# Leonardo Fibonacci

### Mathematician — ~1170–1250 — Republic of Pisa, Italy

> _"In the history of computation, Fibonacci stands as the bridge — the figure who carried the Hindu-Arabic numeral system from the Islamic world to Europe, making modern arithmetic, accounting, and ultimately computing possible."_

---

## Why This Matters

You cannot understand the history of computation without understanding Fibonacci. Before him, Europe calculated with Roman numerals — try multiplying MCMXCIV by XLVII and you'll see the problem. Fibonacci's _Liber Abaci_ (1202) didn't merely introduce new symbols; it introduced **positional notation**, the insight that a digit's position determines its value. This single conceptual shift made arithmetic tractable for merchants, bankers, and eventually engineers. When you write "1024" and understand that the "1" means one thousand, you are using the system Fibonacci brought to Europe. Every spreadsheet, every computer's binary arithmetic, every calculation you've ever performed descends from this transformation.

---

## Quick Reference

| Attribute | Value |
|-----------|-------|
| **Registry #** | 9 |
| **Born** | ~1170, Pisa, Republic of Pisa (modern Italy) |
| **Died** | ~1250, Pisa |
| **Active Period** | ~1200–1240 |
| **Fields** | Mathematics, Number Theory, Arithmetic |
| **Known For** | _Liber Abaci_ — introduced Hindu-Arabic numerals to Europe; Fibonacci sequence |
| **Influenced By** | Indian mathematicians (Brahmagupta, Bhaskara); Islamic mathematicians (al-Khwarizmi, Abu Kamil) |
| **Influenced** | European mathematics; commercial arithmetic; modern computing |

---

## Table of Contents

1. [Origins & Formation](#1-origins--formation)
2. [Intellectual Genealogy](#2-intellectual-genealogy)
3. [The Work: Chronological](#3-the-work-chronological)
4. [Core Ideas & Contributions](#4-core-ideas--contributions)
5. [Impact & Legacy](#5-impact--legacy)
6. [Study Guide: The Mental Model](#6-study-guide-the-mental-model)
7. [Going Deeper: Sources](#7-going-deeper-sources)

---

## 1. Origins & Formation

### A Note on Historical Sources

> **On Uncertainty:** Fibonacci's biographical details are better documented than many medieval figures, but gaps remain. Our primary source is the autobiographical prologue to _Liber Abaci_, where he describes his education. Beyond this, we have a few legal documents, dedications in his works, and mentions in contemporary records. The dates ~1170–1250 are scholarly estimates; his birth year is inferred from his statement that he was young when sent to North Africa.

### Early Life & Context

> _Etymology: **Fibonacci** is a contraction of "filius Bonacci" (son of Bonacci). His father was named Guglielmo Bonacci. Leonardo sometimes signed himself "Leonardo Pisano" (Leonardo of Pisa) or "Leonardo Bigollo" (the traveler, or possibly the blockhead — a self-deprecating nickname)._

Leonardo was born in **Pisa**, one of the four great maritime republics of medieval Italy. In the late 12th century, Pisa was a commercial powerhouse — its ships dominated Mediterranean trade, its merchants established colonies and trading posts across the known world, and wealth flowed through its harbor.

**Pisa in the Late 12th Century:**
- A major maritime republic rivaling Venice, Genoa, and Amalfi
- Trading networks extending to North Africa, Egypt, Syria, and Constantinople
- Commercial culture demanding practical mathematics for accounting, currency exchange, and profit calculation
- Intellectual contact with the Islamic world through trade and crusades

Leonardo's father, Guglielmo Bonacci, was a prosperous merchant and customs official. Around 1192, Guglielmo was appointed to represent Pisan merchants in **Bugia** (modern Béjaïa, Algeria) — a major North African trading port. He brought his son with him.

### Education & Training

| Period | Context | Focus | Tradition |
|--------|---------|-------|-----------|
| Youth (Pisa) | Commercial Pisan environment | Basic literacy, Roman numeral arithmetic | Italian merchant tradition |
| ~1192–1200 (North Africa) | Bugia under Almohad rule | Hindu-Arabic numerals, Islamic mathematics | Arabic mathematical tradition |
| ~1200–1220 (Mediterranean) | Travel and study | Comparative mathematical systems | Cross-cultural synthesis |
| ~1202 onward (Pisa) | Mature work | Mathematical treatises | Independent scholarship |

**The North African Education:**

In Bugia, Fibonacci encountered a mathematical world far more sophisticated than anything available in Christian Europe. Islamic scholars had preserved and extended Greek mathematics while developing their own innovations. Most crucially, they had adopted the Hindu numeral system — including zero and positional notation — and developed algebra.

