Mathochism: Random mathy thoughts
One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?
We’re still working on ways to use the derivative to graph functions, and have moved on to the Mean Value Theorem. Both it and Rolle’s theorem seem very straightforward to me, but I am wary. Let’s see how I do with the nastier expressions, full of roots and absolute values.
At any rate, I’ve been having some random mathy (mathesque? math-related?) thoughts. I number them below:
1) Happy Ada Lovelace Day! I raise a derivative to her, and women in STEM everywhere, past and present. And that includes the amazing women I have already had the pleasure of interviewing for the book, as well as the ones I still hope to interview. And more than that, here’s to scholarly women in general. Sadly, we live in a world where a 14-year-old can be shot in the head for daring to demand an education. Her struggles make me grateful for the privilege of being allowed to study safely, and at all. May we fight for a global society where every woman (and man) gets that privilege.
2) I have managed to find several calculus texts that appear to feature more challenging problems. One is Stewart’s Calculus (thank you, Antonia!). Unfortunately, I have neither the cash ($246? Really?) nor the shelf-space to get my own copy, but happily, my college library had a battered specimen that is mine until Halloween. I also found a book called “Math for the frightened,” for some light reading. And another reader recommended “Overcoming Math Anxiety.” I was able to find that at my local public library. As a big public library supporter, this makes me happy.
3) A stupid question, perhaps, but WHY EXACTLY is a vertical line, i.e. x = 1, undefined? And does undefined automatically mean non-existent? And WHY is 0/0 undefined? I do grasp the logic, but never felt I got an adequate explanation. These are both concepts that have been handed down as gospel since Algebra, and I’ve just accepted them, kind of like I accepted that the Earth orbits the sun. And I’m not going to challenge either. I guess I was just wondering if someone could direct me to a proof of these postulates, because I don’t remember ever seeing one. And if these proofs are in any of the books I mentioned, or the books kind readers have recommended in the past, please forgive me. I’m still catching up on my book list.
4) I fear my conversations with my young classmate Phil about the calculus class probably being a weeder have bummed him out. When I talked to him yesterday, he was so forlorn. When I was his age, older people often bummed me out with their wise cynicism. And I’ve always tried not to be that older person, but being relentlessly upbeat can be annoying too. Maybe I’ll stick to talking to him about physics and philosophy.
All text copyrighted by A.K. Whitney, and cannot be used without permission.
On your #3, first of all, the line isn’t undefined, its slope is. The line is perfectly well defined: it’s the vertical line passing through (1,0). The slope is undefined, because we (intentionally) haven’t defined what division by zero means. That doesn’t mean it doesn’t exist – among other things, it leaves the door open for someone to extend the system by giving it a definition – but it turns out that it’s pretty hard to give it a useful mathematical definition that preserves the other abstract properties of a number system that we want to work with and still is descriptive of the real numbers.
For example, I could say “I’ll define system X, which is just the ordinary rules we’ve been talking about, plus I define x/0 to be 7 for all x.” That’s theoretically permissible, because I’m just adding a definition for something that was previously undefined. But now, all kinds of other undesirable consequences start cropping up. We lose some cancellation rules – it’s no longer the case that if (a/c) = (b/c), then a=b, since we could have a=1, b=2, c=0. Likewise, a point and a slope no longer uniquely identify a line, because there are now two lines going through (0, 0) with a slope of 7: the line y=7x and the line x=0. Working through all the consequences of this particular definition is challenging, and doesn’t seem to lead anywhere particularly useful or interesting, at least for my proposed definition above. You still wind up having to special case division by zero a lot when working out your proofs and derivations, so it doesn’t seem to have any real advantages over the conventional system.
(By the way, these are perfectly reasonable questions to bring up with your professor at office hours, at least if there’s nobody waiting to ask about class material they don’t understand. I was asking these kinds of questions of my math teachers back in high school, and got into some really interesting discussions about the nature of math as a result. They are certainly not stupid questions – but they may get into some fairly deep waters in order to get a meaningful answer.)
For an example of an extension that is useful, consider the algebraic properties of the real numbers. Over the real numbers, the answer to “What is the square root of -1?” is undefined. But if we allow ourselves to extend the system by adding a definition, and calling the answer i, and if we further assume that almost all the properties of the real numbers apply to i as well, we get the rules that apply to complex numbers. We still give up some of the old properties when we do this – it’s no longer the case that for any two (complex) numbers a and b either a is less than b, equal to b, or greater than b – but in this case, we generally consider the tradeoff to be worth it, because there are a lot of useful results that come out of the complex numbers.
As far as why you haven’t seen a proof of the various postulates, postulates and axioms are supposed to be the things we accept without proof in order to bootstrap a mathematical system. If you could prove it from the other postulates, you would generally drop it as a postulate and just make it a theorem. That tends to get confused in most basic Geometry texts, where they call a bunch of things postulates where what they really mean is “we don’t have time or space to prove this from first principles, so we’ll just call it a postulate so we can move on,” but that’s the way it’s theoretically supposed to work. Euclid wrote the whole of the Elements based on just five postulates, and mathematicians spent centuries trying to prove the fifth postulate (the parallel postulate) from the other four, before they finally understood that it really was independent of the others.
Hope this was helpful.
That was great. Thank you! Part of the reason I was thinking about this was because of imaginary numbers. I kind of miss being able to work with them in this course, though they complicate matters. And my current professor often tells us it’s not enough to just know something in math, we have to prove it. So I was wondering about the proof of undefined.
