Which is the correct answer for 48÷2(9+3): 2 or 288?

[jhae2.718]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
9/2x \equiv \frac{9}{2x}?

 

[JaredJames]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
9/2x \equiv \frac{9}{2x}?

Doesn't matter does it? It's reduced the same either way.

 

[jhae2.718]

Well, from what I remember from arithmetic, if we had (9/2)*x it would reduce to 4.5*x, and 9/(2x) would reduce to 4.5/x.

So the 9/2 would always go to 4.5, but the power of x would be either 1 or -1 based on the grouping.

 

[JaredJames]

Well, from what I remember from arithmetic, if we had (9/2)*x it would reduce to 4.5*x, and 9/(2x) would reduce to 4.5/x.

So the 9/2 would always go to 4.5, but the power of x would be either 1 or -1 based on the grouping.

Got me thinking now, think I've got it *** about face.

(Maths isn't my strong point, hence my need to try and follow rules as much as possible - or, well, this happens.)

 

[Borek]

And what about 48÷(9+3)2?

 

[uart]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
9/2x \equiv \frac{9}{2x}?

Yes 9 \div 2x \equiv \frac{9}{2x}. In my experience that is how written mathematics is invariably interpreted.

Also if you have a calculator that can handle implied multiplication (most made in the last few years should allow this) then try typing in something like 12 \div 2\pi, you'll find that it is interpreted exactly as I say.

BTW. I just checked on my aging "Casio fx-82MS" and

12 \div 2\pi returned 1.909859 and 48 \div 2(9+3) returned 2.

In other words, don't just try this on C or MATLAB or python or anything else that doesn't allow algebraic implied multiplication, because it's irrelevant. Try it on a calculator that does allow algebraic implied multiplication if you really want to do a proof by calculator.

 

[jhae2.718]

48/(9+3)*2 = 48/12*2 = 4*2 = 8

 

[uart]

And what about 48÷(9+3)2?

Hey I don't like this notation either Borek but I'm just calling it as I see it commonly interpreted. This is why the fraction notation is preferred by most people as the "fraction bar" provides a well recognized "grouping symbol" and removes any ambiguity.

For that one my old calculator says "syntax error" (it wants either an explicit divide or times symbol after the bracketed expression). But if I was forced to make a call I still say implied multiplication and the answer is still 2, but I really wouldn't use that notation myself.

 

[uart]

48/(9+3)*2 = 48/12*2 = 4*2 = 8

Yep, it certain does if you put an explicit multiplication symbol in there. The whole point of this thread though is about what happens when (and what are the potential ambiguities that can occur when) we use the algebraic implied multiplication in an expression.

 

[Dembadon]

After thought, you might actually be right. For example:
a/bc

Is it \frac{ac}{b} or is it \frac{a}{bc}?

EDIT: Perhaps the ambiguity of the question is getting to me, and I was correct initially: a/bc = \frac{ac}{b}

Since parenthesis weren't used around b and c, I would interpret a/bc as \frac{ac}{b}.

If parenthesis had been used, I would assume \frac{a}{b}*\frac{1}{c}.

 

[jhae2.718]

I hold that implicit multiplication is evaluated as any explicit operation. Unfortunately, cases like this are extremely ill-defined.

However, the best way is to avoid the issue entirely and use \frac{}{} or additional parentheses for grouping.

 

[uart]

Since parenthesis weren't used around b and c, I would interpret a/bc as \frac{ac}{b}.

If parenthesis had been used, I would assume \frac{a}{b}*\frac{1}{c}.

What about a \div bc which was the notation used in the original question here?

 

[jhae2.718]

What about a \div bc which was the notation used in the original question here?

Same thing as a/bc.

 

[uart]

Same thing as a/bc.

Ok then I say your interpretation differs from that of at least 99% of written mathematics (maths science engineering textbooks and papers etc).

I don't like this notation either, I also find it wide open to ambiguity and of course many books and papers etc will avoid using it for that very reason. But if you look hard enough you will find textbooks or papers etc that do use notations like f = \omega \div 2\pi and when they do so then it pretty much always means f = \omega \div (2\pi) and not f = (\omega \div 2) \times \pi

 

[jhae2.718]

Then I would argue that such usage is contrary to conventional interpretation of order of operations.

Of course, as long such books/papers/etc. are consistent in their convention of operator precedence, I see no problem.