Fibonacci wrote in _Liber Abaci_:

> _"When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else..."_

This "art of the Indians' nine symbols" would transform European civilization.

**Mediterranean Travels:**

Fibonacci didn't stop in Bugia. He traveled extensively — to Egypt, Syria, Greece, Sicily, and Provence — comparing mathematical methods and learning from diverse traditions. He was not merely a student but an investigator, collecting and synthesizing the best techniques from each culture.

### Formative Influences

**Islamic Mathematical Tradition:**

- **al-Khwarizmi** (9th c.) — The father of algebra; his name gives us "algorithm" and his book title gives us "algebra"
- **Abu Kamil** (9th-10th c.) — Extended al-Khwarizmi's algebra; Fibonacci explicitly cites him in _Liber Abaci_
- The entire Arabic tradition of practical arithmetic for commerce, surveying, and inheritance calculations

**Indian Mathematical Tradition (via Arabic transmission):**

- **Brahmagupta** (7th c.) — First to treat zero as a number with operational rules
- **Bhaskara I** (7th c.) — Decimal positional system
- The concept that a digit's value depends on its position

**Greek Mathematical Tradition (via Arabic preservation):**

- Euclid's _Elements_ (Fibonacci wrote a commentary on Book X)
- Diophantine number theory
- Systematic proof-based mathematics

---

## 2. Intellectual Genealogy

### The Lineage: Who Influenced Fibonacci

```
Indian Mathematicians (Brahmagupta, Bhaskara)
        │
        ▼
┌───────────────────────────────────────┐
│ Islamic Transmission & Extension      │
│ (al-Khwarizmi, Abu Kamil, et al.)    │
│ Algebra, Hindu-Arabic numerals        │
└───────────────────────────────────────┘
        │
        ▼
    ┌───────────┐
    │ FIBONACCI │
    └───────────┘
        │
        ▼
┌───────────────────────────────────────────────────────────────────┐
│ European Commercial Mathematics → Renaissance → Scientific Rev    │
│                                                                   │
│ Italian Abacus Schools → Double-Entry Bookkeeping → Accounting    │
│                                                                   │
│ Number Theory → Euler → Gauss → Modern Mathematics                │
│                                                                   │
│ Positional Notation → Mechanical Calculation → Digital Computers  │
└───────────────────────────────────────────────────────────────────┘
```

**Direct Influences on Fibonacci:**

- **Hindu-Arabic Numeral System:** The core innovation he transmitted
- **al-Khwarizmi's _Algebra_:** Systematic equation-solving methods
- **Abu Kamil:** Explicit source for many problems in _Liber Abaci_
- **Euclid:** Geometric and number-theoretic foundations
- **Practical Mediterranean Commerce:** The problems merchants actually needed to solve

**Contextual Influences:**

- **Pisan Commercial Culture:** Created demand for efficient calculation
- **Mediterranean Trade Networks:** Enabled cross-cultural mathematical exchange
- **Crusades:** Brought European scholars into contact with Islamic learning
- **Translation Movement:** Arabic texts being translated into Latin (Toledo, Sicily)

### The Lineage: Who Fibonacci Influenced

**Immediate Successors:**

| Figure | Era | Contribution |
|--------|-----|--------------|
| **Italian Abacus Schools** | 13th-15th c. | Institutionalized Fibonacci's methods for merchant education |
| **Luca Pacioli** | ~1445-1517 | _Summa de Arithmetica_ — synthesized Renaissance arithmetic, double-entry bookkeeping |
| **Countless Unnamed Merchants** | 13th c. onward | Actually used the system for commerce |

**Long-Term Mathematical Influence:**

- **Number Theory:** The Fibonacci sequence became a foundational object of study
- **Golden Ratio:** Connections between Fibonacci numbers and phi explored by later mathematicians
- **Recursive Sequences:** A paradigm case for mathematical analysis

