I know there’s enough of you math munchkins out there to address this. While I do a lot of grunts in the trenches math every day, I don’t frequent the frontiers. So I must humbly ask the answer to this question.
What if, instead of division by zero being undefined (“we won’t go there”) it was defined as equalling infinity?
I’m not claiming some revelation or anything, I just want one of you who can speak to the subject to let me understand.
Our regular math (i.e., quantifiable small numbers) wouldn’t change. Carpenters would still be able to get a set of stairs put up as before.
What cosmological cataclysm would this change in assuumptions entail?
Maybe not equal to infinity, but why not equal to zero? If X0=0, then why does not X/0=0? Is it because XY can never equal X/Y, assuming the same value for X and Y in each equation?
The overwhelming majority of people have more than the average (mean) number of legs. – E. Grebenik
First, you might want to know how division is defined in the first place. Roughly speaking, we define division by its relation to multiplication: division is the inverse (or opposite) of multiplication. For example, the reason we say that 10/5 = 2 is BECAUSE 25 = 10…multiplication is the “main event”, and division simply undoes multiplication. Now, apply this to division by zero. Suppose we say that 2/0 = infinity. Fine, then by defintion, we are saying that 0 * infinity = 2. Now, suppose we say that 3/0 = infinity…uh oh, 0 * infinity = 3! Which is it? Hence, the term “undefined”–it can’t be defined in a way which is consistent for all problems. (The same argument holds for divsion by zero not equaling 0 either: 2/0 = 0 means that 00 =2).
What we do have to express the idea that dividing by something very small should result in something very large is the concept of a limit (remember that from freshman calc?). So, we can say that as the denominator of a fraction APPROACHES zero, the value approaches infinity. So, we do “go there” – we just never reach where we’re going!
(oh, and hi! i’ve been lurking here for a couple weeks now waiting for a good math question to jump in on )
The reason, I think, is pretty straight forward. If you’re going to divide, you have to divide by SOMETHING. Zero, by definition, is nothing. Therefore, division by zero is not actually division.
“The day after tomorrow is the third day of the rest of your life.” -George Carlin
Lynne, your definition is wonderful:
>>First, you might want to know how division is defined in the first place.<<
However, good as it is, can I offer a dumbed down one, from my second grade teacher?
If you have four apples and you divide them between two people, each gets two; if between three people, each gets one and one-third. If you divide four apples “among” one person, that person gets all the apples. If there are NO people to divide the apples among, you don’t divide them at all, you put them back in the bowl, or whatever.
To summarize: when there are no people among whom to divide the apples, then you are NOT DOING DIVISION. If you say “What about the bowl?” That’s dividing by one-- one bowl.
x/0 just isn’t possible.
–Rowan
Shopping is still cheaper than therapy. --my Aunt Franny
The problem is even deeper than that. It can be proven that if x/0 is defined as being equal to anything at all – you can even call it “foo”, and try to treat it as an extension to the number system of the same kind as imaginaries – then math breaks down altogether, and 1=2. The only way for math to work is to declare that x/0 is always illegal.
John W. Kennedy
“Compact is becoming contract; man only earns and pays.”
– Charles Williams
Lynne’s answer was great. I’d just add one little thing. When you do problems where you are mathematically modeling the real world, and division by zero comes up, it actually means (almost always) that your model doesn’t work, i.e there is something about the behavior of reality that forbids the situation that gave you division by zero. Saying that division by zero “is forbidden” cannot be defined away without losing this important aspect of mathematics, even if you could put it on a logical footing. Which you can’t for the reason Lynne showed.
This is an important conceptual distinction between 1/0 and lim(x->0) of 1/x. The former gives you problems, the latter need not.
Actually, Athena, that isn’t necessarily the case. Based on L’Hopital’s rule, infinity/infinity could be 1, could be something else entirely. Actually, the more proper way to phrase that would be that as the numerator and denominator approach infinity, the answer could approach 1, or it could approach some other value. L’Hopital’s rule concerns itself with 3 scenarios that I remember off the top of my head, 10 years after Calculus II (someone fill me in if I miss anything). They are:
(N/D) --> ? as N --> 0, D --> 0
(N/D) --> ? as N --> infinity, D --> infinity
1^X --> ? as X --> infinity
In each of the above cases, N, D, and X may be variables or expressions. Someone who’s a little more fresh on this, please correct me if necessary (as if I have to tell anyone here to do THAT!).
Great, now we’re going to have to use a message board forum to teach people how to take derivatives.
“I wept because I had no shoes, then I met a man with no feet. So I took his shoes” - Dave Barry
Actually, L’hopital’s rules aside, there are different sizes of infinity. Counterintuitive, I know, but check out G. Cantor’s set theoretic notions of transfinite numbers. Kind of neat, and completely useless, as far as I know.
Strainger, you’re right on target with L’Hopital’s rule. The two basic forms are
(N/D) --> ? as N --> 0, D --> 0
(N/D) --> ? as N --> infinity, D --> infinity
and there are a few other forms (like that 1^x) that can be rearranged into the two basic ones (hey, anybody want a refresher in logarithmic differentiation? yechh.)
That whole idea of approaching (as in limits) is absolutely critical in both theory and application. Infinity is a way to symbolize a very useful mathematical concept (the idea of growing without bounds), but it isn’t a number, and you have to be very careful to not act like it is(trying to cancel infinities and so on)…or at least, I do, since I actually use/teach this stuff…the rest of you needn’t stay awake nights worrying that you’ve improperly used infinity! (More than likely, infinity will NOT come seeking revenge and/or monetary compensation).
How’s this for a tag line?
“The good Christian should beware of mathematicians and all those who make
empty prophecies. The danger already exists that mathematicians have made
a covenant with the devil to darken the spirit and confine man in the
bonds of Hell.” – Saint Augustine
Hey, hey, hey! Y’all should just shut up about this L’Hopital’s rule and limits and stuff. Don’t you know that calculus is useless in the real world???
Thanks to lynne, et al; sometimes this board works the way it’s envisioned - I got an answer that works. I like handy’s comment. With a math that quantifies things between zero and infinity (check the monkeys thread), zero is when we’re fresh out of quantities.
Athena: I said not to start on the infinity thing. Now we’ve got math geeks all over the place, and you know how hard it is to get them out of the upholstery. =B^)
)