# [plt-scheme] Natural numbers

Bill wrote:
>* Natural numbers starting at 1:
*Long list of references.
>* Natural numbers starting at 0:
*Long list of references.
I realize I am late to the party, but I am surprised none of you
mentioned the history of zero. Although it might seem natural
to use 0 as a starting point, when counting, the use of zero
in a postional number system is quite another matter. The problem
of extended the arithmetical operations to 0 (and negative numbers)
took a long time to solve.
In India the first written record is from 876. Brahmagupta (598 - 670)
writes:
"Zero divided by a negative or positive number is either zero or is
expressed as a fraction with zero as numerator and the finite quantity
as denominator. Zero divided by zero is zero."
Two centuries later Mahavira almost get it right:
"... a number multiplied by zero is zero, and a number remains the
same when zero is subtracted from it."
"A number remains unchanged when divided by zero."
From India the concept travels to the islamic and arabic world.
al-Khwarizmi (~790 - ~840) (whom algebra is named after) wrote about
the positional number system with 0,1,2,3,4,5,6,7,8, and 9.
In Europe Fibonacci (1170 - 1250) was one of the first to embrace
the new idea.
It took *centuries* before the new idea was accepted in Europe.
Cardan (1501 - 1576) solved equations of degree 3 and 4 without
using 0. [He did use square roots of negative numbers in his calculations]
The conclusion? Well, there is a reason some don't find zero
a natural concept.
The above is a very short version of:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Zero.html
--
Jens Axel Søgaard