**Ideas That Persist:**

| Fibonacci Concept | Modern Manifestation |
|-------------------|---------------------|
| Positional notation | All modern number representation, including binary |
| Zero as a digit | Essential for computing, placeholder arithmetic |
| Algorithmic arithmetic | Computer arithmetic operations |
| Fibonacci sequence | Data structures, algorithm analysis, natural modeling |

---

## 3. The Work: Chronological

### Master Timeline

| Year | Work | Type | Significance |
|------|------|------|--------------|
| 1202 | _Liber Abaci_ (first edition) | Arithmetic/Algebra | Introduced Hindu-Arabic numerals to Europe |
| 1220 | _Practica Geometriae_ | Geometry | Applied Arabic geometry to practical problems |
| 1225 | _Flos_ | Number Theory | Solutions to challenge problems from Emperor Frederick II's court |
| 1225 | _Liber Quadratorum_ | Number Theory | Theory of squares; sophisticated Diophantine analysis |
| 1228 | _Liber Abaci_ (revised edition) | Arithmetic/Algebra | Expanded and improved second edition |
| ~1225 | _Letter to Master Theodorus_ | Number Theory | Additional mathematical correspondence |

### The Central Work: _Liber Abaci_

> _Etymology: **Liber Abaci** means "Book of Calculation" (not "Book of the Abacus"). The Latin **abacus** here refers to calculation generally, not the bead-frame device._

**What It Is:**

_Liber Abaci_ is a comprehensive treatise on arithmetic, algebra, and practical mathematics using the Hindu-Arabic numeral system. At over 600 pages in modern translation, it is simultaneously:
- A textbook introducing the new numerals and their operations
- A manual of commercial mathematics (currency exchange, profit/loss, interest)
- A collection of recreational mathematics problems
- A treatise on algebraic equation-solving

**Structure:**

| Chapter | Topic |
|---------|-------|
| 1 | The nine Indian figures and zero; place value; reading and writing numbers |
| 2–5 | Multiplication of integers |
| 6 | Addition and subtraction |
| 7 | Division of integers |
| 8 | Fractions |
| 9–11 | Commercial arithmetic: barter, exchange, profit |
| 12 | Series; **the rabbit problem** (Fibonacci sequence) |
| 13 | Recreational problems (puzzles, games) |
| 14–15 | Roots; geometric problems |
| Remaining | Algebra; Diophantine problems |

**What Makes It Revolutionary:**

1. **Complete System:** Not just the digits — the full apparatus of positional arithmetic including algorithms for all operations.

2. **Zero as Placeholder:** Fibonacci explicitly teaches the use of zero (which he calls _zephirum_, from Arabic _sifr_) as essential to the system.

3. **Worked Examples:** Hundreds of problems with solutions, from simple to advanced. This is a teaching text, not a reference.

4. **Practical Focus:** Problems merchants actually face — How much profit on this trade? How to split an inheritance? What's the fair price when exchanging currencies?

5. **Sophisticated Mathematics:** The later chapters contain serious number theory, including material that would challenge modern undergraduates.

**The Rabbit Problem (Chapter 12):**

> "A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also."

The answer generates the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

Each term is the sum of the two preceding terms. This was not Fibonacci's invention — he was illustrating a point about series — but it would become his most famous legacy.

### Other Major Works

**_Practica Geometriae_ (1220):**

A practical geometry text synthesizing Euclidean methods with Arabic surveying techniques. Problems include measuring heights, distances, and areas — the mathematics of engineering and construction.

**_Liber Quadratorum_ (1225):**

"The Book of Squares" — Fibonacci's most original work, a sophisticated treatise on number theory. He proves results about sums of squares, Pythagorean triples, and what we now call Fibonacci's identity:

> (a² + b²)(c² + d²) = (ac + bd)² + (ad - bc)² = (ac - bd)² + (ad + bc)²

This shows the product of two sums of squares is itself a sum of squares — a deep result in number theory.

---

## 4. Core Ideas & Contributions

### The Central Insight

Fibonacci understood that the Hindu-Arabic numeral system was not merely a notational convenience but a **computational technology**. Roman numerals were adequate for recording quantities but terrible for calculating with them. Positional notation makes arithmetic **algorithmic** — reducible to a series of simple, mechanical steps that anyone can learn and execute.

