Fibonacci

How to use this course

Most people meet Leonardo of Pisa through a sequence of numbers he did not invent and a nickname he never used. His real achievement was larger. His Liber abaci of 1202 taught Latin Europe to compute with the Hindu-Arabic numerals and with the algebra of the Islamic world, and his Liber quadratorum of 1225 is the most original work of number theory written in Europe between antiquity and Pierre de Fermat.

The biographical record is thin. Almost everything we know comes from three kinds of source: what Leonardo says about himself in the prologue to the Liber abaci and in his dedications, the surviving manuscripts of his books, and one decree of the Commune of Pisa from 1240. The lessons below say which source supports each claim and mark later tradition as such. All dates are CE. Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, arguments, and exercises with folded solutions. Leonardo wrote everything in words; the symbols are ours.

Timeline

Pisa and Bugia

Leonardo was born around 1170, probably in Pisa. No birth record exists; the date is inferred backwards from the 1202 Liber abaci and from his father’s career. Pisa was then one of the great maritime republics of Italy, a rival of Genoa and Venice, with trading colonies across the Mediterranean. Its merchants bought wool, leather, wax and alum in North African ports and needed officials there to represent them.

His father, Guglielmo, belonged to a family known as the Bonacci. In the prologue of the Liber abaci Leonardo tells us that his father was a public scribe or notary at the customs house of Bugia, on the coast of what is now Algeria, acting for the Pisan merchants who traded there. Bugia (modern Béjaïa) was a prosperous port under the Almohad caliphate. Leonardo says his father sent for him while he was a boy, because he wanted him to learn something useful for his future.

The lesson in the customs house

At Bugia, Leonardo writes, he was taught for some days the art of the nine Indian figures, and he came to like it above everything else. In paraphrase, he found the new methods so superior that he considered the older Latin practices, and even the methods of Pythagoras, as errors by comparison. This is the earliest and most reliable fact we have about his education, and it comes from his own pen.

The name

He called himself Leonardus Pisanus, Leonardo of Pisa, and sometimes Leonardus Bigollo. The meaning of “Bigollo” is disputed; suggestions include a traveller and a person of little account. “Fibonacci” is a contraction of filius Bonacci, son of Bonacci. Leonardo never used it. A Pisan notary wrote “Lionardo Fibonacci” in 1506, and the historian Guglielmo Libri made the name standard in 1838.

Mediterranean apprenticeship

The prologue continues: after Bugia, Leonardo travelled on business to Egypt, Syria, Greece, Sicily and Provence, and in each place he studied the calculating methods used there and debated them with local practitioners. He gives no dates. The journeys probably fill the 1190s, and he says he returned to Pisa afterwards; c. 1200 is the usual estimate.

These were the trading routes of a Pisan merchant, and they were also the routes along which Arabic mathematics reached Europe. In Egypt and Syria he would have met calculators who wrote numbers with the Indian digits. Sicily, ruled by Norman and then Hohenstaufen kings, had Greek, Arabic and Latin communities side by side. Provence lay on the way to Spain, where translators at Toledo had been turning Arabic science into Latin since the mid-twelfth century.

What he read

His books show his sources more clearly than his biography does. The algebra of the Liber abaci follows the six standard equation types of Muhammad ibn Musa al-Khwarizmi, and many of its problems and methods follow the Egyptian algebraist Abu Kamil, whose work extended al-Khwarizmi’s. His geometry rests on Euclid’s Elements and on On Divisions, a Euclidean work known through Arabic. He also knew Latin translations of Arabic arithmetic. He refers to the double false-position method by its Arabic name, elchataym.

Whether Leonardo read Arabic is not known. His commercial problems use the coins and goods of North Africa and the Levant, which suggests direct contact with Arabic-speaking calculators rather than only Latin books.

