Introduction

    In this essay I will write about a mathematician, a rabbi, a philosopher, a poet, a linguist and an astronomer. No, this is not the start of a bad joke. And this essay is not about a big group of people, but rather one very intriguing figure, Abraham Ibn Ezra, who excelled in all the above listed professions. And while this essay will focus mostly on his works in the field of mathematics, there is a connection between all the above fields of study, and this mixture which Ibn Ezra had in him affected how he viewed the field of mathematics.

    I have divided this essay into three sections. I will first cover the biography of Ibn Ezra. Piecing together his biography was not an easy task because of numerous contradicting sources. After shedding light on the question of “who was Ibn Ezra” I will survey the mathematical background of the period he lived in, and why his works were important to advance the mathematics in Europe. Finally, I will analyze two books which he wrote. While Ibn Ezra has written dozens of books on many subjects, including poetry, linguistics, grammar and astrology, I will focus on two of them that are most strongly related to mathematics: Sefer Haehad (Book of the Unit) and Sefer Hamispar (Book of the Number). Sefer Haehad deals with the properties and peculiarities of the Indian numerals 0 through 9, while Sefer Hamispar introduces the decimal system and shows how to perform complex calculations.

Biography

    There are many sources relating to the life of Ibn Ezra, many of those sources contradict each other on various account. The main source I used to reconstruct the life of Ibn Ezra, the source which I believed is most accurate is a book called Abraham Ibn Ezra- Kobetz Maamarim Al Yetzirotav and Tolodotav (A compilation of essays on his works and life), a compilation of various biographical essays by Hebrew scholars. Most notable is Yehuda Leib Fleischer, a Hebrew scholar who studied the life of Ibn Ezra and other Jewish Rabbi’s, and edited many of Ibn Ezra’s works. I believe Fleisher’s expertise and research on the life of Ibn Ezra provides the most accurate information, and while I do complement the information with other sources, I focus on Fleischer’s works to filter out any white noise related to Ibn Ezra’s history.

    Abraham Ibn Ezra was born in Tudela, Spain. I typed “Ibn Ezra” in the Google Search engine, and was surprised to see most of the websites list his year of birth as 1089, which after my research I believe is wrong. Fleischer cross referenced and analyzed reports of many Jewish scholars who lived at the same period Ibn Ezra lived in, and his conclusion is that all evidence points to the fact that Ibn Ezra was not born in 1089 but rather in 1092. 1 Fleischer mentions that a few scholars mistakenly state that Ibn Ezra died at the age of 78 instead of 75, and I believe that is the source of the inaccuracy in this matter that has spread across the Internet. I can speculate that one site took those reports, made the math, and reached the conclusion that Ibn Ezra was born in 1089 instead of 1092- and other sites kept feeding on this misinformation spread around the internet like the plague.

     I could not locate information about his family, only piece of information I found is that according to one source members of his family held important official posts in Andalucía2. Fleischer and many others like to divide his life into two periods: The first part consists of his life in Spain, from birth and up until 1140. The second period, from 1140 to his death is his life in exile.

    Not much is known about his life up until 1140. As Fleischer puts it, “The first period is covered almost entirely by the fog of the past, with no sources to lean on…”(3) Ibn Ezra lived in Spain and in that period he wrote a big amount of poems, some of religious nature and other secular. While most Jews in Spain wrote in Arabic, Ibn Ezra wrote in Hebrew, though his poetry was influenced by Arabic verse. He also engaged in studying various fields such as astronomy, astrology, grammar and mathematics. He married, and had a son which converted to Islam and died young, which caused Ibn Ezra a lot of grief.

