nm
The wikipedia article on extended reals that I linked to says this:
–Mark
Isn’t our first problem that multiplication is defined as an operation between two real numbers? I understand we can include infinity with the reals but this requires a special definition of infinity. Would this not also require a special definition of multiplication?
Zero Times Infinity would be a terrific title. I’m thinking science fiction anthology.
I once read that log 0 is minus infinity. (I did not take this up with my older brother, who is the math whiz in our family.)
The limit of the function log(x) as x approaches zero is -∞. Or more accurately, log(x) decreases without bound as x approaches zero. You can’t actually evaluate log(0), because there is no exponent n for which e[sup]n[/sup] = 0.
It’s in some contexts useful to consider it so. The log function is only real-valued for positive arguments, so if we’re sticking to real numbers, then the only limit at zero we can consider is the one from the right. And it’s true that the limit of log(x) as x approaches 0 from the right is -infinity.
But it’s not always useful, interesting, or wise to restrict ourselves to the real numbers. You can take the log of a negative number, too, and that’ll always give you something with an imaginary part of pi. Or more precisely, the equivalence set (2n+1)*pi, where n can be any integer: The log function as applied to negative or complex numbers isn’t single-valued (actually, it’s not even single-valued as applied to positive numbers, but it’s much easier to sweep under the rug there).
Actually, to extend that a bit: Not only can you approach 0 from the positive numbers or from the negative numbers, but you can approach it from complex numbers, or even on any path through the complex numbers (once you have two dimensions to work with, the path need not be a straight line). When we’re dealing with complex numbers, in order for the limit to exist at a point, it has to be the same limit no matter what path you take to reach that point. And in order for a function to be continuous everywhere, it’s necessary that the limit of the function as you approach any point via any path be the same as the value of the function, no matter what the path is. The same is also true of the first, second, etc. derivatives, all of which are also defined as limits: A function all of whose derivatives exist at every point, considered in the complex numbers, is said to be “analytic”.
And it turns out that whether a function is analytic is significant even if you’re just looking at the real numbers. For instance, the power series expansion of an analytic function converges everywhere, but it doesn’t for a non-analytic function. My favorite example of this is the function defined as f(x) = 0 for x = 0, and f(x) = exp(-1/x^2) everywhere else. If you just look at the real numbers, this function is continuous everywhere, as are all of its derivatives… but at 0, every derivative is equal to 0, and so the function’s power-series expansion is just 0 as well.
What happened there? Well, it turns out that when you look at the complex numbers, that function isn’t continuous at the origin: It’s got what’s called an essential discontinuity there. Its limit is zero as you approach the origin along the real axis, but there are other paths you can follow that make the limit anything you want.
Multiplication doesn’t require real numbers; the imaginary number i can be multiplied. For example, i x i = -1. But infinity is not even a number; it’s a concept.
Of course it isn’t a definitive answer to OP’s question, but many computers, including those which comply with a modern standard, will get an answer of +∞ or -∞ when you divide a non-zero by zero, and get an answer of +Nan or -Nan (or “Indefinite”) when you divide zero by zero, or multiply zero by ∞.
The last time I tried that on a computer I got a message something like “ILLEGAL QUANTITY…”
I was trying to keep my question real … [giggle] … but yes we can multiply complex numbers. (2+7i) x (5+3i) = (-11+41i)
It’s certainly possible to use infinity as a number just like another real number. In fact, there are some interesting analogies between zero and infinity. Take these rules, for example:x + 0 = x
x + inf = inf
0 + inf = inf
0 + 0 = 0
inf + inf = inf
x - 0 = x
x - inf = -inf
0 - inf = -inf
inf - 0 = inf
0 - 0 = 0
inf - inf = nan
x * 0 = 0
x * inf = inf
0 / x = 0
inf / x = inf
x / 0 = inf
x / inf = 0
0 / inf = 0
inf / 0 = inf
0 / 0 = nan
inf / inf = nanwhere 0 < x < inf, and nan is “not-a-number” or undefined.
Whether or not these rules are useful depends on the context, no different than deciding whether or not any mathematical construct is applicable to the situation.
Another interesting parallel between zero and infinity is in physical measurements, neither absolute quantities of zero nor infinity can be measured with arbitrary precision. (For example, one say an object has length 1.023 +/- 0.034 meters, where there is no theoretical lower limit on the size of the error bars.) When measuring a theoretical zero quantity, the best one can do is put an upper limit on the value. When measuring a theoretically infinite quantity, the best one can do is put a lower limit on the value.
The same is true of a line of length 1, so the same reasoning would conclude zero * infinity = 1. The same is true of a line of length 7, so the same reasoning would conclude zero * infinity = 7. The same is true of a line of any length you like! You can split a line of length L up into n many intervals of length L/n, and as n gets closer to ∞, L/n gets closer to 0, so that we may think of any length L as decomposing into ∞ly many bits of length 0.