The Liber abaci, 1202

Back in Pisa, Leonardo wrote the Liber abaci, the “book of calculation”. Despite the title it is not about the abacus board. It teaches computation with written Indian numerals, the method that would replace the counting board in Italian commerce. He completed it in 1202. That first version is lost. What survives is a revised version of 1228, dedicated to the court scholar Michael Scot, in at least nineteen manuscripts, three of them complete.

The book is long, about six hundred pages in Sigler’s translation, and it has fifteen chapters:

1. Reading and writing numbers with the Indian figures (chapter 1);
2. multiplication, addition, subtraction and division of whole numbers (chapters 2–5);
3. calculation with fractions and mixed numbers (chapters 6–7);
4. pricing of goods, barter, partnerships and alloying of coins (chapters 8–11);
5. a large collection of problems, including the rabbits (chapter 12);
6. double false position, elchataym (chapter 13);
7. square and cube roots and Euclid’s binomials and apotomes (chapter 14);
8. geometric rules and algebra (chapter 15).

The audience was merchants and their sons, but the book is not a simple manual. Leonardo proves his rules, often geometrically in Euclid’s style, and he cannot resist hard recreational problems. This mixture of commerce and proof is the subject of Part II.

The nine Indian figures

For the general reader. Roman numerals are fine for recording numbers and hopeless for calculating with them. Europeans calculated on counting boards and wrote down only the answers. The Indian system, with nine figures and a zero, lets you calculate on paper, and the written work can be checked afterwards. Leonardo’s first chapter opens by naming the nine figures and the sign 0, which he says the Arabs call zephirum; the word gave us both “cipher” and “zero”.

Place value

Proof. Existence: divide, \(N=qb+d_0\) with \(0\le d_0<b\), and apply induction to \(q<N\). Uniqueness: reducing modulo \(b\) fixes \(d_0\); subtract it, divide by \(b\), and repeat. With \(b=10\) this is Leonardo’s system. The algorithms for addition and multiplication are the same at every position, which is what made them teachable to merchants’ sons.

Checking a calculation

Leonardo checks multiplications by remainders, the casting out of nines, and also by other numbers such as 7 and 11. The principle is a congruence.

Proof. \(10\equiv1\pmod9\) and \(10\equiv-1\pmod{11}\), so \(10^k\equiv1\) and \(10^k\equiv(-1)^k\) respectively. Reduction modulo \(n\) respects products. The check is one-sided: a wrong product with the right residue passes, for instance a transposition of two digits.

Trial division

The Liber abaci contains the first known description in Europe of factoring by trial division. The method rests on a simple bound.

If \(N=ab\) with \(1<a\le b\), then \(a^2\le N\), so \(a\le\sqrt N\), and any prime factor of \(a\) is also at most \(\sqrt N\).

Fractions, composite and Egyptian

For the general reader. Medieval money and measures were not decimal. A Pisan pound had 20 shillings and a shilling 12 pence. Leonardo invented a fraction notation suited to such systems, a kind of mixed-radix number, and he also taught how to write any fraction as a sum of distinct unit fractions, the old Egyptian practice that was still alive in his sources.

Composite fractions

Leonardo writes fractions to the left of whole numbers and reads them from right to left. A composite fraction is a row of numerators over a row of denominators under one bar. In modern terms,

So a sum in pounds, shillings and pence is a composite fraction with denominators \(12,20\): a quantity of \(s\) shillings and \(p\) pence is \(\frac{p\ s}{12\ 20}\) of a pound. Every rational number in \([0,1)\) whose denominator divides \(d_1\cdots d_k\) has exactly one such expansion, by the same division argument as Lesson 04.

Egyptian fractions

Proof. \(\frac pq-\frac1n=\frac{np-q}{qn}\). Since \(n-1<q/p\), we have \((n-1)p<q\), so \(np-q<p\). The numerators strictly decrease, so the process stops. The unit fractions are distinct because, after subtracting \(1/n\), the remainder is less than \(\frac1{n-1}-\frac1n\le\frac1n\) when \(n\ge2\), so the next denominator is larger than \(n\). The method is now called the Fibonacci–Sylvester expansion, after James Joseph Sylvester, who studied it in 1880. Leonardo gives it as one of several rules and prefers other decompositions when they give smaller denominators.