This is the insight that underlies:
- All modern arithmetic education
- Mechanical calculators (from Pascal to Babbage)
- Digital computers (binary is positional notation in base 2)
- The very possibility of automated computation

### Key Concepts

#### Positional Notation

> _Definition: A numeral system where the position of a digit determines its value. In "352," the "3" means 300, the "5" means 50, and the "2" means 2._

**Why It Matters:** With Roman numerals, multiplication requires tables of results or physical counting devices. With positional notation, multiplication reduces to a fixed algorithm using only single-digit multiplication facts. The same applies to division, root extraction, and more complex operations.

**Example:** Compare multiplying 47 × 23 in Roman versus Hindu-Arabic:

Roman: XLVII × XXIII = ??? (requires extensive procedure or devices)
Hindu-Arabic: Apply the standard algorithm → 1081

**Modern Application:** Every computer uses positional notation (binary). Every programming language assumes it. Every digital circuit implements it.

#### Zero as a Digit

> _Etymology: English "zero" comes from Italian "zero," from Medieval Latin "zephirum," from Arabic "sifr" (meaning "empty"), from Sanskrit "sunya" (meaning "void")._

**Definition:** Zero is not merely the absence of quantity but a symbol that holds a position, enabling positional notation to work.

**Why It Matters:** Without zero, you cannot write "102" — the zero tells you there are no tens. Zero enables compact representation of arbitrarily large numbers and makes the algorithms work uniformly.

**Modern Application:** Computing would be inconceivable without zero. Binary (1s and 0s) is literally built on it.

#### The Fibonacci Sequence

> _Definition: The sequence where each term equals the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55..._

**Mathematical Properties:**
- The ratio of consecutive terms approaches the golden ratio: φ = (1 + √5)/2 ≈ 1.618
- Closed-form expression: F(n) = (φⁿ - ψⁿ)/√5 where ψ = (1 - √5)/2
- Appears in countless mathematical contexts (binomial coefficients, continued fractions, etc.)

**Modern Applications:**
- Algorithm analysis (Euclidean algorithm worst case, Fibonacci heaps)
- Data structures (Fibonacci heaps, Fibonacci search)
- Natural modeling (phyllotaxis, population dynamics)
- Art and design (golden ratio compositions)

#### Algorithmic Thinking

Fibonacci didn't just teach results — he taught **methods**. His book is full of step-by-step procedures that anyone can follow mechanically to get correct answers. This is algorithmic thinking, and it's the foundation of programming.

**Example:** Fibonacci's method for division by repeated subtraction and scaling is exactly how a human might implement division without knowing multiplication tables — it's a robust algorithm.

### Theoretical Framework

Fibonacci's approach operates as a **knowledge transfer and synthesis engine**:

```
INPUT:  Islamic/Indian mathematics + European commercial needs
           │
           ▼
┌─────────────────────────────────────┐
│ Fibonacci's Synthesis:              │
│ 1. Translate concepts               │
│ 2. Develop pedagogical exposition   │
│ 3. Apply to practical problems      │
│ 4. Extend with original results     │
└─────────────────────────────────────┘
           │
           ▼
OUTPUT: Accessible European mathematics
```

### Innovations & Contributions

| Contribution | Description | Prior State | What Changed |
|--------------|-------------|-------------|--------------|
| Hindu-Arabic numerals in Europe | Full system with algorithms | Roman numerals dominant | Computation became tractable |
| Commercial mathematics textbook | Comprehensive worked examples | Scattered knowledge | Systematic merchant education |
| Fibonacci sequence | First appearance in Western mathematics | Unknown in Europe | New mathematical object |
| Number-theoretic results | Original theorems on squares | Limited European number theory | Advanced mathematics in Latin |

---

## 5. Impact & Legacy

### Immediate Impact

**In Fibonacci's Lifetime:**

Fibonacci achieved recognition at the highest levels. Emperor Frederick II, the most intellectually curious ruler of his age, heard of Fibonacci's abilities and invited him to demonstrate at court (~1225). The emperor's philosopher, Johannes of Palermo, posed challenge problems; Fibonacci's solutions appear in _Flos_.

The Republic of Pisa awarded Fibonacci an annual stipend in 1240, acknowledging his service to the city's commercial community.

