Basic question about dividing fractions of functions

I know I should know this, but I can’t get my head around it. Let’s start with:

1/1/2. Simple - one takes the reciprocal of 1/2, so it is 1*2/1=2.

Okay, but the worksheet I’m going over for calculating derivatives asks to figure out the derivative of f(x)=1/(x-1) from first principle.

So I get this far:

the limit as h approaches 0 of -h/(x+h-1)(x-1)/h. OK, so if 1/1/2 is 1*2/1, why aren’t we taking the reciprocal of (x+h-1)(x-1)/h and coming up with -h(h)/(x+h-1)(x-1). The next step of course is -h/h(x+h-1)(x-1), the h’s cancel out, and you get a cromulent derivative.

Talk slow.

As always, not homework - just an old guy trying to come to terms with the math he glossed over in his youth.

Where are you getting that extra h in the denominator?

The problem is that without parentheses the inline syntax a/b/c is ambiguous. It can mean a/(b/c) = ac/b or (a/b)/c = a/bc.

The interpretation you want for your problem is (-h/(x+h-1)(x-1))/h = -h/(h(x+h-1)(x-1)) = -1/((x+h-1)(x-1)) which goes to -1/(x-1)^2 as h -> 0, which is the right answer.

In more conventional mathematical formatting (i.e. not limited the available text formatting) the syntax is unambiguous.

If it helps, I found a page with this particular problem worked out.

Thank you for finding that, Thudlow. it will make it easier to frame my question.

OK, take a look at line three of the solution. We can recast that as line (leaving out the limit part):


Here’s what I don’t get: let’s recast this as

b = (x+h-1)(x-1)
c = h

So we would have a/b/c. In line four of that solution, a/b/c becomes a/b*c.

But, if a = 1, b=1, and c= 2 (that is 1 divided by 1/2), we would recast this as a*c/b.

So I don’t understand why it’s different in the two situations.

And yes, I realize I should have figured this out in, oh, grade 9, but I didn’t, so I appreciate the help.

What we have is (a/b)/c.

This is not the same as a/(b/c): division is not associative. Using your numbers, it’d be (1/1)/2, not 1/(1/2).

And (a/b)/c (or, a/b divided by c) is equivalent to (a/b) * (1/c), which is equivalent to (a1)/(bc).

Thank you! So just out of curiosity, at what point should I have learned that?

If a/b/c is interpreted to mean (a/b)/c then that’s the same as a/(b*c).

(a/b)/c = (a/b)(1/c) = (a1)/(bc) = a/(bc).

ninja’d by Thudlow Boink.

If you’re enough of an “old guy,” when you originally learned about fractions and/or division you probably didn’t have to worry about the ambiguities and confusion introduced when trying to type such things into a computer message board (or even a calculator, maybe), so there’s that.

At some point in a relatively basic algebra class, if not earlier, you should have been introduced to the associative property (of addition and of multiplication). Your teacher or textbook probably should have pointed out, but might have neglected to do so (or you might have forgotten) that one reason it’s important is that subtraction and division don’t work that way: for them it matters what order you do things in.

Working with relatively complicated fractional algebraic expressions (like the ones in your example) is a topic that maybe falls through the cracks sometimes. In many of the College Algebra or Precalculus books I’ve looked at, it’s included in a “Review” or “Preliminary” chapter; yet in my experience it’s something that students often haven’t learned coming into such a class and often find difficult.

Back when when you learnt ( a/b ) = 1 / ( b/ a ) … so you can reverse it

1/(a/b) = (b/a)

Huh. So based on the explanation of lack of associative property in division, I just figured this out after 3 decades. I was always unclear how the acceleration due to gravity (on Earth) could be expressed as both 9.8m/s^2 or 9.8m/s/s. So, turns out that it’s really 9.8 (m/s)/s.

I’m not stupid! Really! :smack:

That’s actually a really good example. Acceleration is how fast velocity is changing. If your velocity is measured in meters per second (m/s), the rate at which it’s changing would be how many meters per second it’s changing by per second ((m/s)/s). But that’s mathematically equivalent to, and often shortened to, m/s[sup]2[/sup].

1/1/2, written like that, would actually be 1/2, following the order of operations. You’d need to write it out as 1/(1/2) to get 2 (or some other way where it’s clear that the 1/2 is to be treated as a single unit.)

