Something I have wondered since at least high school algebra: What is zero times infinity?
A couple of things I already know, that may be significant. Of course, zero times ANY number, is zero. But infinity is not your average number. Actually, I don’t think it is technically a number at all.
Also, when you have a line of infinite length, you have an infinite number of points on it. Each point has zero length. So that seems to suggest that zero times infinity, is just infinity.
Also, I used to think one divided by zero is infinity. Actually, I learned on these boards (you learn so much here:)), one divided by zero is ±∞. In any event, I do know that one divided by infinity, is zero. So zero times infinity, COULD(?) be also defined as zero divided by zero (I hope I haven’t lost some of you by now;)). Zero divided by zero is undefined, because there is no such number. Again, I am just more confused.
So I submit it to my fellow board members: What on earth is zero times infinity?
First, 1/0 is undefined. Because the limit can be either + or - infinity depending on which direction you come from. Two different things can’t be one thing.
Hence 0 * infinity is also undefined. It is also ambiguous in terms of limits. E.g., the limit of x * 1/x is 1 as x goes to 0 (in either direction). But x * 2/x goes to 2, etc.
Some things just don’t have answer. Get used to it.
The more interesting question here is whether the product of “a function that approaches infinity” and “a function that approaches zero” can have a distinct finite value. The answer is yes (sometimes) but it will be different depending on what those quantities are, and you need calculus (specifically L’Hopital’s rule) to figure out what it.
Trying to simply multiply the numbers 0 and ‘infinity’ won’t get you anywhere though, it’s undefined.
There are many different things in mathematics one might mean by “infinity”, and nearly as many different things one might mean by “zero”. You need to define your terms before the question is meaningful.
Zero is a member in good standing of many commonly used sets of numbers, including the real numbers, the rational numbers, the integers, the complex numbers… (Infinity, not so much.) Mathematicians don’t really have a definition of what is or is not a number, just of what is or is not a particular kind of number.
Longer answer, that requires calculus to fully explain: 0∞ is an example of an indeterminate form, which means that if you have an expression made up of one factor that’s approaching zero, times another factor that’s approaching (positive or negative) infinity, just knowing that doesn’t determine what the whole expression is approaching. There are examples where it’s 0; there are examples where it’s infinite; there are examples where it’s 42.
A: (x²)(1/x) as x approaches 0
B: (x)(1/x²) as x approaches 0
C: (x)(42/x) as x approaches 0
Zero is most definitely a number. In the most common everyday cases, “infinity” is not a number, it’s just the idea of something that goes on forever. Multiplication is well-defined for many different kinds of numbers, but not for abstract concepts.
There are alternate number systems where infinite and infinitesimal values are defined and operated upon. But they are rarely used, so you have to be clear about your context if you’re working within those systems.
As stated above, in normal arithmetic, 0 multiplied by infinity is undefined. Infinity is not a number and can’t be used with arithmetic operators. However there is the Extended Real Numbers, in which infinity and negative infinity are added to the set of reals and can be used in some ways like normal numbers. However, even in this system, zero times infinity is usually undefined (as is 1/0), although there are a few uses where zero times infinity is defined to be zero.
The human brain seems to be hard-wired to understand zero (or nothing), and the first few numbers (1, 2, 3, not sure where it ends but somewhere around 3 or 4 IIRC), “a few”, and “many”. In earliest times, zero, or nothing of something, wasn’t a number. The Babylonians had a zero of sorts as a placeholder, but it wasn’t a true number because it was never used alone to represent a zero number of items.
The ancient Greeks used to argue at great length about the philosophy of zero. Could nothing of something be something? There were arguments on both sides. Greek mathematicians did have a zero placeholder that was used like a number, and was used to represent nothing of something. Later, in India, they started using zero as a number in the modern sense, which is how we use it today.
So your claim that zero isn’t a number is slightly outdated.
IMHO you would have a hard time coming up with any answer other than zero for just about any reasonable definition of infinity.
0/x = 0 * 1/x is zero for all x where x ≠ 0. As x approaches 0 from either side 0 * 1/x remains zero. It is thus possible to obtain 0 here too.
If infinity is the cardinality of the set of natural numbers, well, by induction we should be able to get zero as the answer as well. Thus even for transfinite numbers, multiplication by zero must yield zero.
Yeah, but you’re kind of fixing the race by having 0 be zero but having 1/x only approach infinity.
I don’t think it works that way. Induction can be used to prove that something is true for all (or arbitrary) natural numbers, but you don’t get to then assert that it’s true for transfinite numbers as well.
Francis Vaughan, if you’re going to treat infinity as a limit, then why not do the same for zero? x/x = x*(1/x)= 1 for any x ≠ 0. Just as 1/x approaches ±infinity, so does x approach 0. So one could say, by that argument, that 0*infinity should be 1.
Or you could instead look at 2x/x and get 2, or look at x^2/x and get 0, or sqrt(x)/x and get some sort of infinity, or…
I’m trying to avoid i/0 being infinity, but it occurred to me that the objection to 1/x goes away for the case where it is 0/x as we don’t have the problem of it having different signs on each side - we can collapse onto 0 from both sides. Thus we can side step the need to worry about an undefined number.
For infinite and transfinite I would be making an argument that we are using the 1-1 correspondence definition of cardinality. For every possible pairing of numbers the answer is zero. The transfinite numbers are built from the powerset of infinity. It is a constructive process. I would say that if multiplication by zero has essentially the semantics “none of these” it doesn’t matter how you construct the transfinite number, none of them is zero.
