48÷2(9+3)=?

[Lyric]

48÷2(9+3)=?

 

I saw this on some other forum, and it sparked a HUGE debate among it's members concerning the solution. I'm curious as to what spidweb members think about this expression.

 

 

Click to reveal..

One might view the problem as (48÷2)(9+3) or as 48÷(2(9+3)) thus leading to different results.

Clearly on paper one would rarely/never see such a poorly formulated expression, but is there a correct way to interpret the problem? Is there an "authority" that clearly states that the problem should be viewed as (48÷2)(9+3) or as 48÷(2(9+3)) ?

[Nicothodes]

Order of operations states that you solve what's in the parentheses first, then multiply and divide from left to right. So first you would add the 9 and 3, then divide 48 by 2, and multiply the result by 11 to get 264.

 

EDIT: This is how I would figure it out. I don't think using parentheses to indicate multiplication overrides using the sign, but it has been a long time since I've taken a maths class.

[Randomizer]

A computer program would interpret the equation as Nicothodes said except 9+3 = 12.

[dave s]

Yes, PEDMAS clearly states the order that you would approach this.

 

Parenthesis, Exponents, Multiplication AND Division, Addition AND Subtraction.

 

Parenthesis refers only to things INSIDE parenthesis, not things next to each other in parenthesis, that falls under multiplication.

 

When dealing with things of the same rank, like multiplication and division, you go left to right.

 

So you get 48÷2 = 24

24 (9+3) = 288.

 

That's the only correct way to interpret this, and if you put it into a graphing calculator exactly as shown or into google, that is the answer you will get.

 

Now, that may not be how someone writing it might intend it to be interpreted, but that doesn't change the fact that this is the correct way to interpret it as written.

 

http://www.purplemath.com/modules/orderops.htm

[Lyric]

9+3 would equal 12, not 11.

 

But you raised a curious point, why do i have to go from left to right ? Is that stated somewhere in the math books and i utterly failed to read it ?

 

Edit: dave, computers do that because in most programing languages the multiplication and division operators are stated to be left associative, though i am not sure this is "hardcoded" in mathematics, and this is what i want to know.

 

google found me this article as well: http://mathforum.org/library/drmath/view/57021.html

[*i]

The ordering is as dave s states. Within each grouping, they are done left to right. One could define other rules, but these are the ones everyone has agreed to follow, so it is best to use them.

[dave s]

From the link I posted:

 

"When you have a bunch of operations of the same rank, you just operate from left to right"

[The Ratt]

Should we mess things up and throw in reverse Polish notation?

[Kelandon]

Going from left to right is a standard thing that discussions of Order of Ops sometimes leave out because it so rarely matters. But consider the expression 2 - 3 + 5; if you read this right to left, you might be tempted to say that this is equal to 2 - 8, which is -6 and clearly wrong. Now, technically, you could read this as 2 + (-3) + 5 to get around this problem, but you're safer just reading left to right.

[Lyric]

If i read from right to left it still gives the same result: 5+(-3)+2.

 

Could someone provide a reputable source that clarifies my dilemma? Even a book that i could later read.

 

 

edit: i started browsing through the physics book on my desk, it describes the radius of the circle created by a an electron that enters a magnetic field perpendicular on the field lines as: r = mv/qB. If i were to follow PEMDAS(or however it's called, i only learned that multiplication and division had higher priority than addition and subtraction, no mnemonics ) i should read it as r = (mv/q)B, but it is actually r = mv/(qB)

 

This is becoming annoying :|.

[Kelandon]

Originally Posted By: Lyric

If i read from right to left it still gives the same result: 5+(-3)+2.

Um, when I say "read from right to left," I don't mean, "Literally reverse the order of the numbers." I mean, "evaluate the stuff on the right before you evaluate the stuff on the left." Clearly addition is commutative but subtraction is not, so you have to do some manipulation even to understand what you've done there. (Plus unary operators, etc.)

 

In doing this, though, you're already assuming that you know how subtraction works. So why don't you already assume that you know how division works and be done with it?

 

At any rate, I'm tempted to provide a Let Me Google That For You link, but instead I'll point out that if you Google "Order Of Operations," literally the first link provided discusses this very issue, and so do most of the others. It's not hard to find. (It's a little buried on the Wikipedia page on the subject, but it's there, and it cites the Danica McKellar math texts — whether they are "reputable" is, I guess, a matter of opinion.)

