Al-Khwarizmi’s Mathematical Doctrines in Ibn-Ezra’s Biblical Commentary r 181numbers. One is to consider nine different numbers, while the other is
to think of ten numbers. Yet 10 is not really a new number requiring a
new symbol, as it is in fact like 1. Ten is an inclusive name of the one-digit
numbers and the first member of a new series of numbers resembling
those of the basic series.
It is thus clear that Ibn-Ezra doesn’t have here zero as a number in its
own right with its own symbol. He does have it in other writings, using
the same symbol we use today, borrowed from Indian mathematicians.
He calls it Small Wheel (Galgal Katan גלגל קטן), and the idea presum-
ably is that the emptiness of this symbol represents the nothingness of
zero.
What Ibn-Ezra has here, without spelling it out, is the notion of zero
as an empty space within a number of two digits or more. Moreover, be-
tween the lines we read here three of the most fundamental doctrines of
arithmetic as we know it today, though they date from antiquity and from
the Middle Ages. These are
- The Decimal System
- The Notion of Zero
- The Position Principle
Ibn-Ezra states here that ten is like one and twenty is like two ones and
so on. Similarly, one hundred is against one and two hundred is against
two ones and so on; and this too is the case for thousands and myriads
and beyond. The head of each system of numbers is like 1 (and the second
member of each system is like 2, etc.). And the “sages of the number”
(mathematicians) tell us that in this way all numbers are combinations of
units and multiples of 10.
The decimal system is presented here together with the maxim (stem-
ming from the position principle) that no new symbols are needed for
numbers from 10 and on. All one has to do is look at the head of each
number (or system of numbers) and at the way the number is built from
units and multiples of 10. Thus when we see, for example, the number
3333we know that from left to right the first digit represents three thousand,
the second three hundred, the third three tens, and the fourth three
units. The same symbol (3) appears here four times, but each time it is
read differently according to its position. This way of writing numbers is