Commercial problems and false position

For the general reader. Before symbolic algebra, the standard way to solve a linear problem was to guess. Guess an answer, see how far off it is, guess again, and combine the two errors. Leonardo called this method by its Arabic name, elchataym, and devoted chapter 13 of the Liber abaci to it. Chapter 12 collects hundreds of problems, many of them older than Leonardo and still told today.

Proof. \(e_i=a(g_i-x)\) where \(x\) is the true solution. Then \(g_1e_2-g_2e_1=a(g_1g_2-g_1x-g_2g_1+g_2x)=a x(g_2-g_1)\) and \(e_2-e_1=a(g_2-g_1)\). Divide. For non-linear \(f\) the same formula is the secant method, which is not exact but converges.

Remainders

Chapter 12 asks for a number which leaves given remainders when divided by 3, 5 and 7. The rule Leonardo gives, also found in the Chinese Sunzi suanjing centuries earlier, multiplies the remainders by 70, 21 and 15, adds, and reduces modulo 105.

For moduli 3, 5, 7, the idempotents are \(e_1=70\), \(e_2=21\), \(e_3=15\): each is a multiple of the other two moduli and is \(\equiv1\) modulo its own.

The rabbits and the sequence

For the general reader. In chapter 12 of the Liber abaci, Leonardo asks how many pairs of rabbits one pair produces in a year, if each pair bears a new pair every month starting in its second month. The answer is 377, and the monthly totals form the sequence now named after him. It was one recreational problem among hundreds, and it was not new: Indian scholars of poetic metre had met the same numbers centuries before.

The problem as Leonardo sets it

He starts with one pair, which gives birth in the first month, so that after one month there are 2 pairs. He lists the totals month by month: 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, and he explains that each number is the sum of the two before. He adds that the method works for any number of months. With the modern indexing \(F_0=0,\ F_1=1,\ F_{n+1}=F_n+F_{n-1}\), his total after \(n\) months is \(F_{n+2}\), and 377 is \(F_{14}\).

Why the recurrence holds. Let \(A_n\) be the number of adult pairs and \(Y_n\) the number of young pairs after \(n\) months. Each adult pair produces one young pair, and young pairs become adults a month later: \(A_{n+1}=A_n+Y_n\), \(Y_{n+1}=A_n\). The total \(T_n=A_n+Y_n\) then satisfies \(T_{n+1}=A_{n+1}+Y_{n+1}=T_n+A_n=T_n+T_{n-1}\).

Closed form

Proof. Both \(\varphi^n\) and \(\psi^n\) satisfy the recurrence, since \(\varphi^{n+1}=\varphi^n+\varphi^{n-1}\) follows from \(\varphi^2=\varphi+1\). So does any linear combination. The combination \((\varphi^n-\psi^n)/\sqrt5\) takes the values 0 and 1 at \(n=0,1\), and a solution is determined by two initial values. Since \(|\psi|<1\), \(F_n\) is the integer nearest to \(\varphi^n/\sqrt5\), and the ratio tends to \(\varphi\). The formula is named after Jacques Binet (1843) but was known earlier to Abraham de Moivre and Daniel Bernoulli. Johannes Kepler remarked on the limit of the ratios in 1611.

Identities

Proofs. (a) by induction, multiplying by the matrix once more. (b) take determinants in (a): the matrix has determinant \(-1\). (c) compare entries of \(Q^{m+n}=Q^mQ^n\), where \(Q\) is the matrix in (a). (d) From (b), consecutive Fibonacci numbers are coprime. From (c), \(F_n\mid F_{kn}\) by induction on \(k\), and \(\gcd(F_{m+n},F_n)=\gcd(F_{m+1}F_n+F_mF_{n-1},F_n)=\gcd(F_mF_{n-1},F_n)=\gcd(F_m,F_n)\), using \(\gcd(F_{n-1},F_n)=1\). So the pair of indices can be reduced exactly as in the Euclidean algorithm, ending at \(F_{\gcd(m,n)}\).

