# What is “multiplication by juxtaposition”?

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

 = 16 ÷ 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?

• 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.

• Parentheses are nothing to be miserly about; certainly not in a time where disk space is cheap. – J. M. is a poor mathematician 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
• I'll trade you numerology for a Good Answer badge. – Asaf Karagila May 9 '16 at 13:08

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 $2 \times x^2$.

• 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. – user9638 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

$a/bc$, 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.

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