 

[uart]

Then I would argue that such usage is contrary to conventional interpretation of order of operations.

Of course, as long such books/papers/etc. are consistent in their convention of operator precedence, I see no problem.

jhae, do you own a calculator that is less than about 5 years old? If so try something like 12 \div 2\pi (without any explicit multiplication symbol between the 2 and the pi). You may get a surprise.

 

[Studiot]

Micosoft comes up with 288 in the windows calculator.

Excel is more interesting in that if you try to type it straight in that *** paperclip corrects you.

If you follow clippy's advice you get 288

 

[jhae2.718]

I have a TI-84 that is about 6 years old, I think, and a cheap Casio scientific calculator I bought a few months ago. The TI-84 gives 6pi, and the Casio can't handle 12/2pi being input without an explicit operator. Calculators vary on precedence used; how repeated exponentiation is treated is a good example.

I prefer to use the standard* order of operations I learned years ago. *I think we've learned from this thread that there really isn't a uniform standard defined for operator precedence.

 

[uart]

I have a TI-84 that is about 6 years old, I think, and a cheap Casio scientific calculator I bought a few months ago. The TI-84 gives 6pi, and the Casio can't handle 12/2pi being input without an explicit operator. Calculators vary on precedence used; how repeated exponentiation is treated is a good example.

I prefer to use the standard* order of operations I learned years ago. *I think we've learned from this thread that there really isn't a uniform standard defined for operator precedence.

I'm surprised about that. All recent scientific calculators from both Casio and Sharpe that I've seen have been able to handle that type of operation. Are you sure you used the divide "\div" key to enter that expression and not some kind of calculator fraction notation? I don't think your Casio calculator would even have a "/" key.

BTW. In all of this discussion can we please stick with the divide "\div" notation as per the original question. The alternative "/" is not used in well formatted typeset text such as in papers or textbooks. It exacerbates the ambiguity even further as it doubles as both a divide symbol and a half baked fraction bar as well. The original question was expressly about the divide "\div" symbol.

 

[jhae2.718]

I did use the \div key. Keep in mind my Casio is a really cheap and basic scientific calculator; what happens is that it replaces the 2 in the expression with \pi unless an explicit operator is used.

I have never seen "\div" used in any paper I have read.

 

[uart]

I did use the \div key. Keep in mind my Casio is a really cheap and basic scientific calculator; what happens is that it replaces the 2 in the expression with \pi unless an explicit operator is used.

I have never seen "\div" used in any paper I have read.

Ok but just to make that clear, are you saying that they always use the alternate "/" symbol instead, or are you saying that they always forgo the divide symbol for a proper well formatted fraction bar?

AFAIK "\div" is the correct symbol for divide so I can't think of any good reason to completely forgo it.

 

[jhae2.718]

I recall seeing \frac{}{} used most of the time*, though there have been a few uses of "/" I can recall, but almost always with something in the form a/b, i.e. only two arguments.

Of course, I'm sure there are plenty of authors who use "\div"; I just haven't read any.

Wolfram Mathworld lists both symbols for division. http://mathworld.wolfram.com/Division.html

*Most of the expressions are typeset in the equation environment in the papers I've read.

 

[Temad]

48÷2(9+3)=288

48÷[2(9+3)]=2

 

[tak08810]

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.

The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy:

Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask! - http://www.purplemath.com/modules/orderops2.htm

 

[Mark44]

Registered on this fine forum just because of this. :)

Regarding calculators, here is something interesting:

See the thumbnails in post #46.

This is a sad state of affairs when two models of calculators (TI 85 and TI 86) from the same company report different answers for exactly the same simple arithmetic expression.

 

[uart]

The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations.

Arh, proof by large font. I think that even trumps my proof by calculator.

I agree with it though.

 

[RJS]

And again it's a 50/50 split. This is obviously the most difficult math problem ever conceived.

 

[DR13]

And again it's a 50/50 split. This is obviously the most difficult math problem ever conceived.

On Tuesday I'm going to go to office hours and ask my math professor.

 

[DR13]

I think the problem comes down to whether or not 2(9+3) is the same as 2x(9+3)

 

[fraga]

See the thumbnails in post #46.

This is a sad state of affairs when two models of calculators (TI 85 and TI 86) from the same company report different answers for exactly the same simple arithmetic expression.

Thank you.
At least some one noticed.