Whether or not infidecimal numbers existed

[JonF]

I’m reading this book on math history, and its pretty interesting.

There is this one part where the book goes into two mathematicians (sorry forgot which two) that argued about whether or not infidecimal numbers existed. The book said whether or not they do exist wasn’t proved till much later in history, but it never told if they did or didn’t exist! So do infidecimal numbers exist?

If I remember correctly the definition of a infidecimal number is:

A number that is not Zero.
A number that is so small, that when it is multiplied by any finite number their product will never be a number greater then one.

 

[Integral]

Perhaps you mean infinitesimal?

This is not a feature of the Real Number system.

 

[Hurkyl]

What do you mean by "exist"?

As Integral mentions, there are no infinitessimals in any of the standard number systems (such as integers, rationals, reals)

Using abstract algebra, it's a simple exercise to make a new ordered field (a "number system" with +, -, *, /, and <) by adding a number to the real numbers which is decreed to be infinitessimal.

With some tough mathematics, one can arrive at non-standard analysis which can often "enlarge" things to include transfinite and infinitessimal numbers in a practical way. (But in some sense, this is no more powerful than doing things the ordinary way)

 

[quartodeciman]

In the seventeenth century, Leibnitz treated differentials as if they were infinitesimal and represented bonafide numbers. But he offered no proof of their existence.

bio - Gottfried Wilhelm von Leibnitz --->
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leibniz.html

By the middle of the nineteenth century the idea of infinitesimals was effectively squelched by mathematicians as unreal and unnecessary.

By the second half of the twentieth century, Abbie Robinson presented a rigorous theory of extended numbers containing regular real numbers, infinitesmals and transfinite numbers too. If x is an infinitesimal number not zero, then 1/x must be a transfinite number (larger than any regular real numbers).

bio - Abraham Robinson --->
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Robinson.html

So far, Robinson's theory has been used to find out general facts in the extended number system and translate those facts back into ordinary standard real numbers. A complete theory that deals with this extension as a unique and definite system of its own has eluded researchers.

 

[Kalimaa23]

Hm, funny. At my university the mathematicians scoff at the physicists for using the term infinitissimal at all, because it is "archaic". True enough, it is out of style in the way calculus is taught these days, but it is a handy conceptual tool in physics (so that if you have a dV you think of it is as an itsibitsi piece of volume...)

 

[quartodeciman]

The Robinson theory is usually called Non-Standard Analysis. Here is a summary of the essentials of the subject, including a list of mathematicians and other specialists interested in it.

Phillip Apps: What is NonStandard Analysis? --->
http://members.tripod.com/PhilipApps/nonstandard.html

Quart