Notes from Limmud 2006

All for nought? Abraham Ibn Ezra's numerical innovation

Kineret Sittig

Abraham Ibn Ezra (born 1092/3 in Spain, died 1167) probably travelled through the north of Africa, also Italy, France and even spent some time in London. He is well known for his commentary on the Bible.

In his commentary on Exodus 3:15, as part of a long commentary on the name of G-d, he wrote:

Note that the number one is the secret and foundation of all numbers, and that two is the first of the even numbers and three the first of the odd numbers. Now from one point of view there are only nine numbers. However, from another perspective there are ten numbers. If the numbers one to nine are written in a circle and the last number is multiplied by each of the other numbers, we find that the ones are on the left and the tens, which looks like ones, are on the right. Now when we come to the number five which is in the middle, we find the reverse, i.e. the tens become ones and the ones become tens. ... ודע כי האחד סוד כל המספר ויסודו׃ ושנים תחלת מספר הזוגות ושלשה תחלת מספר הנפרדים׃ והנה כל המספרים הם תשעה מדרך אחת והם עשרה מדרך אחרת׃ ואם תכתוב התשעה בעיגול ותכפול הסוף עם כל המספר תמצא האחדים שמאלים והעשרות הדומות לאחדים לפאת ימין׃ ובהגיעך אל חמשה שהוא אמצעי׃ אז יתהפכו המספרים להיות העשרות אחדים והאחדים עשרות׃ ...

This shall (hopefully) be rendered clearer below...

His ספר המספר The Book of Number (not to be confused with the Book of Numbers) is known in many manuscripts mostly from the fourteenth to the sixteenth century. There is an introduction and seven chapters about multiplication, division, addition, subtraction, fractions, ratios and square roots.

In the division chapter—you can read the Hebrew original here— we encounter the following (paragraphing mine):

Since G-d created nine large spheres in the upper world that circle the earth—the lower world—and since, as the author of The Book of Creation says, the ways of wisdom are counting, writing and recounting, counting is performed by means of nine numerals, for nine is the end of every number. These are called the ones, they are in the first position. For ten is like one and twenty is like two—twenty is two tens. It would have been appropriate to call it עשריים—twice ten—as מאתים is called after מאה and אלפים after אלף. Only because of its companions that that come after it, שלשים until תשעים, is it handled in their manner. But שלשים is from the root שלש, etc.

One hundred is similar to one, also to ten; and two hundred is similar to twenty and also to two. And in the same way one thousand and ten thousand, they are the beginning of decades (כללים) for the numbers that follow them, thus one, ten, one hundred and two, twenty, two hundred.

And the proof for this: if one draws a circle and writes the nine numbers around it and multiplies nine by itself—and the meaning is that it will be a square with equal width and length—one will see, and it is thus. Nine squared equals eighty-one and and behold, the one is on the left hand side of the nine that is the beginning of the ones and eight that stands for eighty in the decade is on the right.

And if one multiplies nine by eight the product will be seventy-two and the two is on its left and the seven that stands for seventy is on its right.

And if one multiplies nine by seven the product will be sixty three and behold the three is on its left and the six that stands for sixty is on its right.

And if one multiplies nine by seven the product will be sixty three and behold the three is on its left and the six that stands for sixty is on its right.

And if one multiplies nine by six the product will be fifty-four and behold the four is on its left and the five that stands for fifty is on its right.

And because the number five is the middle number of the nine numbers it is called the round number for it circles around itself: its square has the number five in it.

And when one multiplies nine by five the matter is turned around on the circle since the ones will be on the right and the tens will be on the left. For the product is forty-five, and now the five is on the right and the decades on its left that is four instead of forty.

And when one multiplies nine by four the product will be thirty six and now three stands for thirty.

And when one multiplies nine by three the product will be twenty seven and the two stands for twenty.

And if one multiplies nine by two the product will be eighteen and the one stands for ten.

Therefore the proof of a number that it is multiplied by itself or by a different numbers that it is multiplied by itself, is the number nine.

This is true for every divisor of n–1 in an n-based system. It's not a proof, but it's not nonsense. The number when squared can be split into two digits, which when added, add up to the original number. 9² = 81; 8+1 = 9.

Ibn Ezra continues:

Therefore the wise men of India made all their numbers by nine and made shapes for the nine numbers and they are:

but I write instead: א ב ג ד ה ו ז ח ט.

This is because he is more familiar with these numerals.

If we have a number with ones before the decades that are the tens, write first the number of ones and then the number of decades.

I.e. אח ≠ חא. This is the place value system we are used to; but at the time it was completely new in Europe. The Babylonians were using this nearly five thousand years ago, and the Maya and Chinese had also been using this for hundred years ago, but in Europe it had not been known, and even after Ibn Ezra was not very well known.

(Even in the eighteenth century, the place-value system with a notification for empty rows was not universal. Consider, in Britain, the office of Chancellor of the Exchequer. The Exchequer is a large chequered piece of cloth on which sticks or pebbles can be moved around to do calculations—the same system as an abacus. It's a representation, but it's a physical representation, something you can pick up and move around! And there's no representation for an empty row.)