[Lyric]

From the Wikipedia article:

Quote:

Similarly, care must be exercised when using the slash ('/') symbol. The string of characters "1/2x" is interpreted by the above conventions as (1/2)x. The contrary interpretation should be written explicitly as 1/(2x). Again, the use of brackets will clarify the meaning and should be used if there is any chance of misinterpretation.

 

I see.

 

 

Though, contrary to what the Wikipedia article states, i must confess that I am more drawn towards 1/(2x) when reading 1/2x .

[dave s]

Originally Posted By: Lyric

Though, contrary to what the Wikipedia article states, i must confess that I am more drawn towards 1/(2x) when reading 1/2x .

Well, hopefully you aren't in a profession like engineering, programming, physics of math that requires people to follow the state rules then.

Those of us that are must be careful about these things as an error will mess up a calculation and could cause a bug in a program for example.

[A less presumptuous name.]

Originally Posted By: Lyric

r = mv/qB. If i were to follow PEMDAS(or however it's called, i only learned that multiplication and division had higher priority than addition and subtraction, no mnemonics ) i should read it as r = (mv/q)B, but it is actually r = mv/(qB)

This is the problem with simply typing in problems. Ideally we would all use some sort of formatting agent that would allow us to put fractions into our writing.

Also, one can argue that subtraction and division are false operations are are more accurately described as adding negative numbers and multiplying by reciprocals.

[Lyric]

Originally Posted By: dave s

Well, hopefully you aren't in a profession like engineering, programming, physics of math that requires people to follow the state rules then.

In fact, i am studying engineering, and if you read the post above that discusses the circle described by an electron when entering perpendicularly into a magnetic field, you'll notice that physics books have trouble in keeping with the "Standards" as well. Sadly, i have now found another occurrence: http://i.imgur.com/LXGap.png

[Sullust]

Originally Posted By: dave s

Originally Posted By: Lyric

Though, contrary to what the Wikipedia article states, i must confess that I am more drawn towards 1/(2x) when reading 1/2x .

Well, hopefully you aren't in a profession like engineering, programming, physics of math that requires people to follow the state rules then.

Those of us that are must be careful about these things as an error will mess up a calculation and could cause a bug in a program for example.

If you write 1/2*x instead of x/2 you're doing it wrong. I mean, if you can put it in the numerator why don't you? The same goes for the original problem. It should really be written as 48(9+3)/2; otherwise I'm going to look at it as 48/(2(9+3)).

[nikki.]

Originally Posted By: dave s

Yes, PEDMAS clearly states the order that you would approach this.

Parenthesis, Exponents, Multiplication AND Division, Addition AND Subtraction.

I learned it as BODMAS: Brackets over Division, Multiplication, Addition and Subtraction. I mean they're both the same, but BODMAS sounds far more jollier.

Also, no amount of time since your last maths class justifies getting 9+3 wrong, Nicothodes.

We'll have words later.

[Dantius]

I guess the rule is that it the expression is clear enough to be recognized as a important equation or law or expression, you don't have to bother with the parentheses to insure that it is technically correct, you just assume the reader will recognize them. Off the top of my head, e^i*pi=-1 and dy/dx both come to mind, since they are both common enough to recognize as the special case of Euler's formula and the expression for the derivative of y with respect to x without needing more clarification.

 

I guess it's just the mahtematical equivalent of being able to interpret typos correctly, and having done so for so long that when you see the actual word spelled right, it looks strange.

 

EDIT: Also, when I see / written, I tend interpret it as a vinculum grouping everything before into the numerator and everything after as the denominator, where I just interpret ÷ as plain old division. Then again, I'm a sloppy and lazy mathematician, so oh well.

[Nicothodes]

Originally Posted By: Randomizer

A computer program would interpret the equation as Nicothodes said except 9+3 = 12.

I fail. Clearly, I am not awake on weekends.

Also Nikki, unless you want me to share embarrassing stories, no, we won't.

[Dantius]

Originally Posted By: Nicothodes

Originally Posted By: Randomizer

A computer program would interpret the equation as Nicothodes said except 9+3 = 12.

I fail. Clearly, I am not awake on weekends.

