Saturday, January 5, 2013

Whether 6 ÷ 2(1 + 2) = 1 or 6 ÷ 2(1 + 2) = 9 depends on….

Subjective Intent in Mathematics.

January 5th, 2013

Whether 6 ÷ 2(1 + 2) = 1 or 6 ÷ 2(1 + 2) = 9 depends on who is asking and why.  Ultimately, the answer depends on the subjective intent of the person who wrote the equation.  Subjective intent in mathematics is something we obviously don’t hear very often, especially when speaking of mathematical equations and mathematical operators. 

For the various arguments involved as to whether the answer is nine (9) or one (1), the best place to look is Ask a Mathematician / Ask a Physicist at   Also, Wikipedia has a good synopsis on the order of operations in mathematics which seems to be correct, located at  By any reader of this blog entry reviewing the arguments at those other sites listed above saves me the trouble of having to repeat it all here, by the way.  :)  Nevertheless, as we can see, this matter appears as a heated debate among the internet mathematicians. 

A.  The ultimate answer depends on who is asking, why they are asking, and who or what is performing the mathematical operations.

Starting in reverse order:

1.  The answer is dependant on who or what is performing the mathematical operations represented by the equation.  As in the real world we mostly have been taught to clear the parentheses first and then proceed on from there - additionally, it is likely that a second set of parentheses is implied surrounding the two (2) multiplying the quantity of one plus two, ie: 6 ÷ (2 (1+2)).  This would yield an answer of one (1).  But a computer program would likely proceed from left to right and would divide six (6) by two (2) and then multiply by three (3, or 2 + 1 as given in the equation) which would then yield an answer of nine (9).  :)   But then again, a computer program would likely need a mathematical operator to be inserted between the two (2) and the parenthetical sum of two plus one (2 + 1).  Also, the operation represented by the division sign would likely be an issue for a computer program as well - as most programming languages use a slash for the division operator.

2.  Then we have to look at why someone is asking this question regarding the numerical answer of this poorly written equation and in light of the confusing notation of this equation as well.  Are they asking because they have a genuine need to decipher the poorly constructed equation or are they asking because it’s a trick?  Most likely they are asking because it is all a trick and therefore the answer they are seeking would probably be nine (9).  Because as I said above, the typical answer in the real world to humans would probably be one (1).  So if one were being asked by a jokester or trickster, the trickster would likely be looking for the answer of nine (9).

3.  Ultimately, the answer is dependent on who is asking or who wrote this equation because it all depends on the subjective intent of the individual who wrote or transcribed the equation.  What did they intend to be the order of the operations when they wrote the equation?  As I mentioned above, in the human world, such notation probably intends an answer of one (1).

But this all begs the question that in math, there can only be one right answer.  Right?  If math is done properly, shouldn’t we all get the same answer?  Well, maybe, but maybe not – as displayed by this equation in this matter/blog.

So, in this blogger’s mathematical world, the answer could be one (1) or nine (9).  But most likely to us humans the answer would normally be one (1).  That is to say, if a human wrote this poorly clarified equation, he/she probably was looking for an answer of one (1), unless they are a trickster as explained in paragraph A.2. above.  But if a computer displayed this equation, regardless of the syntax errors for the typical computer program, the answer could be nine (9) as contrasted in paragraph A.1. above. 

The answer is particularly and ultimately dependent on the subjective intent of the creator of the equation, as discussed in paragraph A.3. above.   So, to make an analogy for those who were told there was no math involved in blog reading, it’s sort of like the use of certain punctuation (and wording) in grammar.  With such an analogy being, when is it proper to use a comma, when is it not, and when is it up to the discretion of the writer as to use of the comma (or colon, or question mark).     

Therefore and as such, the answer is:  It depends.  Start at the top of this blog and reiterate through the paragraphs herein again, if this answer bothers you.  If so, it will probably only bother you more as you repeat any iterations of this blog.  :) 


6 ÷ 2(1 + 2)
6 / 2(1 + 2)

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