infinity times infinitesimal - what happens?

Question

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?

Answer 1

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.

Answer 2

Normally, multiplication of a real number with infinity is undefined, multiplication is only defined between two real numbers. Also, dx is not a real number. So, basically, you are trying to multiply two entities for which multiplication is not defined in the sense that they are defined for real numbers.

However, you can look at the definition of integral. Integral of plain dx

$\int_{a}^{b}dx$ = b-a

means summation of infinite number of infinitesimals over a finite period.

We can then conclude that sum of infinite number of infinitesimals is any finite number. This also suggests an infinitesimal is not a real number because

$\lim_{n\to \infty} (\sum_{i=0}^n \epsilon) = \infty$ where $\epsilon$ is any real number, no matter how small it is.

However, this is any real number in case of infinitesimals.

$\lim_{n\to \infty} (\sum_{i=0}^n dx) = c$ where c is any real number.

whereas

$\lim_{n\to \infty} (\sum_{i=0}^n 0) = 0$

Answer 3

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

Answer 4

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.

Answer 5

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.