It's OK. Today's WSJ has a little blurb on the side of an article about math asking the reader to evaluate -5x+7 at -1 and I said 13. People like us with bad arithmetic skills should stick together .

[Rowen]

I am amazed at how natural and easy math comes to some people. I look at numbers and I see monsters that wish to eat me. I hide from them in my liberal arts majors.

[nikki.]

Originally Posted By: Rowen

I am amazed at how natural and easy math comes to some people. I look at numbers and I see monsters that wish to eat me. I hide from them in my liberal arts majors.

Yes, to paraphrase, people like us who forwent any useful learning should stick together!

(Although I was actually pretty good at maths until I had to choose between doing that and English Literature at A Level)

[Sullust]

(9+3)/3 = 11.

[Rowen]

Originally Posted By: Lt. Sullust

(9+3)/3 = 11.

What? How does 4=11?

[Dantius]

Originally Posted By: Rowen

Originally Posted By: Lt. Sullust

(9+3)/3 = 11.

What? How does 4=11?

He's probably using a different base. I'm too lazy to figure out what it is, though.

EDIT: Base 3. (9+3)/3 equals 11 in base 3.

[Alorael at Large]

It's not acceptable to express your equation with different bases on the left and right sides of the equality.

 

—Alorael, who considers the first equation in the thread a bad equation. 48÷2*(9+3) would be fine. 48÷2(9+3) implies, but is not, 48÷(2(9+3)). Lesson: clarity is sometimes at least as important as being technically correct.

[A less presumptuous name.]

Originally Posted By: Alorael, who has a nice name today.

48÷(2(9+3)). Lesson: clarity is sometimes at least as important as being technically correct.

 

The real moral of this story is to use more parenthesis.

(48)/(2(9+3)).

 

When inputting expressions into my graphing calculator, I tend to err on extra parenthesis, as long as all sets close properly.

 

Also, as my tech ed teacher (who is really a mathematician who just applied too late for the opening) says, you don't really use numbers in math. You use variables and operations. Numbers are only for examples.

[Aoslare]

You could just as easily say: you don't really use variables in math. You just use numbers and operations. Variables are only for generalizations.

 

But really, you use both in math: variables for platonic forms and numbers for specific instances.

[Dikiyoba]

Originally Posted By: Master1

The real moral of this story is to use more parentheses.
((48)/(2(9+3))).

FYT (Dikiyoba fixed your typos).

[Dintiradan]

You're both wrong! Everything in mathematics must be defined in terms of sets!

[Aoslare]

Hmm. Maybe mathematics is just a specialized application of semantics.

[Lilith]

Originally Posted By: Dikiyoba

Originally Posted By: Master1

The real moral of this story is to use more parentheses.
((48)/(2(9+3))).

FYT (Dikiyoba fixed your typos).

i wrote a TI-83 program once where i ended up having to nest parentheses 6 or 7 levels deep. writing out big equations on a tiny screen is kind of a pain.

it's even worse if the program doesn't give the correct results and you have to look back through the code hunting for the typo you made

[Sullust]

Originally Posted By: CRISIS on INFINITE SLARTIES

Hmm. Maybe semantics is just a specialized application of mathematics.

FTFY

[A less presumptuous name.]

Originally Posted By: Lilith

i wrote a TI-83 program once where i ended up having to nest parentheses 6 or 7 levels deep. writing out big equations on a tiny screen is kind of a pain.

Some of the new software for the TI-84s lets you enter division as a fraction. Such equation editors remove at least some of the parenthesis in large expressions. There is also a format for putting in definite integrals that uses the standard symbolic notation rather than fInt(f,x,a,

[Aoslare]

Yeah, it works that way too, Sullust.

 

Maybe everything is just a subset of everything.

[Dintiradan]

Way to go, Slarty. You just destroyed all of reality with Russell's paradox. I hope you're satisfied.

[Aoslare]

I'd like to thank Anaxagoras and in his honour present this reality-delimitational video:

 

[Sullust]

I would argue that written/spoken mathematics is a subset of semantics. But 'pure' mathematics is only a subset of itself.

[Dantius]

Originally Posted By: Master1

Originally Posted By: Lilith

i wrote a TI-83 program once where i ended up having to nest parentheses 6 or 7 levels deep. writing out big equations on a tiny screen is kind of a pain.

