Les Schaffer schaffer at SPAMoptonline.net
Fri Apr 20 20:25:07 MDT 2001

jenyan said:
> x->x*5  this map is bijective so reversible.
> x->x*0   map is not bijective so not reversible.

in line with this, even my suggestion in the first post that would
give a reasonable definition for 1/0 as being a number very big still
leaves a quandary when we go 1 / -1/10000 , 1 / -1/10000000, etc,
gives us two infinities, one positive, one negative.

charles said:
> Who invented "inverse" and "reciprocal?

good question, i'll sniff around some for a history. should be

> Didn't arithmetic come about before the theory explaining it with
> inverses and reciprocals

its important to keep in mind that there are two different algebras
here, numbers and addition, and numbers and multiplication.

yes, people were adding numbers long before dedekind. and people were
multiplying numbers and dividing two whole numbers. but a quick look
through "The universal history of numbers" by Georges Ifrah, i dont
see any mention of concern about _dividing_ numbers by zero or some
small positive number less than 1.

Jim Craven is quite right that zero has been around a long time, but
then again, zero poses no problems under addition. nor under
multiplication -- though as jenyan points out, once you join zero, the
identity element for addition, in with multiplication, you loose any
unique inverse.

to me, the interesting question would be at what point did fractions
appear as potential candidates for divisors, because then we have
division by smaller and smaller numbers. which then brings us to, when
did fractions first appear. and then, even before this, its logically
and historically important to back and look at the counting (whole)
numbers, and the first uses of large numbers and intimations of
infinity. after all, diving by something small is akin to multiplying
by larger and larger numbers.

if you look back through history, early fractions often used a change
of scale scheme. so for example, measurement of angles use
degrees-minutes-seconds scheme, and then rather than use the quirky
'/' symbol, numbers were written as (like)

30 deg. 20' 45" etc.

which is implicitly an addition scheme: 30 of the degrees, plus 20 of
those minutes (of which there are 60 in a degree) and 45 of those
seconds (of which ...). so built in is division by _whole_
numbers. Ifrah has a set of pictorials representations from Babylonian
astronomers which have a notation for this.

i did a quick web search and found this:

An Introduction to Egyptian Mathematics:

and this: Egyptian Fractions:

[Note the connection with Fibonacci in the first URL.] What i would
really like to see but havent found yet is the specific representation
scheme the Egyptians used for their fractions. Ifrah shows pictorially
the numbering scheme from the elamite scribes, and they do fractions
by using specific symbols, rather than a 1/30, eg, which involves two
numbers and an (implicit) operation between them. again though, their
arithmetic is addition and some multiplication. but no division by
less-than-one fractions as far as i can see. i scanned in an example
page from his book and put it here:


which is about 180 kB and a higher quality version:


but beware, its about 530 kB large.

In terms of _division_, for example on page 122-124 Ifrah details
division of whole numbers by the Sumerians, circa 2650 BCE, as
discerned through interpretations of a tablet apparently discussing
the division of grain between a number of people. And on page 285 and
287 Ifrah shows how the Chinese used a numerical checkerboard to
construct division of two whole numbers. he also claims, without
showing the steps in the computation, how to solve a system of three
linear algebraic relations on this checkerboard. this is equivalent to
finding the inverse of a matrix, today's parlance. Time period: Han
dynasty (206 BCE to 220 CE). I need to hunt this down some more to see
if they knew about singular (non-invertible) matrices, a
multidimensional analogue to dividing by zero, though not quite

In line with Jim's remark about the Mayans, Ifrah makes an argument
that the Babylonians had a zero some time between 3-rd and 6-th
century BC.

As for the Mayans, there is much that is amazing. some snippets from

They were, in the first place, astronomers of far greater precision
than their European contemporaries. As C. Gallenkamp (1979) tells is,
the Maya used measured sight-lines, or alignments of buildings that
served the same purpose, to make meticulous records of the movements
of the sun, the moon, and the planet Venus. (They may also have
observed the movements of Mars, Jupiter, and Mercury.) They studied
the solar eclipses in sufficient detail to be able to predict their
recurrence. They were acutely aware that apparently small errors could
lead in time to major discrepancies: the care they took with their
observations allowed them to reduce margins of error to almost
nothing. For example, the Maya calculation of the synodic revolution
of Venus was 584 days, compared to the modern calculation of 583.92.

The Maya also made their own very accurate measurement of the soloar
year, putting it at 365.242 days. The latest computation give us the
figure of 365.242198: so the Maya were actually far nearer the true
figure than the current Western calendar of 365 days (which, with leap
years, gives a true average of 365.2425).

[ snip similar for lunar cycle: today's value: 29.53059 days, Mayan:

Even more fascinating is the Mayas' use of very high numbers for the
measurement of time. On a stela at Quirigua [sorry for lack of proper
punctuation], for instance, there is an inscription that mentions the
last 5 alautun, a period of no less than 300,000,000 years, and gives
the precise start and end of the period according to the ritual
calendar. Why did they count in terms so far beyond any human
experience of life? Perhaps that will always remain a mystery; but it
suggests that the Maya had a concept if not of infinity, then of a
boundless, unending stretch of time.

It is even more puzzling that the Maya measurements were done without
any tools to speak of. They had not discovered glass, so there were no
optical instruments. They had no clockwork, no hour-glasses, no idea
of water-clocks (clepsydras), no means at all of measuring time in
units  less than a day (such as hours, minutes, seconds, etc); nor did
they have any concept of fractions. It is hard to imagine how to
measure time without at least basic measuring devices.

The tool that the Maya used for measuring the true solar day was the
very simple but utterly reliable device called a gnomon. It consists
of a rigid stick or post fixed at the center of a perfectly flat
area. The stick's shadow alters as the day progresses. When the shadow
is at its shortest, then the sun is at its meridian: that is to say,
the sun has reached its highest point above the horizon, and it is
"true noon".

more details on this would be interesting, to see how they did the
computation. doubtless, division of whole numbers is implicated.

see also:


and finally, two comments to finish off this already long post:

1.)  one can play off the style of jenyan's take on division and
reciprocal of zero to see how the construction of the hyperreal number
system can be brought about.

2.) The history of plain old negative numbers is amusing. European
mathematicians started using them only fairly recently. See, for
example, Morris Kline's "Mathematics: The Loss of Certainty", in
Chapter 5.

les schaffer

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