication by juxtaposition rather than explicitly using a '*' symbol (though really some parentheses would be nice too).

However if we remove that potential source of confusion and specifically put in the * symbol then what you're saying is that 6÷2*3 was at one point treated as 6÷(2*3) and = 1 ?

or to use a new example with say addition for a clearer difference (dunno if this helps?) you'd be saying that 6÷2+3 was at one point in time treated as 6÷(2+3) and equal to 1.2 rather than 6

I'd be rather skeptical that this was the case ergo my question of when this happened?

I don't think you'd find an old calculator that treats the ÷ symbol as simply applying to everything on the right though I'm happy to be corrected

(on another note it is rather refreshing to see everyone being civilised in this thread - last time we had this thread come ups a few years ago there were some really obnoxious posters getting quite angry and throwing around insults over this stuff)

muon said:
The issue is that there is no order for the division and multiplication.

that's not really the issue IMO - while they're both of the same order of precedence they're both also left associative, if a * is used instead of juxtaposition then the order is clearer. It it is the use of multiplication be juxtaposition that can cause confusion as it is sometimes given higher precedence.

(edit - spelling)

Diddums x · 16 Oct 2017 at 07:29

Four pages? :eek:

Nice to see this being discussed like this, I love reading a healthy debate

h4rm0ny · 16 Oct 2017 at 08:59

ubersonic said:
The accepted method used to be that using the divide symbol meant that you divided by the entire product of the right equation.

Put simply the current accepted method goes like this:
6÷2(1+2)
= 6÷2(3)
= 6÷2×3
= 3×3
= 9

However the previous convention was to do it like this:
6÷2(1+2)
= 6÷2(3)
= 6÷(2(3))
= 6÷6
= 1

Hence it being an example of how the changing usage of mathematics conventions over time has changed the answer deemed "correct" by the masses. The reason it causes arguments is because many older people who were taught the previous method and even older scientific calculators will still give the now depreciated answer.

Uh, no matter how many times you use the term "accepted method", it doesn't make it so. I think maybe some people have come to think so because that's how basic calculators do it if you just type in the sums from left to right as read. But this is news to me and it seems like everyone else. So "accepted" doesn't really seem the right word to use, does it?

h4rm0ny · 16 Oct 2017 at 09:05

dowie said:
that isn't strictly correct, it depends on whether multiplication by juxtaposition is treated as a higher precedence than division or not (which is what I referred to in the post on the previous page), some calculators would treat it like that but I suspect that most wouldn't

I was offering it more as an example of how such a sum would be expressed properly in mathematics rather than the primary and secondary school version. As written normally (i.e. by people who perform mathematics routinely) there is no ambiguity. I haven't used a division symbol in a long time! I agree with you that calculators have a hand in this. Also, you are definitely correct that it was never the accepted method that the division symbol meant divide by the product of everything to the right of this.

danlightbulb · 16 Oct 2017 at 09:09

dowie said:
I've dug out two Casio calculators and as it happens one actually does give multiplication by juxtaposition higher precedence:

I think this is the main source of confusion in the question
)

I'm genuinely shocked at this.

What do your calculators do if the sum is written as: 6*2(3)^(-1) ?

And what do your calculators do if you 9229668*6 and turn it upside down?

bledd · 16 Oct 2017 at 09:17

^Also very surprised at that..!

I feel like I just divided by zero.

dowie · 16 Oct 2017 at 09:30

danlightbulb said:
I'm genuinely shocked at this.

What do your calculators do if the sum is written as: 6*2(3)^(-1) ?

both give 4, I don't think that there is any convention whereby multiplication by juxtaposition exceeds the precedence of exponents and roots, it's just given higher precedence than explicit multiplication or division in some cases... so the juxtaposition of the 2 with the brackets doesn't matter in this case

the calculator on the right sees it as (6*2)*(1/3) =4

and the one on the left sees it as 6*(2*(1/3))= 4

And what do your calculators do if you 9229668*6 and turn it upside down?

ubersonic · 16 Oct 2017 at 10:17

h4rm0ny said:
Uh, no matter how many times you use the term "accepted method", it doesn't make it so.

Eh? Obviously it's not the accepted method anymore, that convention has changed with time (which was my original point in response to your assertion that that had never happened in the history of mathematics).

dowie said:
I've dug out two Casio calculators and as it happens one actually does give multiplication by juxtaposition higher precedence:

It's pretty crazy seeing two so similar calculators giving correct/incorrect answers isn't it lol.