This is related to the idea that zero * infinity is, in continuous contexts, an “indeterminate form”: just knowing that x is very close to zero (i.e., very small), and that y is very close to infinity (i.e., very large), doesn’t tell us very much about what x * y could be (it could be very small, or very large, or just about any value at all), if x are continuously varying quantities.
Both of those statements are true, and to the extent they are interpreted as distinct claims, it’s just that they each hold true in different contexts. Math is not a set of rules handed down from on high, one-size-fits-all for all occasions. Different systems of rules arise (and can be devised anew) to model different situations.
The sense in which zero divided by zero is undefined is that it is “indeterminate”, much like we previously spoke of zero times infinity as an indeterminate form, but in this particular case, we can see how this indeterminacy arises in a perhaps more familiar fashion. Zero divided by zero is indeterminate because, clearly, there are multiple values which multiply by zero to yield zero; 1 * zero = zero, 2 * zero = zero, 3 * zero = zero… So none of these (1, or 2, or 3, …) can be singled out as the unique result of dividing zero (as in the right-hand-sides of these equations) by zero (as from the left-hand-sides of these equations).
It depends on your context, what you are using these numbers to represent and what multiplication means to you.
If you have infinitely many farms, each farm with zero pigs, well, you have zero pigs overall. In the context of cardinality, or other similar discrete contexts, we generally have unambiguously that 0 times ∞ is zero.
If f(x) and g(x) are positive fractionally valued functions, but f(x) approaches 0 and g(x) approaches ∞ in some limit, then the asymptotic behavior of f(x) * g(x) may be to approach any particular limit in the range [0, ∞], or to oscillate wildly, or any number of things, but at any rate, it will never go negative.
If f(x) and g(x) are fractionally valued functions, not necessarily positive, but f(x) approaches 0 and g(x) approaches +∞, then the asymptotic behavior of f(x) * g(x) may again be to do all kinds of things, only now over the wider range [-∞, +∞].
If we let them take on complex values, or matrix values, or what have you, wider possibilities arise. Everything is always context dependent.
(But the reason that 0 * ∞ is tricky and 0 * 5 is not is because there are good contexts in which 0 * ∞ takes on various different values, while 0 * 5 = 0 in just about every context anyone has any reason to use the familiar language of counting and multiplication)
All numbers are concepts. There is no reason to single out ∞ as somehow non-numbery but i and -3 and 1/2 and 0 and 7 and all the rest of them as perfectly numbery.
There’s no one fixed notion of what “numbers” are; long ago, we used the word only to refer to finite counting numbers, but we’ve long since expanded its use beyond that. Any abstract entity in any calculational system with enough family resemblances to this archetypal origin might well be considered a number of some sort. And ∞ certainly fits that bill. One can fruitfully standardize reasoning with the concept of infinity into various calculational systems for various purposes, just as was previously done with i and -3 and 1/2 and 0 and 7 and all the rest of them.
Right. And the reason they were designed to do so is the reasoning that goes into calling 0/0 and 0 * ∞ indeterminate forms.
dougie_monty’s observation that log(0) = -∞ might well be taken with conversely interpreting e[sup]-∞[/sup] as 0. That’s a perfectly cromulent thing to do.
The response that these are merely “limits” and not worthy of being reified into actual honest-to-god values is a bit like saying “No, no, the limit of x[sup]2[/sup] as x goes through 1.4, 1.41, 1.414, 1.4142… is 2, but there’s no actual square root of 2”. That would be a true statement about the rationals, sure, but not of use in chastising someone who never claimed to be restricting themself to the rationals, and similarly dougie_monty clearly did not intend to restrict themself to finite quantities so it is no error at all that they considered an infinite one.
There are forms of induction that legitimately have transfinite scope, though it’s not clear that Francis is employing these. Regardless, ignoring induction entirely, it is easy to establish that, in the context of cardinality multiplication, zero * infinity is, like zero * anything, equal to zero. Which is to say, if A is an empty set and B is an infinite set, their Cartesian product is, in turn, empty; there are no ordered pairs whose first element is drawn from A and whose second element is drawn from B.
In discrete contexts, the rule “0 * x = 0, unambiguously” is usually absolute. It is typically only in continuous contexts where this begins to fail.
“…isn’t a number. It’s a representation…”
Just what the hell do you think numbers are?
Missing words reinstated in bold. Not terribly important, and all other typos will be left uncorrected. That’s it from me for now; I end my posting spree.
+++divide by cucumber error. please reinstall universe and reboot+++
How many readers will get that reference?