Maths Test! 2(1+2)2/(1+2)

[Leoreth]

I think the main result of this thread is that whoever designed the capabilities of computer typesets clearly hated math.

I've been doing algebraic equations for a while now, it's been holding up just fine.

Why your reluctance to distribute parentheses then? I cannot imagine you get far without it.

 

[Mise]

Well, I'd write \frac{3}{2} x quite often. When restricted to plain text and students who wouldn't understand LaTeX notation, I might write 3/2 x, although I'd be more likely to write 1.5 x. It would be very unlikely that I'd write 3x/2, even though it is unlikely to be misinterpreted.

Perhaps I'm not someone who "actually does maths" though, as I primarily do physics.

Right, yeah, you put the space between the 3/2 and the x. That's what I was saying in my previous post: the space is important! 3/2 x, to me, is unmistakably multiplying 3/2 by x; it makes the 3/2 a single "word" almost. "Three-over-two". Similarly, 3/2x, to me, is (slightly mistakably) dividing 3 by 2x: "two-x" is the single unit here.

Spoiler unnecessary thoughts on preferred notation of fractions :

Incidentally, 3/2 x, to me, represents something different to 3x/2. 3x/2 is halving 3x, whereas 3/2 x is increasing x by half. 3 x/2 is multiplying x/2 by 3. And so on. So depending on the context, I'd use different notation. E.g. if x was a wavelength or something and I was describing what a curve looked like at different intervals, I'd use 1/2 x, x (or maybe 2/2 x) and 3/2 x. But if the wavelength was supposed to be x/2, then I'd do x/2, 2 x/2 and 3 x/2. They're like words.

I find it hilarious the thread has got to this length.

It's serious god damn business.

 

[Takhisis]

 

[Thedrin]

C+H aside, that is my all time favourite edition of any comic.

 

[Defiant47]

The answer is 1, at least once you leave elementary school.

The order of operations is as follows:
1. Brackets/parentheses - e.g. (3+4), (1+(3*8))
2. Exponents - e.g. x^2, 4^3
3. Implicit Multiplication - e.g. 2x, 9(1+1)
4. Multiplication and Division - e.g. 3*9, 4/2
5. Addition and Subtraction - e.g. 1+2, 4-1

The "BEDMAS" framework is used in elementary school, but is only meant for younger children, which is why it's left simplified. Pointing to it is no different than pointing to the alphabet learned in school and complaining that other letters exist when we write words (such as from other languages for words that include umlauts or other accents).

To solve the original operation, follow these steps:
6/2(1+2)
A. First solve the operation in brackets, since it has precedence (under Order 1 above)
=6/2(3)
B. There are no more brackets. There are no exponents. But there is implicit multiplication. 2 times 3 is an implicit multiplication, so be careful that you remember it. It would be easy to put a * symbol, but then it would be hard to distinguish and remember this. So implicit multiplication next takes precedence (under Order 3)
=6/6
C. Complete the equation
=1

Any questions?

 

[muhtesem insan]

The answer is 1, at least once you leave elementary school.

The order of operations is as follows:
1. Brackets/parentheses - e.g. (3+4), (1+(3*8))
2. Exponents - e.g. x^2, 4^3
3. Implicit Multiplication - e.g. 2x, 9(1+1)
4. Multiplication and Division - e.g. 3*9, 4/2
5. Addition and Subtraction - e.g. 1+2, 4-1

The "BEDMAS" framework is used in elementary school, but is only meant for younger children, which is why it's left simplified. Pointing to it is no different than pointing to the alphabet learned in school and complaining that other letters exist when we write words (such as from other languages for words that include umlauts or other accents).

To solve the original operation, follow these steps:
6/2(1+2)
A. First solve the operation in brackets, since it has precedence (under Order 1 above)
=6/2(3)
B. There are no more brackets. There are no exponents. But there is implicit multiplication. 2 times 3 is an implicit multiplication, so be careful that you remember it. It would be easy to put a * symbol, but then it would be hard to distinguish and remember this. So implicit multiplication next takes precedence (under Order 3)
=6/6
C. Complete the equation
=1

Any questions?

Implicit multiplication has no priority over explicit multiplication in mathematics.

 

[Defiant47]

Implicit multiplication has no priority over explicit multiplication in mathematics.

Yes it does, although you could argue that this fact is more from a notational perspective. Especially when you do University-level math, you use implicit multiplication all the time, and it's always meant as one term (i.e. implicitly multiplied taking precedence over regular multipliation).

3/2x will always equal 3/(2x) because "2x" is meant as one term (its resolution comes before all other of its kind).

BEDMAS is very useful for elementary school level math, and is used a lot for younger children, and even in high school. But it's hardly comprehensive enough to cover all aspects of math, such as implicit multiplication encountered later on in one's mathematical career.

 

[muhtesem insan]

Yes it does, although you could argue that this fact is more from a notational perspective. Especially when you do University-level math, you use implicit multiplication all the time, and it's always meant as one term (i.e. implicitly multiplied taking precedence over regular multipliation).

3/2x will always equal 3/(2x) because "2x" is meant as one term (its resolution comes before all other of its kind).

I'd really would like to see a scientific source to this. Untill then I will not accept this information as true. I did study engineering in postgrad level and I have never seen what you said.

 

[Leoreth]

<- equivalent to a math minor and I've used this all the time, and it was standard for all professors and tutors as far as I recall. It was only used as a matter of course with implicit understanding from all sides, so I can't provide you with a source or standard on this.

