# infinity times infinitesimal - what happens?

So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration.

Also, can we divide infinite number by infinite number and get a finite number?

• The answer to this question (or whether your question is even meaningful) depends a lot on what exactly you mean by terms like "infinitesimal", "infinity", and "times". Your first example, e.g., looks like "differential form times ordinal number", and I'm not aware of any context where that makes sense. – Hurkyl Apr 24 '13 at 8:31
• @Hurkyl: If you use the Lebesgue integral of a function which is zero everywhere except a single infinite point, then it looks a bit like having $dx \times \infty$, even though $dx$ cannot be considered a quantity at all, not to say "infinitesimal". Related is the delta function, but again it is a purely formal thing and cannot be integrated normally. – user21820 May 22 '14 at 8:00

## 4 Answers

In the ordinary calculus, there are no infinitesimals.

Abraham Robinson and others, from the $1950$'s on, developed non-standard analysis, which does have infinitesimals, and also "infinite" number-like objects, that one can work with in ways that are closely analogous to the way we deal with ordinary real numbers.

In non-standard analysis, an infinitesimal times an infinite number can have various values, depending on their relative sizes. The product can be an ordinary real number. But it can also be infinitesimal, or infinite. Similarly, the ratio of two "infinite" objects in a non-standard model of analysis can be an ordinary real number, but need not be.

The calculus can be developed rigorously using Robinson's infinitesimals. There are even some courses in calculus that are based on non-standard models of analysis. Some have argued that this captures the intuition of the founders of calculus better than the traditional limit-based approach.

For further reading, you may want to start with the Wikipedia article on Non-standard Analysis.

This is a question for surreal numbers. Surreal numbers are a really amazing thing invented by John Conway that include numbers like 0 and 3/4, but also things like "twice the square root of infinity, all plus an infinitesimal". This question depends on the values of the infinite and infinitesimal, but the way it works is this. The number ω is defined as the number of items in the set {0,1,2,3,4,5...}, so it's infinite. The number ε is defined as 1/ω. So ω*ε is obviously one. If you think about it a bit, it makes sense that 2ε^2 * ω is 2ε, and so on.

 http://en.wikipedia.org/wiki/Surreal_number

• I disagree. w*e is 0. The infinitesimal decreases exponentially while w increases linearly. QED. Mike Maish – user138316 Mar 27 '14 at 1:59
• What do you mean "the infinitesimal decreases exponentially", or that "w increases linearly"? I think this answer is missing quite a few details. – user61527 Mar 27 '14 at 2:30

For $x$ > 0, define an infinite number by the divergent geometric series: $\displaystyle\sum_{i=0}^{n\rightarrow\infty} \left(\frac{x+1}{x}\right)^i$

and define an infinitesimal number as the difference between a convergent geometric series and its sum:

$x+1 -\displaystyle\sum_{i=0}^{n\rightarrow\infty} \left(\frac{x}{x+1}\right)^i$

If the x is the same in both the infinity and the infinitesimal their product will converge to the finite number x(x+1) as n increases without bound. If the x in the infinity is smaller than the x in the infinitesimal their product in the limit will be an infinity. If the x in the infinity is larger than the x in the infinitesimal their product in the limit will be an infinitesimal.

Division of infinity by infinity as defined by these divergent geometric series will result in the limit (1) an infinity if the numerator has a smaller x, (2) an infinitesimal if the numerator has a larger x, (3) the finite value 1 if numerator and denominator have the same x.

Nothing happens, forget infinitesimals and infinity as numbers. Math was built with limits in the last centuries so we don't need to try to define these things.

• Non-standard analysis is a thing.. – Cameron Williams Mar 28 '15 at 4:33
• @CameronWilliams yeah but he's probaly learning calculus and it would be really helpful if he stops to think about infinitesimals by now. – Lucas Zanella Mar 28 '15 at 4:38
• That's fair. Though I don't think OP will see this given that the account has been abandoned. – Cameron Williams Mar 28 '15 at 4:40
• Math was also built with infinitesimal. Leibniz and Newton more or less discovered calculus - but didn't use limits. Euler proved astonishing (and correct) things with infinitesimals and infinities. I think you have a valid answer to the question (since $dx\times \infty$ doesn't make a real lot of sense intuitively, whereas valid uses of infintesimals do), but it needs to be fleshed out more than a categorical "Don't work with infinities and infintesimals" – Milo Brandt Mar 28 '15 at 4:40
• @Meelo While it can be done, clarifying what really does and doesn't work is typically harder than just defining limits. For instance, "Euler-style" arguments like Euler's "proof" that $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$ (based on "factorization" of an appropriately chosen "infinite polynomial") can easily be used to "prove" false statements if one is not careful about convergence issues. In many cases these techniques are better used as tools for calculation: when everything is "nice", they get the correct answer, but you must check that the situation is "nice" to use them. – Ian Mar 28 '15 at 4:52

## protected by user147263 Mar 28 '15 at 4:44

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