Before Leonardo

Indian prosodists counted the rhythms of a line of verse made of short syllables (one beat) and long syllables (two beats). The number of patterns lasting \(n\) beats is \(F_{n+1}\). The idea goes back to Pingala’s treatise on metre (dated variously between c. 450 and c. 200 BCE). Virahanka (c. 700) gave the rule explicitly, and Gopala (c. 1135) and Hemachandra (c. 1150) restated it. There is no evidence that Leonardo knew this tradition.

Algebra and the cubic of Flos

For the general reader. The last chapter of the Liber abaci teaches algebra in the Arabic manner, with words instead of symbols and geometric proofs for the rules. Around 1225 Johannes of Palermo, a scholar of Frederick II’s court, challenged Leonardo with a cubic equation. Leonardo proved the solution could not be found by the usual means, then gave a numerical answer correct to about ten decimal places, without saying how he found it.

Quadratics in words

With only positive coefficients allowed, al-Khwarizmi had distinguished six types: squares equal roots, squares equal numbers, roots equal numbers, and three mixed types such as “squares and roots equal numbers”, \(x^2+bx=c\). Leonardo calls the unknown res or radix and its square census; he also uses causa, apparently the first Latin author to do so. For \(x^2+bx=c\) the rule is to halve the roots, square, add the number, take the root, and subtract half the roots:

justified by completing a square of side \(x+\tfrac b2\) in a figure.

The cubic

Johannes’s problem, in modern form, was to solve \(x^3+2x^2+10x=20\). The same equation appears in the algebra of Omar Khayyam, who solved such cubics by intersecting conic sections. Leonardo argued in several steps that the root is not an integer, not a fraction, not the square root of a fraction, and not one of the other irrational lines classified in Euclid’s Book X. He then stated, in sexagesimal fractions,

while the true root is \(1.3688081078214\ldots\). The error is about \(3\times10^{-11}\). He did not explain his method. Historians have suggested a method of the kind used in Arabic sources for roots, but there is no direct evidence.

The Book of Squares

For the general reader. Johannes of Palermo also asked Leonardo for a square number which stays a square when 5 is added or subtracted. Leonardo answered with \(\left(\tfrac{41}{12}\right)^2\), and around the problem he built the Liber quadratorum (1225), dedicated to Frederick II. It survives in a single fifteenth-century manuscript in Milan. It is a book of number theory in the line of Diophantus, though Leonardo shows no knowledge of him, and its results anticipate Fermat and Leonhard Euler.

Odd numbers and triples

The book starts from the fact that \(1+3+\cdots+(2n-1)=n^2\), so that \(n^2+(2n+1)=(n+1)^2\). If \(2n+1=m^2\) is an odd square, this gives a Pythagorean triple. For example, \(9=2\cdot4+1\) gives \(16+9=25\).

The identity was known to Diophantus implicitly and to Brahmagupta in a more general form; it is often called the Brahmagupta–Fibonacci identity. Leonardo proves it for integers and uses it to produce numbers that are sums of two squares in several ways.

Congrua

A number \(c\) is a congruum if there is a square \(x^2\) with \(x^2-c\) and \(x^2+c\) both squares: three squares in arithmetic progression with common difference \(c\). Leonardo found all of them.

Proof of the identity. Expand: \((m^2-n^2\pm2mn)^2=(m^2-n^2)^2\pm4mn(m^2-n^2)+4m^2n^2\), and \((m^2-n^2)^2+4m^2n^2=(m^2+n^2)^2\). Geometrically, \(m^2-n^2,\ 2mn,\ m^2+n^2\) is a Pythagorean triple and the congruum is four times the area of that right triangle. Leonardo’s own statement uses the product \(ab(a+b)(b-a)\) with \(a+b\) even, which is the same family after substituting \(a=m-n\), \(b=m+n\).