Some of the new software for the TI-84s lets you enter division as a fraction. Such equation editors remove at least some of the parenthesis in large expressions. There is also a format for putting in definite integrals that uses the standard symbolic notation rather than fInt(f,x,a,

The TI n-spires fix just about every single formatting issue that every previous model has- instead of trying to render your equation as clunky code (ArcSin(abs(sqrt(3))/2)), to fake a formula) and having it scream at you and force you to go through your entire equation to hunt for the one parentheses you didn't close, you can just input the equation in PrettyPrint and it'll work everything out perfectly for you. It's a massive leap forward in calculating technology.

Of course, there's also a massive leap backwards from the TI-89- it's cant plot differential equations, functions of two variables, several commands have been removed, it's programming capabilities have been just about removed except for the most basic function definition, and it's physically larger and runs clunkier due to its massive OS. Then again, from what I heard from my woman on the inside, the next generation will fix most of those issues except for the 3D graphs, which aren't really that useful anyways, and it even comes with a color screen. Color! On a handheld calculator! I can't believe we've have to wait until 2012 to even get color or our calculators!

[Alorael at Large]

Some technology never marches on.

 

—Alorael, who rests confidently knowing that mathematics is almost certainly a subset of itself. He's also pretty sure that it's a subset of the set containing mathematics and tuna salad, though, so be careful before leaping to conclusions.

[The Mystic]

Originally Posted By: Lilith

i wrote a TI-83 program once where i ended up having to nest parentheses 6 or 7 levels deep. writing out big equations on a tiny screen is kind of a pain.

it's even worse if the program doesn't give the correct results and you have to look back through the code hunting for the typo you made

Been there, done that. In my case, though, it was on a TI-85.

When I was in college, I programmed a rather large menu containing most of the formulas and theorems. Each menu item had a submenu, and each submenu item in turn had a submenu, which led to different pieces of information. Debugging all the various menus and submenus was such a pain that by exam time, I had memorized most of what I needed to know.

[Danny the Fool]

Originally Posted By: Dantius

Then again, from what I heard from my woman on the inside, the next generation will fix most of those issues except for the 3D graphs, which aren't really that useful anyways, and it even comes with a color screen. Color! On a handheld calculator! I can't believe we've have to wait until 2012 to even get color or our calculators!

Don't need no woman on the inside for that, they've already announced the Nspire CX a month ago and you can read about it on Ti's web site.

It won't be the first calculator with a colour screen though, Casio has made them for ages, possibly others too.

I wonder if they thought of colouring formulas according to evaluation order...

[Andraste]

Speaking of Casio, this is the origin of the OPs question. More interesting this way.

 

[Venom]

Uh, how does your calculator get 2 from that equation?

[Aoslare]

I'd guess that the division operator shown above is set to only accept a single term as the divisor. It is odd that it doesn't use the regular / and * symbols for those operations, and it makes me wonder if the symbols it is using represent non-standard implementations of those operators.

[Sullust]

Originally Posted By: Venom

Uh, how does your calculator get 2 from that equation?

Maybe the parser treats it as a function call. It makes more sense if you look at it as 48 ÷ f(9+3) where f(x) = 2 * x.

[The Mystic]

Originally Posted By: Venom

Uh, how does your calculator get 2 from that equation?

It interpreted the expression this way:

Code:

48÷(2*(9+3))

By doing the division last, the final operation becomes 48÷24, which equals 2.

[MrClick]

Originally Posted By: Port of Black Rivers

It's not acceptable to express your equation with different bases on the left and right sides of the equality.

—Alorael, who considers the first equation in the thread a bad equation. 48÷2*(9+3) would be fine. 48÷2(9+3) implies, but is not, 48÷(2(9+3)). Lesson: clarity is sometimes at least as important as being technically correct.

<analretentive>Those are expressions, not equations</analretentive>

[Aoslare]

Yes: an equation is made up of expressions and an equals sign. Alorael was calling the equation bad on the basis of its bad expressions.

[Alorael at Large]

You know what really peeves me? People claiming to be anal-retentive or picky and then proceeding to nitpick a non-error.

 

—Alorael, who is also pretty annoyed by being ordered to solve an expression. Expressions are slippery things. You can simplify them, and you can even evaluate them, but solve exactly? Impossible.