If there’s no context, you should clarify it either way. It would be wise to use parentheses or some other visual clues (like making one of the horizontal lines much bigger than the other, as was done in the link from post #4) to indicate whether you’re trying to say (a/b)/c or a/(b/c). It would be unwise to simply write “1/1/2” and assume that everyone in your audience knows that you meant (1/1)/2.

In the case of saying “meters per second per second”, it’s clear that they mean “(meters per second) per second” because the alternative, “meters per (second per second)”, is gibberish.

Well for other operations, after all order of operations issues have been resolved, operations evaluate left to right (meaning always (a/b)/c). It’s just that “/” meaning division is kind of non-standard. I think if it was written a ÷ b ÷ c instead it would be clearly thought of as meaning “left to right” the same way a - b - c would be.

I don’t have anything to say about “should” in this context. But, let’s note that you could have learnt that without anyone teaching it to you. You can see it just by looking at examples: for example, (60/6)/2 = 10/2 = 5, but 60/(6/2) = 60/3 = 20. Just looking at how division works on whatever random examples you like, you’ll see that almost never does (a/b)/c come out to the same thing as a/(b/c). Go ahead; pick any three numbers and see what happens.

[The only times the two match is when the third number is 1, but that fact might be hard to recognize immediately. Still, the moral is that you should* rarely just believe some rule about how to shuffle symbols unless you actually know and understand a reason for that rule. There’s no reason to have thought that the rule (a/b)/c = a/(b/c) would work in the first place, and so there’s no mystery in the fact that it doesn’t.]

The great thing about math is that you never have to take anyone else’s word for anything; you can figure it out/see what’s going on all for yourself.

Also, note that this is a fact that has nothing to do with notation: It’s simply the case that if you take 60 and divide it by 6, and then divide that by 2, you get a different number than if you take 6 and divide it by 2, and then divide 60 by that. And this fact is true no matter how you choose to notate division.

Then there are other facts which only have to do with notation, and nothing to do with anything of mathematical substance. For example, facts about conventions and intentions, like what someone means if they write down “60/6/2”. Do they mean “First divide 60 by 6, then divide that by 2”? Do they mean “First divide 6 by 2, then divide 60 by that”? Are they indicating some kind of date instead of an arithmetic calculation? There are conventions which can help you understand the meaning behind the things the people around you say and write (and there’s no way to derive these arbitrary conventions by pure reasoning; you just have to have someone tell you), but note that these are only facts about how people speak and write. They’re different from facts of the kind in the previous paragraph. They’re like the fact that “d-o-g” refers to a certain kind of four-legged animal, which is not a fact about canine biology, but just a fact about English vocabulary. Similarly, it’s worth distinguishing facts about math notation from facts about actual math.

(Well, this statement isn’t quite true (the third number could also be -1. Or the first number could be zero.), but it’s close to true. It’s true enough. The point is, most random examples will quickly disabuse you of the idea that division is associative, just by seeing what happens.)

Almost. You could also deduce the meanings of these conventions by seeing a large number of examples in use. But that’s not so far from “someone telling you”.

And there are a wide variety of kinds of notational conventions. There are also some that, if not followed, won’t actually introduce any ambiguity, but which are still followed anyway. For instance, nobody ever refers to angular momentum as being measured in joules, but there’s nothing actually wrong with doing so.

In a perfect world, perhaps you could assume that everyone in your intended audience went to the same type of school you did, learned the same rules you learned, and never makes mistakes when applying those rules. On second thought, that doesn’t sound very “nice” at all. Never mind, it doesn’t matter, because that’s not the world we live in. Even when dealing with machines (calculators, computers, etc.) which never have opinions and never get bored, you can STILL end up with different answers from different machines because they were programmed with slightly different versions of the rules.

When in doubt, clarify.

It can be argued that the ÷ symbol means the exact same thing as the / symbol. The ÷ symbol is a picture of a tiny fraction with dots where the numerator and denominator would be. So 2÷3 literally means the fraction two-thirds. Using the ÷ symbol twice in rapid succession like a÷b÷c seems to be simultaneously saying that b is both a numerator and a denominator. It needs clarification.

No mathematician I know would look at a - b - c and think “left to right”. On the contrary, mathematicians are taught to see those subtraction symbols as adding negative numbers. For example, 3x - 4x - 2x is interpreted as (3)(x) + (-4)(x) + (-2)(x) and we can just as easily rearrange them in whatever order we choose by applying the commutative property of addition.