General Discussion
In reply to the discussion: 8/2(2+2) what do you come up with? [View all]fishwax
(29,149 posts)The dispute here is basically one involving different mathematical grammars. It's a result of an evolution in the grammar of mathematical notation that stems from the rise of computer-programmed calculations, the fact that typesetting vertical fractions is difficult, and the fact that basic keyboards have no simple key for the traditional division symbol.
In the beginning there was a horizontal fraction bar, and it functioned as a grouping symbol. Everything above the line (the numerator) was grouped as though with an implied parenthesis and everything below the line (denominator) was grouped. But vertical fractions (with a horizontal fraction bar) are difficult to typeset, and so the diagonal fraction was introduced in the 18th century.
The fraction slash, unlike the horizontal fraction bar, is not a grouping symbol--however, traditionally typesetting for algebraic functions held that the diagonal slash grouped together everything immediately before it (that is to say, everything not separated by a space or a function symbol was the numerator) and everything immediately after it (everything connected to the slash bar and not separated by a space or an explicit function symbol was the denominator). So, for example, b/cd would group the quantity of c times d as the denominator, and therefore that would take place first in the order of operations. In my experience (as a student and then later as a typesetter and editor for educational publishers) that is the convention that textbooks and instructional manuals followed.
That convention, though, doesn't translate to computer programming, where the parenthesis becomes essential because generally (I'm not an expert here on coding, but this is my understanding) programming languages are going to require an explicit function. You can't put cd in a computer code and have the program understand that you are multiplying c times d. You have to include the operator.
The equation in the OP basically follows the pattern of a/bc, where a = 8, b = 2, and c = (2+2) = 4. The ambiguity arises because it isn't specifically clear whether a/bc is meant to be read as (a/b)c or a/(bc). It just depends on which grammar convention one uses.
My instinct (because of my background in typesetting and editorial) is to treat "bc" as the denominator of a fraction. It seems odd to me to say that a/bc = ac/b. Or that 4a/a(a+a) = 8a rather than the (to me) more intuitive 4/(a+a). But, again, that's because my training is rooted heavily in the latter. I don't know how widespread that convention is in educational textbooks today--my guess is that, with the increasing prominence of programming languages, texts would be more inclined to use parentheses to eliminate any potential ambiguity. I believe that there are some fields (for example, academic journals in physics) which still treat implicit multiplication immediately before or after a fractional slash as the numerator or denominator.