This is even funnier, the free built in Windows 10 calculator will give the correct answer of 9, however this calculator (which was one of the best on the planet when my father bought it) doesn't XD

dowie said:
This on the other hand I'm skeptical of, while I don't doubt that the ÷ symbol can cause confusion (just as the / symbol can) I'm not sure that there was ever a convention that meant either operator applied to everything to the right of it?

Simple explanation of the older method here:

Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below).

Under that interpretation:

6÷2(3)
= 6÷(2(3))
(Important: this is outdated usage!)

From this stage, the rest of the calculation works by the order of operations. First we evaluate the multiplication inside the parentheses. So we multiply 2 by 3 to get 6. And then we divide 6 by 6.

6÷(2(3))
= 6÷6
= 1

This gives the result of 1. This is not the correct answer; rather it is what someone might have interpreted the expression according to old usage.

This is why you get old people getting mad/conused when they see young people answering 9 instead of 1

WillyNelson · 16 Oct 2017 at 11:56

7+16÷(-7-1)-16= -11
(-7-1)=-8
16/-8 = -2
7-2-16 = -11

38

3 (bananas) * 11 (sides) = 33 + 2 (hours) + 3 (bananas)

danlightbulb · 16 Oct 2017 at 12:05

ubersonic said:
It's pretty crazy seeing two so similar calculators giving correct/incorrect answers isn't it lol.

This is even funnier, the free built in Windows 10 calculator will give the correct answer of 9, however this calculator (which was one of the best on the planet when my father bought it) doesn't XD

The correct answer is 1 though. Quite simply the term 2(3) is a single term in the equation and as such has to be evaluated first. Brackets comes first in BODMAS and this extends to those terms attached to brackets as well. The reason the windows calculator gives the incorrect answer of 9 is that it is evaluating the formula as you type it in.

Take this slightly rewritten equation.

6÷a(3)

No-one in their right mind would say this evaluated to 6/a *3. The correct answer is of course 6/3a. 3a is a complete term and as such is evaluated first.

The best way to consider this I think is to see BODMAS as term specific. I.e you do BODMAS on each term first, then on the equation that is left at the end. In this example case the absence of the operator between the 2(3) is how you identify that 2(3) is a complete term in it's own right.

wildman · 16 Oct 2017 at 12:17

47, or pankcake

ubersonic · 16 Oct 2017 at 12:28

danlightbulb said:
The correct answer is 1 though

The correct answer using the currently accepted method is 9, hence my using it as an example of an equation where the "correct" answer has changed over the years due to the accepted methodology being replaced by a new one (previous poster claimed this has never happened).

dowie · 16 Oct 2017 at 12:32

ubersonic said:
Simple explanation of the older method here:

I understood what you mean by your interpretation of this apparent 'older method' (apparently everything to the right of the ÷ being, I'm not sure though given your reply that you've paid much attention to my post explaining the multiplication by juxtaposition sometimes being given higher precedence (which is what the calculator (at least the one I own) producing the answer = 1 is doing).

Thus I was asking for an example of your 'older method' - perhaps it does exist but I'd be a bit skeptical... we can apparently test it though as it seems you've got an old calculator

what happens if you type in 6÷2+3 ?

if this 'older method' is as you say then you'd get 1.2 instead of 6

if not then you'll find that the reason you're getting a result of '1' for 6÷2(3) isn't because of this apparent 'older method' as you believe it works but because of what I've already explained above - multiplication by juxtaposition taking president over ordinary multiplication and division

I'd be very interested to see, I'd suspect you're wrong but it would be interesting to find out for sure if old calculators were really really weird like that and it seems you've got the means to show us ether way.

danlightbulb said:
The correct answer is 1 though. Quite simply the term 2(3) is a single term in the equation and as such has to be evaluated first. Brackets comes first in BODMAS and this extends to those terms attached to brackets as well. The reason the windows calculator gives the incorrect answer of 9 is that it is evaluating the formula as you type it in.

Take this slightly rewritten equation.

6÷a(3)

No-one in their right mind would say this evaluated to 6/a *3. The correct answer is of course 6/3a. 3a is a complete term and as such is evaluated first.

The best way to consider this I think is to see BODMAS as term specific. I.e you do BODMAS on each term first, then on the equation that is left at the end.

This is wrong, while the answer can be 1 if you give higher precedence to multiplication by juxtaposition above that of explicit multiplication/division as explained above, but generally it would be 9

however this has nothing to do with BODMAS, only terms inside the brackets have the highest precedence, the next level of precedence goes to 'pOwers', if you're to include multiplication by juxtaposition at the highest level of precedence (above powers) then your question in the previous post (which involved a term: ^(-1))would have had a different result... I don't think doing that would be an established convention anywhere

I also don't think things like 'BODMAS' are particularly useful as they do seem to lead to potential confusion