6÷2(1+2) = ?

Okay, understood.

But does it have any value as an illustration of differing interpretations?

I mean, I see both interpretations and see how one is preferred on a very technical basis, if one is forced to make an interpretation, and I found it somewhat enlightening gaining an insight into how these conventions work and in cases like these, might be better off avoided altogether.

Again, I found it worthwhile just in pondering the circumstances bearing on it.

The answer is 9.

Potential ambiguity can arise if you should interpret it as

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

or

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

The first interpretation is what is built-into the expression evaluators of most programming languages, e.g.

if you do python -c "print(6/2*(1+2))"

you will get 9

The reason is that division and multiplication operators have equal precedence, and we evaluate from left to right, so you do whatever you see first in that.

^ yes. This is the overwhelming real-world application of in-line arithmetic and it's not ambiguous. It's standard.

Do interpreters even need it to be standard? Wouldn't they just tell the machine "/2" is "*.5"? Division eats more flops than multiplication.

Bad Wigins wrote:

Do interpreters even need it to be standard? Wouldn't they just tell the machine "/2" is "*.5"? Division eats more flops than multiplication.

Neither, they'd tell it to shift one bit to the right

1

Hardloper wrote:

Bad Wigins wrote:

Do interpreters even need it to be standard? Wouldn't they just tell the machine "/2" is "*.5"? Division eats more flops than multiplication.

Neither, they'd tell it to shift one bit to the right

Really? Why do that, when they still have to go the more complex route for all non-powers of 2? In which more complex route, multiplication is simpler because it doesn't require bitwise subtraction.

9

1

yabba wrote:

1

People are really giving dissertations about PEMDAS and still being wrong. The answer is 1 and it’s not ambiguous. I take it all the people saying 9 don’t work in math.

Let’s do it this way:

6

———

2(1+2)

This is the clearest way to explain. When you draw the division as top / bottom (numerator and denominator), you can see it easily.

The 2(1+2) is in the denominator and thus would be resolved to 6 before ever looking at the numerator.

6

——-

6

oldoldrunner wrote:

Pre 1917 = 1.

Post 1917 = 9.

They had order of operation wrong back then.

Post 1917 is still 1.

yes yes sir wrote:

yabba wrote:

1

People are really giving dissertations about PEMDAS and still being wrong. The answer is 1 and it’s not ambiguous. I take it all the people saying 9 don’t work in math.

Let’s do it this way:

6

———

2(1+2)

This is the clearest way to explain. When you draw the division as top / bottom (numerator and denominator), you can see it easily.

The 2(1+2) is in the denominator and thus would be resolved to 6 before ever looking at the numerator.

6

——-

6

As you can see from this thread, and half a dozen before it, half the people here would not put (1+2) in the denominator.

The TI-82 would but the TI-83 would not.

This is not a universal convention — that’s what makes it ambiguous. To remove the ambiguity, you should write it the way you did, on multiple lines, to unambiguously show what you mean, or use parentheses.

I did not learn any gimmick like PEMDAS or BOMDAS, but I learned “Order of Operations”. It did not differentiate implied multiplication (by juxtaposition without an operator) from explicit multiplication.

In any of these threads, I have yet to see any authoritative mathematical reference, from anyone who argues that it is unambiguously “1”, that gives multiplication without an operator higher precedence than multiplication with an operator.

Lenny gave a nice story problem but half the people here would write the resulting equation with parentheses.

To drive the point further, just because it is not ambiguous to you, doesn’t mean it is not ambiguous. It depends on what your audience understands, and whether some convention or consensus exists.

By way of illustration, replacing math with English, if I said I am wearing “braces”, I know what I mean and would say it is not ambiguous.

But you may not know what I mean, unless you know whether I am British or American, or whether I have adapted my speech to an American or British audience.

In America, most would understand it as metal on your teeth, but in England they’d understand what Americans call “suspenders”, e.g. “my sister looks cute in her braces and boots”.

If I had said “suspenders”, the English would think I’m wearing “garter belts” to hold up my stockings, and not my “pants”, e.g. “I’m a lumberjack and I’m OK”.

In England, “pants” are not “trousers”, but “panties”.

Getting back to braces and math, the Americans and English can’t agree on braces, brackets, and parentheses. They might not even agree how much a “billion” is.

Going to the bar/pub, they can’t agree on pints.

So when you are writing inline equations, it is best to understand what you audience understands, and try to remove any ambiguity, rather than deny it exists.

I agree with those that say it's ambiguous. I went to high school in the '80s, graduated with an engineering degree from UCLA 30 years ago. I never heard the acronym PEMDAS before this thread, and also was never taught left to right priority. None of the problems I worked from grade school through engineering school to standardized tests were intentionally written to be ambiguous in that way.

I can see left to right evaluation convention maybe becoming more of a thing with inline problems/programming, which I didn't have to think about much in school back then, but if you use enough parentheses, it's not necessary.

If I had to choose an answer, 1 makes more sense because I see the 2 as more attached to the parentheses, and, like I said, left to right was not a rule that I learned.

This thread and its idiocy needs to end. Mods?

bodmas wrote:

This thread and its idiocy needs to end. Mods?

Put it to bed by declaring "QED".

It's a math thing, an abbreviation in Latin, meaning you proved what you set out to solve.

Or we could just link to the Beatles reciting "Number 9" over and over in their classic song "Revolution 9" as some sort of final overture.

zzzz wrote:

I never heard the acronym PEMDAS before this thread, and also was never taught left to right priority.

That may be because it has no basis in math theory.

Division is DEFINED as multiplication by the inverse, if one exists, of an element in a field. It is understood that 6/2 means 6 x 2^-1. Everyone who understands this (i.e. has studied field theory) knows implicitly that

6/2(1+2)

means

6 x 2^-1 x (1+2)

i.e.

6 x 0.5 x (1+ 2)

multiplication in a field is commutative (part of the definition of a field) so it doesn't matter in which order the multiplicative operations take place. 3 x (1+2) = 6 x 1.5.

It's a thing in math education probably because schoolkids don't know this and need a crutch to avoid getting confused.

The division operator is non-associative, i.e., a/(b/c) != (a/b)/c, which limits one's ability to treat a large problem as a set of smaller independent problems that can be solved and recombined in any order.

Transformations for the above trivial cases exist so that only associative operations are necessary, e.g., a%b => a*(1/b), or more generally a*b^(-1) where b^(-1) denotes the group inverse of b, or a-b => a+(-b), where -b denotes the group inverse of b.

feef wrote:

Transformations for the above trivial cases exist so that only associative operations are necessary, e.g., a%b => a*(1/b), or more generally a*b^(-1)

While I miswrote it as "commutative," it's no mere transformation, it's the very definition of division! It's how it is read.

9/2a? that's 9/2 x a,, i.e. 9a/2, unless you put 2a in parentheses. Absolutely no ambiguity about it at all.

It's ambiguous due to implicit multiplication. It's a poorly written expression and therefore not clear on the intent. People that actually understand pedmas have a hard time with this until you are asked to solve something like 2x / 2x. Is that 1 or is it x^2? The 2x is implicitly multiplied and therefore takes precedence.

Most people that say 1 for the original question are half correct for the wrong reason