@DaveW, what an awesome comment. 🙂
About 0/0: it seems to me that this is undefined because it’s a place where two reasonably well defined mathematical concepts collide.
On the one hand, the limit of 1/x as x goes to zero is infinity. And we can write 0/0 as 0*(1/0).
On the other hand, x/x=1.
Which would imply that 0 times infinity = 1.
Duuude. You just blew my mind.
😀 Mission accomplished!
Which kinda demonstrates what “undefined” means, in math: “Dude, that’s bad news. Let’s just not go there, k?”
Heh. And here I thought denial was a humanities thing.
Glad you’ve found Stewart useful! I’m so glad my uni uses it; it’s my favorite so far. But it is painfully expensive. Have you looked into getting an online copy? That might still cost too much, but it’s definitely cheaper than the hard copy.
You’ll see nice examples dealing with 0/0 when you get to L’Hospital’s Rule! That’s when the idea of math coming from definitions and postulates rather than some inherent mathematical “truth” kind of exploded my brain for the first time.
0/0 isn’t undefined; its indeterminate. “Undefined” means we know what’s going on and its the answer is not a real number (e.g. you get infinity, or you get infinite oscillation). “Indeterminate” means we don’t know what’s going on yet, the 0/0 doesn’t tell us what the answer is. You have to do some algebra or apply other methods like l’Hopital’s rule to find out what the indeterminate form comes out to in each case, and it might be undefined but it also might be a number. But yes, it is fair to say that 0/0 is indeterminate because its a clash between two mathematical concepts, multiplying by 0 and dividing by 0, each which would give a different answer, and you can’t have both at once. So its indeterminate, because that form doesn’t determine what your answer is. But that is different than undefined. The distinction between those will be really important when you get to l’hopital’s rule.
A vertical line is defined, its slope is “undefined” in the sense that it rises infinitely for zero run. Its also not a function of x (fails the vertical line test). But you might find that you prefer parametric equations, where that isn’t really a big issue.
I am not disagreeing with you. However, ever since pre-algebra, my various profs have referred to anything/0 as “undefined.” *shrugs*
When people say “anything over 0 is undefined” they mean “anything nonzero over zero is undefined”. If you look in your calculus textbook, that’s what it will say (probably stated in terms of limits). And there’s an obvious problem with arguing that 0/0 is undefined – for instance, what if I take x/x, and then take the limit as x goes to 0? By that logic I should get an undefined answer, but by cancelling the x’s the answer is clearly 1, which is a perfectly well defined number. I can get any number that way – if I want to get 5, I’d take 5x/x and take the limit as x goes to 0, so its indeterminate at first glance, and again cancel the x’s and get 5, which is again defined. So it really does not make sense to say 0/0 is undefined, but it does make sense to say that it doesn’t determine what our answer is or what our limit approaches, i.e. is indeterminate. I’m not trying to be pedantic – its an important distinction for the rest of the first semester of calculus, and it will help you if you keep those distinctions clear. Its worth making a list of forms that are indeterminate vs forms that are undefined, and maybe even also things that look confusing once you’ve been staring at those lists but are actually defined (like 0/1 or 1/infinity). The wikipedia page for indeterminate forms will give you a nice list of those and some explanations, especially for when you study l’hopital’s rule.
I should add that if I define the function f(x)=x/x, the *function* *is* undefined at x=0. But the *limit* as x approaches 0 of f(x) gives you 0/0 at first glance which is indeterminate, not undefined, and then resolves to being equal to 1, which is defined. So the *limit* is defined, even though the function isn’t defined at that point. In an algebra class you wouldn’t be talking about limits, and the algebra book probably has a blanket caveat at the front saying that they’re only dealing with finite quantities so they don’t have to give that caveat every time they say something about dividing by 0. In a calculus class if someones says “0/0” they are nearly always referring to a limit, and in that context it does not make sense to say its undefined. Lots of people get midterm problems wrong because of that confusion – they want to just stop at 0/0 and say the limit is undefined, and that isn’t always true. You have to find out what that limit actually comes out to, and it might be defined.
Thanks, that does clarify things. Some of them did talk about 0/0 also, though, explaining that it didn’t just mean zero. But as you pointed out, this was before calculus, so likely they just didn’t want to confuse everyone.
Right, because the limit doesn’t actually have to reach its destination, just get infinitely close.
Right, the point is that 0/0 doesn’t necessarily mean 0 and it doesn’t necessarily mean undefined – we just don’t know what it means for the limit from looking at that form, and have to figure it out for each case depending on where the 0/0 came from. That’s completely different from something like 1/0, which DOES tell you that your limit is undefined (because its approaching plus or minus infinity, and if you want you can check which in a given case).
As a practical matter, on an exam, if you plug in to a limit and get 5/0 you’re done – you can say the limit is undefined (and you can check if it approaches positive or negative infinity if you care, for instance for graphing asymptotes). If you plug in and get 0/0 you’re not done, and you don’t know anything about what the answer is from that 0/0. I think its important for students to know the difference between what tells you the final answer to your problem and what doesn’t, no matter what you want to call them. (This is where a list of indeterminate forms comes in handy.)
Its technically bad math language to say something like “1/0=plus or minus infinity”, but often people say that in calculus knowing that in the context they mean a limit that approaches 1/0 then approaches plus or minus infinity. A lot of this language only makes sense if we know we’re talking about limits (and remember derivatives are defined as limits, so when you’re talking about derivatives you’re still implicitly talking about limits). So in an algebra class you wouldn’t be talking about limits, so you wouldn’t use this same phrasing.