**Adoption of Hindu-Arabic Numerals:**

The transition was not immediate — Roman numerals persisted in some contexts for centuries. But in commercial arithmetic, where efficiency mattered, the new system spread steadily:

- Italian merchant communities adopted it first (13th-14th centuries)
- Banking centers (Florence, Venice, Genoa) standardized on it
- "Abacus schools" teaching Fibonacci's methods proliferated
- By 1500, the transition in commerce was essentially complete

### Long-Term Influence

**In Commercial Practice:**

- **Double-Entry Bookkeeping:** Impossible without positional arithmetic. Luca Pacioli's 1494 _Summa_, which systematized double-entry, builds on Fibonacci's foundation.
- **Banking and Finance:** Complex financial instruments require efficient calculation
- **Insurance and Actuarial Science:** Risk calculation demands arithmetic

**In Mathematics:**

- **Fibonacci Numbers:** Became a major object of study in number theory
- **Golden Ratio:** Connections explored extensively (Kepler, et al.)
- **Number Theory:** Fibonacci's _Liber Quadratorum_ anticipated many later developments

**In Computing:**

- **Positional Notation:** Binary representation is Fibonacci's system in base 2
- **Algorithmic Arithmetic:** Computer processors implement Fibonacci's algorithms (modernized) in hardware
- **Algorithm Analysis:** Fibonacci numbers appear throughout theoretical computer science

### The Counterfactual

> What if Fibonacci had never existed?

The Hindu-Arabic numeral system would eventually have reached Europe — the mathematics was too superior to ignore, and cross-cultural contact was increasing. But the transition might have been delayed by decades or longer. Fibonacci provided:

1. **Authoritative Exposition:** A single comprehensive source
2. **Practical Orientation:** Immediate utility for merchants
3. **Quality:** Sufficiently sophisticated to serve as a lasting standard

Without _Liber Abaci_, European commerce and eventually science might have developed more slowly. The centuries matter — delayed adoption means delayed Renaissance mathematics, delayed scientific revolution, delayed industrial revolution. The compounding effects are enormous.

### Recognition & Honors

| Era | Recognition |
|-----|-------------|
| 1240 | Annual stipend from Republic of Pisa |
| 19th c. | "Fibonacci" name and sequence popularized by Édouard Lucas |
| Modern | Fibonacci numbers ubiquitous in mathematics and computer science |
| Modern | Statue in Pisa (at the Camposanto) |

---

## 6. Study Guide: The Mental Model

### The One Sentence

> **Fibonacci brought positional notation to Europe, making arithmetic algorithmic and thereby enabling five centuries of mathematical and technological progress culminating in the computer.**

### The Three Things to Remember

1. **The Bridge:** He transmitted Hindu-Arabic mathematics to Europe. The knowledge existed in the Islamic world; he made it accessible to Latin Christendom.

2. **Positional Notation:** The key insight is that a digit's position determines its value. This makes arithmetic mechanical, teachable, and eventually automatable.

3. **More Than Transmission:** Fibonacci wasn't just a translator. _Liber Quadratorum_ contains original, sophisticated number theory. He was a first-rate mathematician in his own right.

### The Visual

```
┌────────────────────────────────────────────────────────────────┐
│                    FIBONACCI'S CONTRIBUTION                     │
│                  (The Computational Bridge)                     │
│                                                                 │
│   BEFORE               FIBONACCI              AFTER             │
│  ┌──────────┐      ┌─────────────────┐      ┌──────────────┐   │
│  │ Roman    │      │ Translation &   │      │ Hindu-Arabic │   │
│  │ Numerals │ ───▶ │ Pedagogy in     │ ───▶ │ Numerals     │   │
│  │ (MCMXCIV)│      │ _Liber Abaci_   │      │ (1994)       │   │
│  │          │      │                 │      │              │   │
│  │ Hard to  │      │                 │      │ Easy to      │   │
│  │ Calculate│      │                 │      │ Calculate    │   │
│  └──────────┘      └─────────────────┘      └──────────────┘   │
│       ▲                                           │             │
│       │                                           ▼             │
│   Computation                               Computation         │
│   Requires                                  Requires Only       │
│   Special                                   Paper & Pen         │
│   Devices                                                       │
│                                                                 │
└────────────────────────────────────────────────────────────────┘
```

