Infinitely large numbers
In a previous post I discussed limits of sequences and I wrote that “plugging in ∞” is not the proper way to evaluate limits. If you read the old masters like Euler and the many Bernoullis then you will see that they regularly use infinitely large and infinitely small numbers. These will be substituted in well-known functions without much ado and after some manipulations these old masters achieved results that still stand firmly.
One of my favorite formulas says that the sum of all numbers of the form 1/n2 (n=1, 2, 3, …) is equal to π2/6. This formula was found by Euler by taking an `infinitely small number z' and an `infinitely large natural number n' such that x=nz is a `finite' number. With these Euler manipulated as is these were normal numbers, except that on his way to the result he occasionally used the equality z=sin(z). U can read this argument in his Introductio in Analysin Infinitorum (Part 1 and Part 2).
The problem with these arguments is that those `infinitely small numbers' cannot exist: their absolute value is less that every positive real number yet the numbers are not equal to zero. Similar comments can be made about the `infinitely large numbers'. An other problem was that lesser mortals would use the same methods and would obtain false results.
In the 19th century these numbers were banished from Analysis and the definitions of limits, continuity and other notions were formulated in terms of real numbers only; the famous ε-δ-definitions for example.
Around 1960 Abraham Robinson found a way to give the old masters' methods a firm logical foundation; that way is now called Non-standard Analysis. The idea is to add, in a well-structured manner, new `numbers' to the set R of real numbers; among these new numbers are also a large cloud of `infinitely large numbers' and around 0 a cluster of `infinitely small numbers'. We can work with these new and old numbers as before and Euler's argument can be justified completely in this context. A word of caution is needed: if you want to handle these numbers with any chance of success you will need to learn some Mathematical Logic and Model Theory. But once you clear that hurdle it is great fun to derive many results from Analysis in the old Eulerian way.