So far so good, but what if there is a large number and no tens, only hundreds and thousands. You could leave a space, and this has been done, but it causes problems if you need to leave multiple spaces.

And if there is no number in the ones but there is on the second level, that is, the tens, put the shape of a wheel, O, in the beginning to show that there is no number on the first level and write the number of tens after it. And if the decades has hundreds and tens, write a wheel in the first position and then the number of tens in the second and the numbers of hundreds in the third, and write the number of thousands, if it is there, in the fourth position and the number of ten thousands in the fifth and the number of hundreds of thousands in the sixth position. For the one and the ten and the one hundred become one thousand in the fourth position, a thousand thousands in the seventh position and in the tenth position a thousand thousand thousands and so on ad infinitum. And if there are ones and hundred but no tens, write the number of ones first and then a wheel and the number of hundred in the first position. Do it this way and keep the position of the wheel according to the position of the number by putting one wheel in the beginning or two wheels, as needed in the beginning or in the middle. And this is the wheel: O, and its meaning is "like whirling tumbleweed, like chaff before the wind" כגלגל כקש לפני־רוח (Psalms 83:14) and it is just to keep the position and in foreign language (בלשון לעז, i.e. Arabic) it is called sifra.

He's not using the zero as a real number, just as a sign to indicate nothing. For him, the sum 2 – 0 would not make sense.

[p.12 in the PDF:] Note that every number is a sum of ones and only the one does not undergo change or increase or division, and it is the reason for all increase and change and division, and only the one is original and every number is renewed through it. And it achieves on one side what all (the other) numbers achieve on their two sides, for two is for the number three its one neighbour that comes before it and four is its other neighbour that comes after it, and their sum is six and that is twice three, and it is like this for every number. The one, however, does not have a preceding neighbour and the neighbour after it is two, and two equals twice one. Now I will talk about every number that has whole ones without fractions...

Does this system help us do our sums? Dividing טשפ״א by רפ״א (i.e. 9381 by 281) in Hebrew numerals (as by Roman numerals) is a lot of work, repeated subtraction. The way to do it is to use an abacus.

[p.13 in the PDF:] Now I will give you a rule how you divide every number, whether it has one or two or many numerals. Write them in a row every one according to its position. And then write the number you want to divide it by in a different row, whether it has one or many numerals, every one according to its position opposite every number according to its position in the upper row, and leave some space between the upper row and the lower row, so that you will be able to write a middle row between them, whether the result will be a single numeral or many numerals, every one you will put down according to its position. The number that is divided by should be smaller than the number it is divided from. This holds for a division of whole numbers but not in case of fractions, which I will explain [later], with G-d's help.

ב

ג 0

0 ה א

ג ו 0 ה

ט ג ח א

ג א

ב ט ו

[p.16 in the PDF, beginning after note 25:] Another example, dividend אחגט, divisor וטב. We want to divide nine by two. Now we cannot give four, since there only remains one and when you set that back one place it yields thirteen together with the three and four times nine is thirty six. Therefore we give three, remain three. We set them all back to the second, that is three, yields thus thirty-three. We give three to the nine, is twenty-seven, remain thereof six over the three. From these we take one and leave five, we set it back over the eight which is the third last of the upper row and with the eight in the second row this is eighteen. We give them all to the six which is the third in the lower row and we write over the five a wheel because nothing remains over the five. We divide again for it has not come out yet. We take from the five that we put over the three that is the second in the upper row two, that is (the quotient is) one. We write this one over the six behind the three that we pout over the nine in the first division and it is the third in the lower row and already it has come out. We take from the three that is over the three one and there remain two. We put the one over the wheel and they are ten. We give to the nine that is on the lower row nine, remains over the wheel one. This we set back over the one whch is the fourth and the first of the upper row. This makes eleven, we give to the six six, remain five over the one. There remain over the upper row five and wheel and two, that is two hundred and five. This cannot be divided further since the three that are on the lower row are two hundred and ninety six. And what was taken for every one (i.e. the quotient) is thirty-one.

Sadly the people who were copying this had no idea what they were copying, and all the manuscripts that have come down to us are riddled with mistakes! Ms Sittig tried to go through this example at Limmud, and the maths didn't work out! The industrious might like to try it with the original Hebrew, to see if the printed version has less mistakes than the English!

What happened to this idea? The text was widely used, and we see references in other texts saying this is what you have to use when you teach your students. [An audience member cited a book by John Sacrobosco (or John Hollywood) (ca. 1195–1256) which referred not to the Hindu-Arabic numerals but the Hindu-Jewish numerals. Apparently this was because Jewish traders moving back and forth played a big role in bringing the numbers from India.] However, if you look in the history books, they will say that the number zero was introduced to Europe by Leonardo of Pisa, a.k.a. Fibonacci.

Why is Ibn Ezra not credited or known about? Partly because Jewish books were not generally accepted by Christians.

In summary, Ibn Ezra's contributions were: