Infinitesimals, or fractions in base `\Omega`

Infinitesimals---quantities smaller that positive "real" numbers but bigger than zero---have a long and productive history.  They require careful handling, but that is true of most mathematics.  Because of such mistakes, late nineteenth century mathematicians banished them altogether, replacing them with limits.  In the middle of the twentieth century, Abraham Robinson found a rigorous way of handling infinitesimals.  Others followed, with different, but also rigorous, ways of using them.

In physics and engineering, on the other hand, infinitesimals never really went away.  They are how calculus is made visible to the mind's eye. Of course if one asks a mathematician these days why they don't use infinitesimals in their calculus courses, they'll say that Robinson showed it was equivalent to the limit way of looking at calculus, as so isn't necessary...

If `\Omega` (uppercase omega) is a scalable infinity, or unit of infinityness, then its reciprocal, `\delta` (lowercase delta), is a scalable infinitesimal, or unit of infinitesimality.

It is usual for mathematicians to consider a number line to be a solid thing, with a real number occupying every point.  It is not the only model, however.  It is possible to think to think of there being space of infinitesimal size around each real number.  Consider the following quantity :

`\qquad\qquad\qquad 3  +  \delta`

This is a `3` with an infinitesimal tweak.  It is infinitesimally larger than `3`, but smaller than any real number greater than `3`, however close the latter number may be to `3`.  The quantity

`\qquad\qquad\qquad 3  + 4  \delta`

is infinitesimally larger, but it is still less than any real number greater than `3`.

Quantities like this add, subtract, and multiply as you would expect :

`\qquad\qquad\qquad (3  +  \delta)  +  (3  + 4  \delta)  =  6  +  5  \delta,`

`\qquad\qquad\qquad (3  +  \delta)  -  (3  + 4  \delta)  =  0  -  3  \delta,`

`\qquad\qquad\qquad (3  +  \delta)  (3  + 4  \delta)  =  9   +  15 \delta  +  4 \delta^2.`

The `\delta^2` is a second-order infinitesimal.  It is as teensy compared to `\delta` as `\delta` is compared to `1`.

Such quantities can be expressed in base `\Omega`.  So, for instance, 

`\qquad\qquad\qquad 3  +  \delta  =  3;_{\Omega}1,`

`\qquad\qquad\qquad 3  + 4  \delta  =  3;_{\Omega}4.`

For adding, subtracting, and multiplying, we can write :

`\qquad\qquad\qquad 3;_{\Omega}1  +  3;_{\Omega}4  =  6;_{\Omega}5,`

`\qquad\qquad\qquad 3;_{\Omega}1  -  3;_{\Omega}4  =  0;_{\Omega}\bar{3},`

`\qquad\qquad\qquad (3;_{\Omega}1)  (3;_{\Omega}4)  =  9;_{\Omega}15_{\Omega}4.`

The reader should compare these equations with decimal arithmetic, i.e.

`\qquad\qquad\qquad 3.1  +  3.4  =  6.5,`

`\qquad\qquad\qquad 3.1  -  3.4  =  -0.3,`

`\qquad\qquad\qquad (3.1)  (3.4)  =  9.^{1}54  =  10.54.`

If we stop the last calculation just before carrying, we have  `9.^{1}54`, i.e. a `9` in the ones column, a `15` in the tenths column, and a `4` in the hundredths column, which looks more the corresponding base `\Omega`.  Base `\Omega` arithmetic is much like ordinary decimal arithmetic---easier, if anything, because there is no need for carrying.

In this respect, it is just like working in base `x`.

The difference between calculations in bases `x` and `\Omega` lies mostly in usage.  Base `x` calculations usually have nothing right of the face value superdigit (no "fractional" part), while base `\Omega` calculations usually have nothing left of the face value superdigit (only a face value digit and a "fractional" part).

This also suits them better to each other :  the marriage of polynomials (base `x`) with infinitesimally tweaked quantities (base `\Omega`) is the wellspring of calculus, as we shall see.

The pitiable arithmetic of one, two, many...

Occasionally one will find accounts of hunter-gatherers whose numbering system goes as one, two, many.  Such accounts usually have just a whiff of superiority about them.

Nevertheless, one, two, many, provides a small but workable arithmetic, especially if augmented with nothing and don't know, e.g. :

`\qquad\qquad\qquad 1 + 1  =  2,`

`\qquad\qquad\qquad 1 + 2  =  "many,"`

`\qquad\qquad\qquad 1 + "many"  =  "many,"`

`\qquad\qquad\qquad 2 + 1  =  "many,"`

`\qquad\qquad\qquad 2 + 2  =  "many,"`

`\qquad\qquad\qquad 2 + "many"  =  "many,"`

`\qquad\qquad\qquad "many" + "many"  =  "many,"`

`\qquad\qquad\qquad 1 - 1  =  "nothing,"`

`\qquad\qquad\qquad 2 - 1  =  1,`

`\qquad\qquad\qquad 2 - 2  =  "nothing,"`

`\qquad\qquad\qquad "many" - 1  =  "don't know,"`

`\qquad\qquad\qquad "many" - 2  =  "don't know,"`

`\qquad\qquad\qquad "many" - "many"  =  "don't know."`

The funny thing is that we ultranumerate moderns have something that behaves a lot like many.  We call this thing infinity.  So, for instance,

`\qquad\qquad\qquad 1 + oo  =  oo,`

`\qquad\qquad\qquad 2 + oo  =  oo,`

`\qquad\qquad\qquad oo + oo  =  oo,`

`\qquad\qquad\qquad oo - oo  =  "don't know."`

In particular, `oo` doesn't scale :  `n  oo  =  oo.`   nothing and `0` don't scale either, but there the behavior is just what we would expect.  With `oo`, as with many, the behavior seems much less legitimate :  do we really expect any large thing to be the same as twice itself?

Why not just invent a well-behaved, i.e. scalable, infinity, or at least a unit of infinityness?  Let us invent one, and call it `\Omega`.  We require that, for instance, `2 \Omega > \Omega.`  In fact, we treat `\Omega` as a new kind of number that is larger than any real number, but is otherwise as well-behaved as we can get it to be.

It is amusing to play with this quantity.  But for real fun, we need to look at quantities in base `\Omega`.  For that will take us to infinitesimals and calculus.