# What is “multiplication by juxtaposition”?

I was reading http://www.purplemath.com/modules/orderops2.htm it shows

 = 16 ÷ 2[2] + 1   (**)
...
= 5


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

However when talking to certain people they all have said there is no such thing as this. There is shorthand which uses normal multiplication order and no "multiplication by juxtaposition" and etc.

Is there a "general consensus among math people" or is this simply incorrect?

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I would say there is only one operation, which is multiplication. It can be indicated in various ways, including $\times$, *, $\cdot$, or juxtaposition. – Ross Millikan Apr 16 '11 at 0:47
I do agree with the interpretation that multiplication written as juxtaposition should be regarded as having higher precedence that multiplication written with $\times$... unfortunately this is not the case in any programming language I have ever seen, and only sometimes the case in written mathematics. – Zhen Lin Apr 16 '11 at 1:09
@Zhen I'm sure your dream language would scare programmers away: Wretchedly declare a variable named Marvin := plus 42, NotAFunction(42x) myFunc(2) divided by OpenTheDoor(. this is a string \.) Capiche? :) – Mateen Ulhaq Apr 16 '11 at 2:38
@ZhenLin: Are there any other programming languages besides Mathematica's which interpret juxtaposition as multiplication? – celtschk Aug 2 '12 at 17:27
My issue is when students write it in a non-programming context. I want to point them to a webpage that says "5/7x" is wrong and they should say either 5x/7 or 5/(7x). Right now, I tell them that anyway, but don't have any sort of authoritative backup. – barrycarter Sep 2 '12 at 17:33

So, the question is whether $a/bc$ means $(a/b)c$ or $a/(bc)$. And the answer is, DON'T WRITE $a/bc$, because it will only cause confusion. Some people/software/whatever will make one interpretation, some will make the other, neither one has been endorsed by the Dalai Lama or any other great leader. Put in enough parentheses to make your writing foolproof.

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Parentheses are nothing to be miserly about; certainly not in a time where disk space is cheap. – J. M. Apr 16 '11 at 16:55
If you want to avoid parentheses, just switch to Polish or reverse Polish notation. – Doug Spoonwood Aug 21 '11 at 17:14
It’s notable that Texas Instruments’ calculators have changed their interpretation from $a/(bc)$ to $(a/b)c$, presumably because of the prevalence of expressions like $1/2x$ where many users intend $\frac{1}{2}x$. – alastair Jun 30 '15 at 15:27

It's simply incorrect. If it were correct, then $2x^2$ would really mean $(2 \times x)^2 = 2^2 \times x^2 = 4 \times x^2$, but it doesn't; it means $2 \times x^2$.

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Maybe the OP should tell Purple Math about this error. I'm sure they'd be happy to fix it. – Mateen Ulhaq Apr 16 '11 at 2:33
@muntoo: I'm to lazy. Really i want to know if 2/2(10) is (2/2)*10 or 2/(2*10). Purple Math says the second. Everyone is telling me its the first. – acidzombie24 Apr 16 '11 at 2:43
Every programming language and spreadsheet I have used agrees 2/2*10=10, but none of them accept 2/2(10). Excel tries to correct 112/560(56) to 112/560*(56)=11.2 I avoided the small numbers so it wouldn't think of dates. – Ross Millikan Apr 16 '11 at 2:56
(With apologies for bringing this back from the grave.) No, because exponentiation has a very high precedence (higher than just about anything else), so it would still be $2 \times x^2$ – alastair Jun 30 '15 at 15:11

This question is more about how we deal with trolling and nuisances.

Then maybe we can start a meta article about identifying and dealing with these threads.

Between the math forums that I moderate and otherwise frequent, and dozens of other forums

(a short list is here: http://www.mymathforum.com/viewtopic.php?f=13&t=20148&p=79150#p79150),

I'd guess that thousands of hours have been wasted on this garbage.

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Sorry, but the question if read well, leads to the recognition that in infix notation you need to have a very clear convention for things, which basically indicates how expression would look if written with all implicit parenthetical expressions made explicit. This more-or-less lands directly on the problem of what constitutes a wff, the very first problem of formal logic. – Doug Spoonwood Aug 21 '11 at 17:33
@Doug, you fail to recognize that I "well read" the problem on three dozen forums and have no confidence in the validity of the question. Either way, I agree with your notational/logical comments. – The Chaz 2.0 Aug 21 '11 at 17:56
The question, as stated, asks if there exists a general consensus about "multiplication by juxtaposition." – Doug Spoonwood Aug 21 '11 at 21:05
This question is actually rather important when writing computer programs that parse mathematics; if they are to implement support for juxtaposition, which seems reasonable, their authors need to know whether multiplication by juxtaposition has a higher precedence than the usual multiplicative operators or not. It’s far from clear which choice is best — and indeed some software has changed its interpretation, so it’s wrong to regard discussion of this topic as “garbage”, even if much time has been spent on it. – alastair Jun 30 '15 at 15:16

$a/bc$, which is $a/b*c$ of course means $(a/b)*c$, and that is for the same reason $a-b-c$ means $(a-b)-c$ and not $a-(b-c)$. The reason being that mathematical expressions are meant to be read from left to right when there is no operator which takes precedence.

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