Multiplication by Juxtaposition

[Battlemage!]

Recently there was that long and ridiculous thread about ambiguity in notation when writing an expression. In the course of that discussion, someone posted a link from purplemath, which claimed that multiplication by juxtaposition was "stronger" than "regular" multiplication. However, this seems to conflict with the notion that the multiplication operation goes from left to right. Can someone enlighten me on this question?

Here is the link posted:
http://www.purplemath.com/modules/orderops2.htm

Here is the comment in question:

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.

which intuitively sounds like nonsense to me that contradicts the convention.

The problem she is in reference to is this:

16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2[2] + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5

I was taught that division was just multiplication, but of the reciprocal. If she is right, that would indicate that multiplication by juxtaposition takes precedence over "regular" multiplication.

So, is she correct? Or should the answer be 17?

 

[tiny-tim]

Hi Battlemage!

Yes, she's correct.

Brackets (parentheses) take precedence over everything … see http://en.wikipedia.org/wiki/BODMAS"

 

[Borek]

Hi Battlemage!

Yes, she's correct.

Brackets (parentheses) take precedence over everything … see http://en.wikipedia.org/wiki/BODMAS"

Tim, operation IN brackets takes precedence, but it doesn't influence orders of operations OUTSIDE of the brackets.

2*(2) is not different from 2*2.

2(2) is different from 22, but that's another problem.

 

[Battlemage!]

Tim, operation IN brackets takes precedence, but it doesn't influence orders of operations OUTSIDE of the brackets.

2*(2) is not different from 2*2.

2(2) is different from 22, but that's another problem.

Yeah, I didn't mean multiplication INSIDE brackets. I mean OUTSIDE of them.

She was arguing that this:

a * b(c)

is different from this:

a * b * c


except in her case, b was the multiplicative inverse of some number d. (say, d = b^-1).


It makes no sense to me on those grounds, since division by x is only multiplication of the multiplicative inverse of x. After all, aren't rings composed of only TWO operations?

So, what I don't understand is, if


a * b(c) is identical to a * b * c,


how can


a ÷ d(c)

not be identical to

a * b * c

where d = b^-1

 

[Unrest]

Sounds like nonsense to me. However, don't forget the rules for notation are just as arbitrary as the rules for language. You can't justify them by saying d=b^-1 because the whole system is just made up! Maybe this is an obscure part of the math culture in some society.

a * b(c) is identical to a * b * c,

What if a, b, and c are something for which multiplication isn't associative? Then maybe it's really:

a * b(c) is identical to a * (b * c)

with the juxtaposition taking precedence over both ordinary multiplication and division.

Anyway, does anyone actually use the ÷ symbol? I just write division like a fraction. ÷ is for schoolchildren without algebra.

 

[uart]

Anyway, does anyone actually use the ÷ symbol? I just write division like a fraction. ÷ is for schoolchildren without algebra.

Yep that's true. The first time I ran into this problem was actually when I was helping my nephew with high-school algebra some years ago. He was constantly getting homework questions asking stuff like : "Simplify $8 x^3 y^2 \div 2xy$".

Of course I wanted to use standard BIDMAS operator precedence and answer $4 x^4 y^3$, but as expected the textbook answer was $4 x^2 y$. It annoyed the hell out of me but in the end I just had to concede defeat and accept that many authors will allow implied multiplication to take precedence over divide ($\div$) in this type of problem.

Since then I have seen many other examples of this practice and many of the scientific calculators that allow implied (juxtaposition) multiplication do follow this convention. I haven't use HP recently, but recent Casio and Sharp calculators I've tested both do so.

For example, on my Casio calculator " $12 \div 2\pi$ " returns 1.909859.
One the other hand " $12 \div 2 \times \pi$ " returns 18.84955.

BTW. I've used "pi" in these example but the same applies to every object for which the calculator allows implied multiplication.

 

[uart]

Also let me repeat something I mentioned in the other thread that got locked. We might like to say well let's just not use the $\div$ symbol then. Be aware however that this is not the only context in which this altering of precedence with implied multiplication can occur.

The same thing can happen with function notation like $\sin 2x$ which would normally be interpreted as $\sin(2x)$ rather than $x \sin(2)$. On the other hand with numbers and an explicit "times" symbol we would usually interpret $\sin 30 \times 10$ as $10\, \sin 30$ rather than $\sin 300$ .

So there's another common example of implied (or juxtaposition) multiplication altering precedence. This is just something that we have to watch out for. Rearrange your equations to make it non ambiguous or use extra parenthesis if there's no other way.

 

[Borek]

We are back to the other thread arguments. Topic locked.