# Maths

Discussion in 'Just Talk' started by Mike58, Aug 6, 2019.

1. ### Mike58Active Member

What is the answer to:

8÷2(2+2) =

And why?

And what is wrong with the expression?

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16

3. ### mcooper2406Active Member

Brackets first so 8 ÷ 2 x 4

Multiplication and division are equally weighted so it should be done left to right.

4 x 4 so 16

Nothing wrong with the question just not written in the easiest way to understand. Could use another set of brackets or 8 on top of a fraction of the intention is to divide 8 by the rest!

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4. ### btiw2Well-Known Member

I can see the ambiguity, but I think most people would read that as:
8
————
2(2+2)

Rather than
(8/2)(2+2)

Because, outside of primary school, the inline division sign is rarely seen and 2(2+2) is a single thing.

If in doubt then add more brackets - the stixall of mathematics.

5. ### Harry StottleWell-Known Member

Our maths teacher at primary school taught BODMAS: Brackets of division, multiplication, addition and subtraction to define the order

The answer is one.

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6. ### candoabitofmoststuffWell-Known Member

1

If it was meant to be;

8 divided by 2 (i.e. 4)
multiplied by
2 +2 (i.e. 4)

giving the answer 16...

it should be written as...

(8 ÷ 2)(2+2)

Because the brackets aren't present around the 8 ÷ 2, the 2 is multiplied by what's contained in the following brackets, (2+2)

So the as it's presented, 8÷2(2+2) = the route to solution is...

8
divided by...
2 multiplied by (2+2), i.e 2 multiplied by 4, ie. 8

so that's
8
divided by
8

equals 1

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7. ### dobbieWell-Known Member

It is a badly written equation the answer could be 1 or 16.

8. ### btiw2Well-Known Member

I doubt they taught that.
The O stands for Order (x^3 has order 3 - it means [in this case] powers).
Multiplication and division have equal precedence and left->right after that - similarly with addition/subtraction. As McCooper2406 said.
So if you apply BODMAS (or BIDMAS,PEDMAS whichever acronym your teacher preferred) strictly then you'd get 16.

BUT..... Here's the thing.
I think most mathematicians would get 1. Why?

We all agree with the 2+2 first, right?
So let's read that as:
8÷2(4)
Let's replace 8 with x, and 4 with y.
x÷2y is clearly x÷(2y).
A number in front of a bracket has a sort of hidden higher precedence.
So I believe people who use equations everyday would read that as 8÷(2(4)). Actually, they'd probably add the bloody missing brackets.

Last edited: Aug 6, 2019
9. ### Mike58Active Member

BODMAS is actually Brackets - ORDERS - Division - Multiplication - Addition - Subtraction

Orders - that is exponents, powers, roots,

and after BODMAS working left to right

10. ### SparkielevWell-Known Member

1 or 16

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11. ### ajohnWell-Known Member

No the answer is 16. The only confusing thing is the implied multiplication via the brackets

So the first step get rid of the brackets gives

8/2*4 worked left to right.

If some one wants and answer of one they would need more brackets

That would result in
8/(2*(2+2))=1

If the above wasn't correct every VPAM calculator on the planet would have to be recalled. Actually that sort of sum is the reason for them. The earlier X<->y swap types needed some thought to come up with the correct answer. Same's true of the even earlier reverse polish ones. I still use an X<->Y type a lot that's a sort of hybrid as it can also use brackets. Sadly my earlier x y one that could also do hex and binary sums broke a long time ago. Only reason I bought a vpam one.

John
-

12. ### diy_nixyActive Member

Using BODMAS rule first, and left to right rule.

13. ### KIABWell-Known Member

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14. ### ajohnWell-Known Member

LOL if some one does it backwards

(2+2)*2/8 the answer is 1.

I suppose some one might stick it into an X<->Y calculator that way round. If so they had better stick to a VPAM one where they would be told to enter left to right exactly as it's written.

John
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15. ### btiw2Well-Known Member

Being obtuse about this would be like sending your mate out for 1/4" ply, and him coming back empty handed saying that they only had 6mm.

This is a fun puzzle (thanks Mike58!) because it's kind of ambiguous.
We can sorta see what's asked for, but it turns out that schoolroom BIDMAS doesn't really do what we meant.
It's sort of like a mathematical koan. That's the spirit in which I'm sure it was posted.
If Mike58 had posted "does a tree falling in a wood make a sound if there's no-one to hear it?" it's kind of dull to just post "Yes. Conservation of energy.".

Calculators are just following an algorithm, but mathematical expressions are also a human language.
Of course, mathematicians hate ambiguity and will do everything to avoid writing it. If it does happen, you sometimes have to rely on seeing what was meant.
If I wrote dy/dx you'd (probably) know that I was taking derivatives. I wouldn't mean ((d*y)/d)*x, where d,x and y are all parameters.
(1+tan^2x)/2tanx is perfectly readable as an alternative to cosec 2x (but is BIDMAS wrong).

Errr no. If we want to put all our faith in calculators then try that expression in more modern calculator.
I just typed '8÷2(2+2)' into my Casio fx-991EX and hit '='
The calculator interprets the ambiguity and corrects the display to 8÷(2(2+2)) and then gives the answer 1.
So if we're looking at what's meant, then Casio's researchers interpret it differently to you.
That doesn't make either of you right or wrong.
That's why it's a fun puzzle.
Similarly, (1+tan(20)^2)÷2tan(20)-1÷sin(40) is corrected to (1+tan(20)^2)÷(2tan(20))-1÷sin(40) to give the answer we wanted of 0, not the answer we strictly asked for (by BIDMAS).

It's interesting that calculator programmers understand the ambiguity and automatically correct it to what's (probably) meant.
How did they determine what people probably meant? Dunno.
I wonder if that the "natural" in "natural VPAM" which was the upgrade from VPAM? VPAM is all Casio marketing nonsense anyway.
It seems daft to argue that Casio's marketing team now define how people read mathematics.

I've tried that expression with a couple of maths graduates. Both saw it was ambiguous, but interestingly each interpreted it differently.

16. ### candoabitofmoststuffWell-Known Member

I'm having second thoughts on this!
Cando.

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17. ### ajohnWell-Known Member

My vpam calculator come up with the correct answer so find it odd that yours doesn't.

John
-

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18. ### btiw2Well-Known Member

The fx-85GT plus also says “1” although it doesn’t change the display to show how it resolved it.

The fx-83ES says 16.

Is yours an old model?

19. ### btiw2Well-Known Member

It’s really interesting that even different Casio VPAM calculators give different answers.

When even a single calculator manufacturer finds the question ambiguous, then it’s certainly not surprising that humans will disagree.

It’s like the arithmetic version of that blue dress/white dress thing that was on the Internet a few years ago.

What an interesting and subtle question Mike58 posted.

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20. ### Mike58Active Member

I passed my Pure-Maths A level when I was 15 ... and studied and took A levels in Pure Maths (twice, both passes), Pure & Applied Maths, Applied Maths and Higher Maths (which in those days was a totally differently level). Things that were drummed into me pre O-level (Maths at 13!) included BODMAS and the need to ensure an expression can only be interpreted one way and even when there is an "implied " multiplier, for example, consider including it to remove any chance of confusion.

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