    From 1140 and up until his death he became a wandered, a nomad, living in exile from his home town in Spain, traveling and living amongst many Jewish communities mostly in Europe. There is a debate as to why he left Spain. Gerch, a rabbi who was also born in Tudela, claims that Ibn Ezra left Spain because Toledo, where he was living, was poor and war-stricken. 4 However Fleischer dismisses poverty as the reason, for even if Ibn Ezra could not support himself in Tudela there were other cities in Spain that were not as poor and Ibn Ezra could have just gone to any of them. Another Jewish Scholar, Cahanah, claims that Ibn Ezra left Spain because of a death of a good friend of his, Rabbi Yosef Ben Amdan. However Fleischer states that he could not find any evidence that Ibn Ezra’s friend indeed died in that year. Fleischer finds evidence in Ibn Ezra’s works and in the works of other scholars that suggest that Ibn Ezra left Spain because of heavy persecution of the Jews by the Muslims, mainly by the Al-Mohaden government, and concludes that this is the reason Ibn Ezra left Spain.(5)

    After leaving Spain Ibn Ezra traveled between Jewish communities around the world and wrote most of his works. Constructing the exact path which Ibn Ezra took was not easy task- different sources put Ibn Ezra in different places at different periods. Again I used the Hebrew essays to try and construct his path and plotted the route on the map. In Spain, he lived in 4 cities: Tudela when he was born, then Toledo, the Cordova and lastly in Lucena. From then he traveled straight to Italy on 1140. From 1140- 1148 Ibn Ezra stayed in various cities around Italy. First at Rome, then he moved to Luca in the Lombardia District near Pisa. He then traveled to Mantoba, and in 1142 he arrived in Verona.6 In 1148 Ibn Ezra left Italy and arrived in Beziere on the Southwest coast of France; after that, Fleischer points out that Ibn Ezra mentions in his works a place called “Rhodus” and it is not clear where it was. One of the possibilities is that Rhodus is a city on the Rhodus Island East of the Mediterranean Ocean. Another possibility is that Ibn Ezra meant the city “Rodez” in the Aveyron district in the South of France. Another possibility is that Ibn Ezra meant the city “Rouen” at the North of France on the Seine River. However Fleischer believes that “Rhodus” actually refers to “Dreux” in North France which was a famous Jewish center. 7 It was there where Ibn Ezra wrote “The book of the Unit” and “The book of the number” which I will discuss later. From there Ibn Ezra Traveled to London, England. The date and place of Ibn Ezra’s death is again not clear, but Fleischer states that Ibn Ezra died in 1164 at the age of 75. Four places are listed as the place of his death: Calhorra in Spain, Rome in Italy, Cabul in Israel, and London in England. By analyzing and cross referencing reports of various Jewish scholars at the time, Fleischer comes to the conclusion that Ibn Ezra must have died in London, England.

Historical-Background

    Struik (1948) sets the 12th century as the period when European cities began to establish commercial relations with the orient. Europe was yet to flourish in the field of mathematics, and in that time Islamic sciences were more advanced.  In 1085 when Toledo was taken from the Moors by the Christians, western students flooded the city to learn the sciences of the Arabs. Those students often used the services of Jewish interpreters to converse and to translate Arabic mathematical manuscripts into Latin. 8 Recall from the biography, Ibn Ezra arrives would later arrive at Toledo before setting on his Journey to Italy. The Arab mathematician Al-Khwarizmi (790-840, gave “Algorithm” and”Algebra” their names) wrote the book Al'Khwarizmi on the Hindu Art of Reckoning in which he mentions the Indian symbols 0 through 9. Ibn Ezra was inspired by Al-Khwarizmi works and tried to introduce the Indian symbols, including the number zero- which he referred to as “Galgal” or “Wheel”- as well as the decimal system in his books Sefer Hamispar (Book of the Number) and Sefer HaEhad (Book of the One). But despite Ibn Ezra’s work, he was ahead of his time in Europe, and his ideas would not become mainstream until a few centuries later.9 One of the reasons being that the Europeans were attached to the Roman numeral System, which did not have a zero. Ibn Ezra preceded Fibonacci, who also tried to push the Decimal system and the Indian symbols in his book Liber Abaci.

Sefer HaEhad and Sefer HaMispar

    As mentioned in the introduction, I will focus on two of his books: Sefer Ha-Ehad (Book of the Unit or Book of the one) which deals with the integers 1 through 9 and Sefer Ha-Mispar (“Book of the Number”) which deals with the decimal system and arithmetic. While some of his explanations might seem trivial to the modern reader, remember that the decimal system and Indian numerals were not so common in Europe in that era.