The problem of 5

We want \(x^2\pm5\) to be rational squares. Multiply by \(k^2\): it suffices to find a congruum equal to \(5k^2\). Take \(m=5,\ n=4\): \(4\cdot5\cdot4\cdot9=720=5\cdot12^2\), with middle square \(41^2\). Indeed \(41^2-720=961=31^2\) and \(41^2+720=2401=49^2\). Dividing by \(144\),

Squares are not congrua

Leonardo stated that no congruum is a square. Sources differ on whether his argument counts as a proof; the first complete proof is Fermat’s, by infinite descent, in the form that the area of a right triangle with rational sides is never a square. Equivalently, \(x^4-y^4=z^2\) has no solution in positive integers. In modern language a number is congruent if it is the area of a rational right triangle, so the result says 1 is not a congruent number. The congruent number problem, for which Leonardo’s 5 was the first worked case, is now tied to elliptic curves and the Birch and Swinnerton-Dyer conjecture through Tunnell’s theorem.

Roots and practical geometry

For the general reader. The Practica geometriae (1220) is a surveyor’s and builder’s geometry in eight chapters, dedicated to Dominicus Hispanus. It begins with definitions and constructions from Euclid, computes with Pisan units of measure, extracts square and cube roots, measures fields and solids, finds heights with similar triangles, and ends with pentagons and decagons. The letter to Master Theodorus adds a pentagon inscribed in a triangle.

Approximating roots

The medieval rule, going back to Heron and used throughout Arabic arithmetic, is to write \(N=a^2+r\) with \(a^2\) the largest square below \(N\) and take \(\sqrt N\approx a+\frac r{2a}\). Repeating the step is Heron’s iteration.

Proof. \(\left(a+\frac r{2a}\right)^2=N+\frac{r^2}{4a^2}\ge N\). Then \(a+\frac r{2a}-\sqrt N=\frac{r^2/(4a^2)}{a+\frac r{2a}+\sqrt N}\le\frac{r^2}{8a^3}\). The iteration identity is direct algebra: \(\frac12(x+N/x)-\sqrt N=\frac{(x-\sqrt N)^2}{2x}\).

For cube roots the same linearization gives \(\sqrt[3]{a^3+r}\approx a+\frac r{3a^2}\). The Practica gives rules for cube roots of this general type; the exact rule Leonardo used is discussed in Hughes’s edition.

Pentagons and decagons

In the isosceles triangle formed by the centre and one side of the decagon, the apex angle is \(36^\circ\) and the base angles \(72^\circ\). Bisecting a base angle produces a smaller similar triangle, which gives \(s_{10}^2=R(R-s_{10})\). Solve the quadratic.

The emperor’s court and the city’s salary

Frederick II of Hohenstaufen, king of Sicily, was crowned emperor in Rome in November 1220. His court gathered scholars of many languages: the astrologer and translator Michael Scot, the court philosopher Master Theodorus, and Johannes of Palermo. Frederick founded the University of Naples in 1224. When the court was at Pisa, around 1225, Leonardo was presented to the emperor. According to the dedication of the Liber quadratorum, it was Master Dominicus who brought him to Frederick’s attention, and Johannes of Palermo put problems to him in the emperor’s presence.

Leonardo answered in writing. Flos (“flower”) contains his solutions to three of the problems and other work; the Liber quadratorum grew out of the problem of 5; and a letter to Master Theodorus treats the birds problem, a pentagon inscribed in an isosceles triangle with sides 10, 10 and 12, and a system of linked equations for five men’s money. In that last problem one variant has a negative solution, which Leonardo interprets as a debt, an early and careful use of negative quantities in Latin mathematics.

In 1228 he revised the Liber abaci and sent it to Michael Scot. Two other works are mentioned by Leonardo and are lost: a book of commercial arithmetic, Di minor guisa, and a commentary on Book X of Euclid’s Elements.

