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Here's a common question among math teachers and students (and math dabblers who just like to raise people's hackles!):
"If you see 3/2x, how do you interpret it? Is it 3 divided by 2x? Or 3 divided by 2, times x? Order of operations says we do division and multiplication left-to-right, which leads to the second answer. However, if you look at the slash as a division symbol, it appears to be the other way: 3 is the numerator and 2x is the denominator."
The correct answer to this question is: it's neither.
That's right. It's neither 3/(2x) nor 3/2 times x.
So no matter which way you were arguing, you're wrong. Let me explain.
Whenever you come across something like this: 3/2, the standard reading is not "three divided by two." You read it as "three over two," (this is considered to be the proper designation for the slash symbol when used in this context). This lends credence to the notion that the slash is being used as a fraction bar, and therefore, our example should be read as a fraction: 3/(2x).
But did you know that there are specific rules for how you write fractions using standard typographic practices? First, you are expected to use a specific slash symbol, which is not your standard "forward slash" on your keyboard - it's a unicode symbol called "fraction slash." The fraction slash is designed with minimal kerning (space between characters), and there's a very good reason for this.
There's another typographical practice we must follow: we superscript the numerator and subscript the denominator. The superscripts and subscripts, combined with the minimal kerning, result in the numerator being above the slash, and the denominator below.
Thus, we would either write: 3/2x or 3/2x, and now you can see that proper typographic practices makes it clear which way we intended it to be interpreted.
In other words, 3/2x is actually just a typographical error, and not a real mathematical expression. It's the result of someone being lazy. (Don't worry, I've done it too!). With the sophisticated word processors we have these days, with powerful equation editors, there's no longer any excuse for any mathematician to type 3/2x. In fact, with equation editors, you can get expressions that appear much nicer than the ones that you create with superscripts and subscripts.
Of course, there is one place where this typographical error still shows up: calculators.
Many calculators are not designed for proper typographical display of fractions. So what do we do? We do one of the following:
- Figure out which way your particular calculator handles this expression, and always do it that way.
- The safer approach: When dealing with a calculator, always clarify your meaning by including parentheses.
Once you've settled on one of these practices, it's time to accept the fact that you haven't been arguing about some standard of mathematics, but about typography. It's now time to do some real math, and leave behind the arguments about typographical quirks!