# [FOM] Infinitesimal calculus

Rob Arthan rda at lemma-one.com
Wed May 27 17:40:22 EDT 2009

```The classical real numbers can be developed very nicely using decimal
expansions (or base N for any N) as shown in:

@article{behrend56,
author="F.A. Behrend",
title="{A Contribution to the Theory of Magnitudes and the
Foundations of Analysis}",
journal="Mathematische Zeitschrift",
volume=63,
pages="345-362",
year=1956}

The ordering relation is easy to define and easy to prove complete;
the definition of addition requires a little bit of thought about how
"carrying" works if you have to work from left to right. This gives a
complete ordered abelian group - the additive group of real numbers.
Multiplication is then be defined by analysing the group of order-
preserving homomorphism of the additive group.

Regards,

Rob.

On 27 May 2009, at 01:24, joeshipman at aol.com wrote:

> I disagree. Real numbers can be explained to high school students
> perfectly soundly in terms of successive rational approximations
> (decimal expansions, binary expansions, continued fractions, etc.) to
> points on a geometric line. When this is done, they have a consistent
> intuition for "what real numbers are", an intuition which makes a
> proof
> of the least upper bound property quite easy to understand. There is
> no
> philosophical difficulty at this level, unless you try to get across
> the point that physical space might not have a structure that is
> precisely captured by this definition.
...
```