The decree of 1240

The last document about him is a decree of the Commune of Pisa from 1240. It grants the “serious and learned” Master Leonardo Bigollo an annual salary, in recognition of his services to the city and its citizens in matters of accounting and teaching. The decree shows that Leonardo had become a public figure: the city valued him for exactly the commercial skills his first book taught. Nothing is known of his death. It is usually placed after 1240 and before about 1250, probably in Pisa.

Abbaco, Lucas and the afterlife of a name

Leonardo’s numerals spread slowly. From the late thirteenth century, Italian cities supported abbaco schools, where masters taught merchants’ sons arithmetic with Indian numerals, commercial rules and some algebra, in the vernacular. Hundreds of abbaco treatises survive. Many draw on the Liber abaci, directly or at second hand, though historians such as Jens Høyrup have argued that the abbaco tradition also drew on other sources from the Mediterranean. Luca Pacioli’s Summa of 1494, the first printed encyclopedia of this mathematics, used Leonardo’s material and named him as a source. Full adoption of written numerals in European accounting took until the sixteenth century.

His number theory had less effect. The Liber quadratorum was barely read in the Middle Ages, and its results were found again by Francesco Maurolico, Fermat and Euler. Its single manuscript was identified and published by Baldassarre Boncompagni in the nineteenth century, together with the other works, in the Scritti di Leonardo Pisano (1857–62).

The sequence and the name

The fame of “Fibonacci” comes from the nineteenth century. The French number theorist Édouard Lucas, working in the 1870s on recurrences and primality, attached Leonardo’s name to the rabbit sequence and studied the companion sequence \(2,1,3,4,7,11,\ldots\), now called the Lucas numbers. Lucas’s methods led to tests for the primality of Mersenne numbers; in 1876 he proved \(2^{127}-1\) prime. In 1963 the Fibonacci Association began publishing The Fibonacci Quarterly, founded by Verner Hoggatt and Alfred Brousseau, devoted to the sequence and its relatives.

Pisa set up a statue of Leonardo by Giovanni Paganucci in 1863; it now stands in the Camposanto Monumentale. Modern English translations by Laurence Sigler (the Book of Squares, 1987; the Liber abaci, 2002) and Barnabas Hughes (the Practica geometriae, 2008) have made all his surviving books readable.

Assessment

• Arithmetic: the most influential European account of Indian numerals and their algorithms, written for practical use and with proofs.
• Algebra: transmission of al-Khwarizmi’s and Abu Kamil’s algebra, and a careful study of a cubic three centuries before the Italian solutions.
• Number theory: the congruum theorem, the two-squares identity and the first worked congruent-number problem.
• Name: a sequence older than Leonardo now carries a name he never used.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Leonardo Pisano Fibonacci, MacTutor History of Mathematics.
2. Fibonacci, Wikipedia.
3. K. Devlin, The Man of Numbers: Fibonacci’s Arithmetic Revolution, Walker, 2011.
4. A. F. Horadam, Fibonacci’s mathematical letter to Master Theodorus, The Fibonacci Quarterly 29 (1991).
5. J. Høyrup, Jacopo da Firenze’s Tractatus Algorismi and Early Italian Abbacus Culture, Birkhäuser, 2007.

Texts and mathematics

1. B. Boncompagni (ed.), Scritti di Leonardo Pisano, 2 vols., Rome, 1857–1862.
2. L. E. Sigler, Fibonacci’s Liber Abaci, Springer, 2002.
3. L. E. Sigler, Leonardo Pisano Fibonacci: The Book of Squares, Academic Press, 1987.
4. B. Hughes, Fibonacci’s De Practica Geometrie, Springer, 2008.
5. Liber Abaci, The Book of Squares and Congruum, Wikipedia.
6. T. Phillips, The rabbit problem in the Liber abaci (Latin text with translation), Stony Brook.
7. Fibonacci and square numbers, MAA Convergence.
8. Flos: the three problems, ProofWiki.
9. Fibonacci sequence, Wikipedia (history and Indian precursors).
10. The Fibonacci Quarterly, the Fibonacci Association.