 

[Terxpahseyton]

As many I thought this thread to be mostly amusing and a little embarrassing, but now I am happy to have been made aware of the different perceptions of implicit multiplication.

 

[NickyJ]

Why your reluctance to distribute parentheses then? I cannot imagine you get far without it.

Where do you see that I am reluctant to do the parenthesis first? My whole point is that you do the parenthesis first, then division/multiplication.

 

[Defiant47]

I'd really would like to see a scientific source to this. Untill then I will not accept this information as true. I did study engineering in postgrad level and I have never seen what you said.

I'm not about to scrounge up all the university math I did years ago just to prove this point (plus, I don't care that much). You're asking me to prove something so elementary and so notational, that everyone just takes it for granted.

Suffice it to say that implicit multiplication takes precedence not because it's "implicit multiplication" - that's a term I've used to help people understand the concept.

When you have 3/2x, there is no "multiplication" of 2 times x. There is no "2 * x" within that. "2x" is its own term. When we write 3/2x, we mean 3/y, where y=2x. For that reason, the "implicit multiplication" of 2*x ends up happening at the beginning or the end; it does not follow normal orders of operation because there isn't a regular operation there - it's just one term "2x" like "4" or "n".

Now, granted, in university level math this issue is rarely ever encountered. It is implicitly understood, and there don't tend to be any problems of confusion, because the writing 3/2x is typically done using a fractional format that is much more obvious. It's rare enough and obvious enough that nobody ever says that "if I get lazy and write 3/2x, I don't mean 3 divided by 2 multiplied by x - I mean 3 divided by 2x".

However, I will accept that maybe in your area of the world, there is no such thing as "implicit multiplication", and 3/2x means 3x/2. Damn weird, though.


Around here, this is a type of issue that if you went to a professor and asked if "3/2x = 3x/2", they'd get pissed off at you for wasting their time, and talk to you like to a child that obviously 3/2x means 3/(2x).

 

[NickyJ]

Oh, by the way, for what I take as the final decider for the question: WolframAlpha, when given problem, says the answer is 9.

 

[ParadigmShifter]

I agree with Wolfram Alpha

But it clearly breaks the rule when dealing with something like 1/2x, especially 1/2pi

http://www.wolframalpha.com/input/?i=1/2pi

This is why brackets should be used

I suspect they have an extra rule to bind values on the right of a number tighter than the operator, going with "the principle of least surprise".

 

[Defiant47]

Oh, by the way, for what I take as the final decider for the question: WolframAlpha, when given problem, says the answer is 9.

WolframAlpha is incapable of evaluating "6/2(1+2)". It is not telling you the answer to 6/2(1+2), because there is no operation listed between 2 and (1+2). WolframAlpha then inserts a multiplication sign, since putting two terms besides each other means multiplication. Computer programs are usually unable to resolve matters of implicit multiplication, and simply don't - they leave it up to the user to properly state the problem. I hope you don't expect programs to always interpret user inputs perfectly all the time.

So WolframAlpha is correct, in a way.

6/2*(1+2) = 9
but
6/2(1+2) = 1

 

[Defiant47]

I agree with Wolfram Alpha

But it clearly breaks the rule when dealing with something like 1/2x, especially 1/2pi

http://www.wolframalpha.com/input/?i=1/2pi

This is why brackets should be used

I suspect they have an extra rule to bind values on the right of a number tighter than the operator, going with "the principle of least surprise".

See, WolframAlpha is smart and actually tells you what your input is.

It showed 0.5pi, when we actually input 1/(2pi) by regular notational implications. In this case, we'd go back and try to explicitly tell it that 1/2pi is 1/(2pi) and insert it as such. Programs aren't perfect, and it's not worth it to deal with such a notational issue within the programming when you can just leave it to the user to explicitly state their input, and remind them of the interpretation of the input.

 

[ParadigmShifter]

It's still interpreting the rule in 2 different ways.

It's doing what (most) users most expect in each situation. In the 1/2pi case, it binds 2pi before binding the result to the division operator.

Look what it does for this

http://www.wolframalpha.com/input/?i=1/pi2

1/pi2 = 2/pi

 

[Defiant47]

It's still interpreting the rule in 2 different ways.

It's doing what (most) users most expect in each situation. In the 1/2pi case, it binds 2pi before binding the result to the division operator.

Look what it does for this

http://www.wolframalpha.com/input/?i=1/pi2

1/pi2 = 2/pi

No, it made the same "mistake" both times. It put a multiplication sign between the 2 and the pi (or the pi and the 2), so that it could actually evaluate and solve the equation.

Unfortunately, under regular standards of notation, 2pi is one term, without an "actual" multiplication sign in between (except for what is implicitly meant when written like that).

 

[ParadigmShifter]

No it didn't, it gave 2 different answers... clearly 1/2pi and 1/pi2 should give the same answer, by your argument.

 

[Polycrates]

On a related note, MATLAB refuses to even answer the question unless you put in the explicit *
Computer programs are not exactly reliable with these things, which is why you always have to be so utterly paranoid about bracketing every last thing.

There's people in the thread referring to what scientists would do, which seems odd because every scientist I've ever known has been utterly terrible at maths; but I have never seen a situation where 2x didn't mean (2 * x). Likewise where 2(x) didn't mean (2*(x)). I would say the answer to the question as written is 1; with a * it would be 9.