One divided by infinity is not zero?

Question

I know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ?

I've got some idea how to explain that: Let's say we have a random-number generator that generates numbers in interval $(0, \infty)$. What is the probability that it generates 5? $\frac{1}{\infty}$. So it cannot equals zero because if it was zero, it wouldn't generate any number (every number has the same probability - $\frac{1}{\infty}$).

Am I right? Or is that wrong idea?

Thanks

Answer 1

It is worth noting that since $\infty$ is not a number, the expression $\frac{1}{\infty}$ is not meaningful. That is, you cannot evaluate this expression. It is not a number. You can think of it in much the same way as $\frac{1}{0}$ --- an expression without any interpretation.

The only means that we have to talk about expressions involving $\infty$ is through the concept of limits. The shorthand $$ \lim_{x \to \infty} \frac{1}{x} = \frac{1}{\infty} = 0 $$ is sometimes used in a careless way by students, but the middle expression in this string of equations isn't strictly meaningful. It is only a device used to remind you that we are looking at what happens as we divide 1 by increasingly large numbers.

So, I suppose that in some sense, you can say that $\frac{1}{\infty} \neq 0$, as the left-hand side here isn't any number at all.

Answer 2

Operations in the real line are maps: \begin{align} +&\colon \Bbb R \times \Bbb R \to \Bbb R \\ -&\colon \Bbb R \times \Bbb R \to \Bbb R \\ \cdot &\colon \Bbb R \times \Bbb R \to \Bbb R \\ \div &\colon \Bbb R \times (\Bbb R \setminus \{0\})\to \Bbb R \end{align}

But $\infty \not\in \Bbb R$, so there is no sense trying to do things like $1/\infty$ in this context. Any pair on which $\infty $ appears is not on the domain of the functions above.

Answer 3

In some contexts it makes sense to say $1/0=\infty$ and $1/\infty=0$. If you're working in a context in which there is just one $\infty$ that can be approached in either the positive or the negative direction, then it makes sense. This makes sense for the values of (but not the arguments to) the tangent and secant functions. For rational functions it makes sense for both arguments and values.

Answer 4

Your argument makes use of stochastics. If you want to do things the mathematically exact way, you would first need to define a probability measure on $(0,\infty)$. Assuming that you choose a continuous distribution, then yes, the Probability that you hit $5$ is $0$ - but there are infinitely many numbers, so there is no contradiction. (Keep in mind that $0 \cdot \infty$ is not well defined, and unless in the case of $\frac1{\infty}$ there is no heuristical way to define it)

Check out Measure Theory: https://en.wikipedia.org/wiki/Measure_%28mathematics%29

Answer 5

If we are to understand "$=$" as a binary relation of identity and "$\neq$ as a binary relation of non-identity, then in most contexts, formally, $1/\infty$ is not identical to zero. Therefore it would make logical sense to write $1/\infty \neq 0$ as long as we "define what we mean by $1/\infty$". This is more in a strictly logical sense than in a mathematical sense.

In order to interpret $1/\infty \neq 0$ mathematically, you have to carefully describe the underlying structure of the statement.

If we are to only consider "$=$" and "$\neq$" for numbers or algebraic expressions involving numbers, then $1/\infty \neq 0$ doesn't make sense because the left side isn't a number. However, if you define $1/\infty$ appropriately (e.g. as a limiting procedure), then the statement makes sense and can be evaluated as false.

Answer 6

First: Can $1/\infty = 0$, or $1/\infty\ne0$? Neither is true, according to ordinary algebraic conventions, but there are two ways in which the former can be correct.

1. Notational abuse. We can't use $\infty$ like a number in our expressions, because it isn't a number in any of the usual senses. But let's be honest, everyone knows what you mean when you say $1/\infty$; you mean $\lim_{n\to\infty}\frac{1}{n}$, which does equal zero.
2. Choose-your-own-conventions. I find that one of the cool things about maths is that you can come up with your own system of definitions, axioms, and notations, and as long as it's consistent, it's maths. (And if people take notice and find it useful, then it's good maths. That's why we have things like imaginary numbers: "You can't take the square root of a negative number." "Okay, but what if we could?")

I'm not aware of any commonly-accepted algebra in which infinity is an ordinary quantity and can be operated upon (as opposed to a geometry in which this is the case: see projective geometry). But that doesn't mean that one couldn't exist... and if it did, I'm pretty sure that $1/\infty$ would be zero rather than non-zero.

Second: If random variable $X$ has infinite range, how can $P(X=x)$ be non-zero for any $x$? This is actually a very important question and does not at all depend on the matter of $1/\infty$.

What it boils down to is this: yes, $P(X=5)=0$. However, the probability that $X$ is in any range of values (e.g. $P(4.99\lt x\lt5.01)$) is non-zero, and you can make those bounds arbitrarily close to 5.

Wikipedia's probability density function article has a pretty good explanation. To quote:

Suppose a species of bacteria typically lives 4 to 6 hours. What is the probability that a bacterium lives exactly 5 hours? The answer is actually 0%. Lots of bacteria live for approximately 5 hours, but there is negligible chance that any given bacterium dies at exactly 5.0000000000... hours.

Answer 7

$\infty$ is not a real number (it's not in $\mathbb R$). It's not even an imaginary or complex number. Basically, $\infty$ isn't one of the things people usually call "number," and, as such, $+$, $-$, $\times$, and $\div$ are undefined for it.

However, there are ways to get around this. Define the set $\overline{\mathbb R}$ to be $\mathbb R\cup\{-\infty,\infty\}$, where $\infty$ and $-\infty$ are just meaningless symbols. Then define what $+$, $-$, $\times$, and $\div$ mean in this set. Really, you could define them to be almost anything, as long as you don't end up with contradictions, but a reasonable definition is outlined here. In the system constructed there, $\frac1\infty=0$. However, it seems kind of artificial, since we've basically just added a symbol, called it "infinity," and made it work like we hope an actual infinity would work.

Another thing is called "potential infinity." Consider the following equation: $$\lim_{x\to\infty}\frac1x=0$$ This is read as "the limit as $x$ goes to infinity of $\frac1x$ equals $0$." What this means is, as $x$ gets larger and larger, $\frac1x$ gets closer and closer to $0$. That's why this is called a "potential" infinity — it asks what happens as $x$ gets closer to infinity, without really getting there. This "limiting" operation happens entirely within $\mathbb R$, the set of real numbers. The above equation is as close to your question as you can get with limits.

Then there are things called "cardinalities," which are too complicated for me to get into. (The basic idea is that they measure sizes of sets, sets which might be infinite.) Division is not defined for cardinalities.