order! - a minipost

Apparently a math puzzler became something of a theme on the Internet..  a half dozen people asked me what I thought the answer was

8÷2(2+2)=?

Some came up with 1, others with 16.  Apparently there were near-religious arguments. A mathematician would tell you both are incorrect - the question posed is ambiguous and not an answerable bit of math ..  at least not generally.

 

The problem is the order in which one evaluates expressions.  It turns out the one you learned in grade school is a bit different from the one you may have had when you learned algebra. If you remembered grade school you probably get 16.. algebra people are more likely to get 1.¹  If your algebra teacher was more careful they would have said the expression is wrong - you need to add parenthesis to make it unambiguous. The exception would be if the only people you communicate with agree on representation. That can be a big if..

 

Taking it a bit further, the simple rules you learn in secondary education don't cover don't cover unary ±, which implicitly have lower precedence than exponents on one side and higher on the other!  For example

-x³ → -(x³)
x^-3 → x(^-3)

This is so well agreed upon that no one uses parenthesis in this case.  Historically juxtaposition was treated as implicit multiplication, but that led to ugliness like

1/3x →  1/3*x →  (1/3)*x
a^2x →  a^2*x →  (a²)*x

uck! 

People decided juxtaposition of numeric literals is like a unary operator ..  so the 2 in 2x is attached to the x like the - in  -x.  

 

Back to the original problem. The fact that people had different interpretations and mentioned rules from school meant a better rule was required.  The problem needs to be unambiguous to anyone who might encounter it.  This is an issue for computer languages - different languages can parse the same written expression differently and come to different results.  Agreement on standards is an issue too ..  the Mars Climate Orbiter was lost as  teams at JPL and Lockheed Martin had different takes on a single critical bit of data - one used metric units, the other english..   when the retrorockets on the spacecraft were fired to go into Martian orbit the spacecraft burned up in the Martian atmosphere - it was literally lost in translation.  And this was code that passed multiple very careful reviews - code written for space probes is some of the carefully crafted software period.  Just wait for complex coordination systems and things that move with people nearby or inside.

 

That said there are some areas were common agreement among practitioners creates a local set of standards..  Einstein summation notation in physics is confusing to outsiders, but all physicists know it. And then there's my favorite - the double factorial.  In math (n!)! is not n!! ...  (note that I also have the problem of people interpreting ! as an exclamation point:-)

 

This local agreement is probably true in most fields - even descriptions of an action in a sport are local to the sport.  We all use local shorthand - it's a form of jargon, but we need to know when we can use it and when we need to be more general even it it's awkward.

__________

¹ In one case 8÷2(2+2) → (8÷2)*(2+2) = 4*4 =16

   in the other 8÷2(2+2) → 8÷(2*(2+2)) = 8÷(8) = 1