### Connecting to Other Figures

| If You Know... | Then Understand That Fibonacci... |
|----------------|-----------------------------------|
| al-Khwarizmi | Transmitted al-Khwarizmi's system to Europe |
| Brahmagupta | Brought Indian mathematical innovations westward |
| Euclid | Extended Euclidean number theory with original results |
| Babbage | Made mechanical calculation conceivable by establishing algorithmic arithmetic |
| Turing | Provided the numerical foundation that computers formalize |

### Common Misconceptions

| Misconception | Reality |
|---------------|---------|
| "He invented the Fibonacci sequence" | He illustrated it in a problem; Indian mathematicians knew it earlier |
| "He invented Hindu-Arabic numerals" | He transmitted an existing system from the Islamic world |
| "_Liber Abaci_ is about the abacus" | The title means "Book of Calculation" — it teaches pen-and-paper methods |
| "The transition was immediate" | Roman numerals persisted for centuries in some contexts |
| "He was just a transmitter" | _Liber Quadratorum_ contains original, sophisticated mathematics |

### Test Your Understanding

1. **Conceptual:** Why does positional notation make arithmetic fundamentally easier than Roman numerals? What specific operations become tractable?

2. **Connection:** How does Fibonacci's work relate to al-Khwarizmi's? What did Fibonacci add beyond translation?

3. **Genealogy:** Trace the line from Indian mathematics to Fibonacci to modern computing — what are the intermediate steps?

---

## 7. Going Deeper: Sources

### Primary Sources

| Source | Type | Access | Notes |
|--------|------|--------|-------|
| _Liber Abaci_ | Arithmetic/Algebra | Sigler translation (Springer, 2002) | Complete English translation |
| _Liber Quadratorum_ | Number Theory | Available in translation | Sophisticated original mathematics |
| _Practica Geometriae_ | Geometry | Latin edition; partial translations | Practical geometry |

### Essential Secondary Sources

| Source | Author | Type | What It Covers |
|--------|--------|------|----------------|
| _Leonardo Pisano Bigollo_ | R. E. Grimm | Biographical Overview | Entry in Dictionary of Scientific Biography |
| _Leonardo of Pisa and the New Mathematics of the Middle Ages_ | Joseph & Frances Gies | Popular History | Accessible introduction |
| _Fibonacci's Liber Abaci_ | L. E. Sigler (trans.) | Translation + Commentary | The standard modern English edition |
| _A History of Mathematics_ | Carl Boyer & Uta Merzbach | Survey | Chapter placing Fibonacci in context |
| _The Man of Numbers_ | Keith Devlin | Popular Biography | Readable modern treatment |

### Modern Introductions

- **For beginners:** Keith Devlin, _The Man of Numbers_ (2011) — accessible narrative
- **For mathematicians:** Sigler's translation of _Liber Abaci_ — the actual text
- **For historians:** Joseph and Frances Gies on medieval mathematics and commerce

### Online Resources

- [MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Fibonacci/) — Comprehensive biographical entry
- The Fibonacci Quarterly — Journal devoted to Fibonacci numbers and related mathematics
- [Mathematical Association of America: Fibonacci Resources](https://www.maa.org/) — Educational materials

---

## Appendix: Handling Uncertainty

> **Note on Sources:** Fibonacci's life is better documented than many medieval figures, but significant gaps remain. Our primary source is his own prologue to _Liber Abaci_. Beyond this, a few legal documents mention him, and the Republic of Pisa's decree honoring him survives. The dates ~1170–1250 are scholarly estimates based on internal evidence.

| Claim | Confidence | Source |
|-------|------------|--------|
| Pisan birth | High | Autobiography in _Liber Abaci_ |
| Education in North Africa | High | Autobiography in _Liber Abaci_ |
| Approximate dates (~1170–1250) | Medium | Inferred from textual references |
| Mediterranean travels | High | Autobiography in _Liber Abaci_ |
| Meeting with Frederick II | High | References in _Flos_ |
| 1240 Pisan stipend | High | Documentary evidence |
| Authored listed works | High | Consistent attribution; internal evidence |
| Details beyond autobiography | Low | Sparse documentation |

---

_Last updated: 2026-03-26. This is a living document._