    Sefer HaEhad- Ibn Ezra starts the book by introducing some properties of the number 1. He states that 1 times itself is 1, and that any number multiplied by 1 remains the same number. Basically those are some identity properties. He further states that the root of or 1 to the power of any number is always 1. These properties makes him see the number 1 as a special number, and when he gets to the number 2 he refers to the number 2 as “The source of all numbers” as if number 1 (and 0) are separate from the rest of the numbers.10

    In the number two he states the property that if you add 2 and 2 or multiply 2 and 2 you get the same result.11

    Ibn Ezra then proceeds to do some geometry when introducing the numbers 3 and 4. through the shapes of triangles and squares. Like Euclid, he describes a triangle on a circle, similar to the proofs Euclid used to show that a triangle is equilateral. He states that no one side of a triangle can be bigger than the other 2 sides combined. Ibn Ezra notes a cyclic property of the numbers 5 and 6 pointing out that they remain as the final digit no matter what power they are raised to, for example 5 x 5 =25, 5 x 5 x 5=125, 6 x 6 = 36 and so on.12

    Ibn Ezra then surveys properties of odd and even numbers. He states that a number multiplied by an even number would always be even, and odd number multiplied by an odd number would always be odd. He also shows how numbers can be factored and canceled out when dividing,13 He brings the example of:
2520/120 = (7 X 6 X 5 X 4 X 3)/(1 X 2 X 3 X 4 X 5)
=(7 X 6) / (1 X 2) = 21

Sefer Hamispar

    In Sefer Hamispar (Book of the Number) Ibn Ezra presents us with the decimal system and arithmetic calculations. He divides the book into two sections which he labels using the Hebrew Alphabet letters, Alef through Zain. The sections are:
Alef: “Kefel” (Multiplication)
Beit: “Hiluk” (Division)
Gimmel: “Hibur” (Addition)
Daled: “Hisur” (Subtraction)
Heigh: “Shevarim” (Fractions)
Vav: “Arachim” (Ratios)
Zain: “Shorashim Merubaim” (Square Roots)

    It is interesting to note that despite having introduced the Indian numerals, Ibn Ezra still uses the Hebrew letters for his calculations throughout the book. For example, Alef, the first letter of the alphabet, is “1” and Beit is “2” and so on. This is because Ibn Ezra was interested in the connection between math, Hebrew linguistics, and religion.

    In the first section, about multiplication, Ibn Ezra notes the interesting property of the number 9 which if you multiply it by another number and add the digits, you would always end up with 9. For example, 9 X 9=81 and 8 +1 = 9,  9 X 11 = 99, 9 + 9 = 18, 1 + 8 = 9. He also teaches how to carry out multiplication of large numbers under the decimal system, the techniques we use nowadays of multiplying and carrying numbers over. He gives a step by step walkthrough with examples on how to carry on multiplication.14

    Similarly, in the next section, on division, he walks as through carrying arithmetic division with multiple digit numbers.15

    In the next section, about addition, he first introduces a simple property that if you take half the number and add it twice, you get back to the original number.  For example, 10 divided by 2 is 5, and 5 + 5 = 10. He also shows that if you take the square root of a number and keep adding it to itself you again get back to the original number. For example, the square root of 9 is 3, and 3 + 3 + 3= 9. Again he walks the reader through how to carry out addition of large numbers by carrying digits over.16

    In the section about subtraction, Ibn Ezra does not acknowledge negative numbers (those did not go into common use until the renaissance). He implicitly says that if you want to subtract 2 numbers, you look at the leftmost digit of each number and put the number that has the higher leftmost digit on top, and the number with the smaller leftmost digit on the bottom, that way you subtract the smaller number from the larger number. He walks us through an example of how to subtract 2379 from 5432, including the concept of  “borrowing” numbers.17

    In the sections about fractions, Ibn Ezra shows the reader how to work with fractions. For example he illustrates that if you want to multiply ¾ times ¾ you first multiply the top part, 3 X 3 = 9, and then the bottom part 4 X 4 = 16 and combine them to get 9/16. 18

    In the last section of his book, concerning square roots, he gives an example of an interesting method to approximate square roots. Here is the algorithm he presents through his example:
1) Let's say you want to compute the square root of a large number i.e.  √600,000
2) Locate the nearest number that you can easily compute the square root of. In this case, 640,000 for which the square root is 800, since the Square root of 64 is 8 and the zeroes are place holders.
3)  √600000 = √ (640,000 - 40000)=800- (40,000/(80*2))=775
4) 775 X 775= 600625, 774 X 774 = 599076 and so the √600000 must lie between those 2 numbers.19
I observed while reading this book that unlike Euclid, who emphasizes his work on proofs, Ibn Ezra does not use proofs much. Instead Ezra provides us with detailed examples, "walkthroughs", that are easy to follow. The examples usually build upon knowledge learned in previous chapters.

Summary

    In the first part of the essay, I tried to piece together the life of Ibn Ezra. Piecing together the puzzle was not an easy task, for there existed numerous contradicting sources. Different sources contradicted others in terms of when Ibn Ezra was born, what routes he took in his travels, and when and where did he die. My framework for building his biography was mostly done by reading the works of the Jewish Scholar, Yehuda Leib Fleischer, who studied Ibn Ezra and edited many of his books. Fleischer, through cross referencing and analyzing testimonies of other Jewish Scholars at the time and period of Ibn Ezra's life, and through comparing it with events and names of places mentioned in Ibn Ezra's works, allowed me to build what I believe is the most accurate picture of who Ibn Ezra was and the path he took.

    I then gave a survey of the mathematical background at that time. European cities were beginning to establish commercial relations with the orient and as a result Arabic mathematical influences began to flow into Europe. Ibn Ezra was strongly influenced by the Muslim mathematician Al-Khwarizmi. Ibn Ezra Sought to push the use of the Indian Symbols 0 through 9 and the decimal system to Europe. And while Ibn Ezra was a bit ahead of his time and it would be a while before those systems would become mainstream, he did help spread them and bring them to the attention of European Mathematicians.

    Finally, I analyzed two of his books: Sefer HaEhad and Sefer HaMispar. In Sefer HaEhad Ibn Ezra introduced the Indian Symbols to Europe. And what better way to introduce it then to show the various properties and peculiarities this system had! In Sefer HaMispar Ibn Ezra described the decimal system and showed the notation of writing the place values from left to right. He then showed how to work with the decimal system to perform complex calculations.

    One main observation I made is that rather than using long proofs, as Euclid did, Ibn Ezra presented us with a plethora of examples, and walked the reader through, step by step, on how to perform his calculations and algorithms, and to prove that they are right. This would allow a broader audience to understand his works and help make the decimal system more common. Since one does not have to follow long proofs, some of which are rather complicated. Instead the reader can follow the examples step by step, as I did while reading his books.

    Ibn Ezra was a most interesting figure, and I enjoyed researching his life, the mathematical background he lived in, and analyzing his compelling works, all of which allowed me to unveil many facets of Hebrew mathematics.

Bibliography

1.Epstein, Meira, Rabbi Avraham Ibn Ezra,  http://www.bear-star.com/article-ibnezra-lifeandwork.htm.
2.McCann, David, The Life & Work of Abraham Ibn Ezra, Skyscript,  http://www.skyscript.co.uk/ezra.html .
3.O'Connor, J. and Robertson, E. F., Abraham ben Meir Ibn Ezra, The MacTutor History of  Mathematics archive, http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Ezra.html .
4.Ibn Ezra, Avraham, Sefer HaEhad, Odessa: Nitscha and Tsaarboim, 1867.
5.Ibn Ezra, Avraham Sefer HaMispar, Jerusalem: Makor, 1969.
6.R' Avraham Ibn Ezr- Kovetz Maamarim Al Toledosav VeYetziraso, Tsion, Tel-Aviv, 1969
7.Struik, Dirk J., A Concise History of Mathematics, New York: Dover Publications, 1948.