Fibonacci

Numbers are all around us. Organising our lives depends on them. Our societies are regulated by them. Science, commerce, government, and education are built upon them.

Consider arranging a holiday, a few days relaxing in the sunshine. Questions immediately arise: How long? How far? How hot? How much? Each response is, of course, a number, whether an estimate or a precise value, but figures that exist within our decimal convention, more formally termed as the Hindu-Arabic place-valued decimal system.

Few of us wonder about the origins of such a numbering system, despite it being a product of human ingenuity. Or more correctly, one such product of our dexterity. Others existed, distinct and independent of today’s standard. The Egyptian pyramids, masterpieces of engineering and project management in their day, could scarcely have been built without a numerical foundation. They used a base-10 method, as we do, that was adequate for addition but imperfect for complicated reckonings. The Babylonians devised a sexagesimal (meaning base-60) system that was more efficient. Remnants endured, giving us 60-second and 60-minute usages, and a circle divided into 360°. Tools formulated 2-3 millennia ago.

Closer to us are Roman numerals, a standard with a lengthy lineage but still in common use within the last one-thousand years. Again, an additive/ subtractive system, so cumbersome in more intricate situations.

Today we employ the Hindu-Arabic concept, also an approach with a long history. Created in what is now India, nearly two-thousand years ago, and repeatedly refined. A later, vital step was the insertion of a symbol for zero. Not just a symbol, a mathematical concept. The ten present-day characters evolved through the endeavours of Arab scholars who disseminated the system in the Middle East and North Africa.

The cleverness lies in the notion of place-value, meaning the position of a symbol in the digit string determines its value (2, 20, 200 all utilise one or two digits but represent three amounts), and the designation of ten Arabic characters, simplifying computations. Figures on either side of the decimal point are handled with the same conventions.

All this, the result of abstract contemplation: pure human genius.

In Europe, circa 800 years ago, this work of beauty was largely unknown and certainly unrecognised. A young man, Leonardo Pisano, nicknamed Fibonacci, was being educated in North Africa. His father, Guilielmo, a diplomat in the territory, represented traders from the Republic of Pisa who were active in the region. An exceptional student with mathematical talent, Fibonacci progressed to become a local teacher. His travels, accompanying his father, introduced him to the Hindu-Arabic methodology, which he discerned as superior to prevailing practises in Pisa and beyond. 

On his return to Europe around 1200 CE, he wrote Liber Abaci – the book of calculation – his most famous work. In it, insights into the Hindu-Arabic standard are revealed, along with much else accumulated during his travels. In the second section, problems familiar to merchants are addressed: currency conversions, pricing, and profits. While the work of an itinerant scholar might be thought of minor interest at a time when European scholarship was a little threadbare, Fibonacci’s efforts were fruitful. He was not ignored. Liber Abaci became popular far beyond Pisa, with multiple copies produced; a challenge before printing was invented. The utility of this new numerical system became obvious to both the commercial fraternity and the wider intelligentsia, including mathematicians and scientists.

A growing reputation, including invitations to the court of Frederick II, the Holy Roman Emperor, served to promote Fibonacci’s later works that explored algebra, geometry, and number theory over the subsequent quarter-century. Liber Quadratorum – the book of squares – is regarded by many as his most impressive work.

His present-day profile arises from the third section of Liber Abaci in which he describes the rate of rabbit population growth through a numerical series, now known as the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, .. – each entry is the sum of the two preceding values.

The sequence has been found to possess wider relevance than rabbit propagation. In natural systems, petals, leaves, and tree branches exhibit Fibonacci numbers. Similarly, the family history of bee populations. More widely, computer science, financial markets, and architecture have applied the series. A well-known mathematical phenomenon, the Golden Ratio, is closely related.

A sequence known to Indian scholars long before Fibonacci’s time.

It might be argued that Fibonacci’s contributions were ignored in the Middle Ages, with ideas resurfacing through the intuition of scholars only after 300 years. His introduction of Hindu-Arabic numerals and a fascinating sequence ensure his continued celebration.

More tangibly, an academic magazine, the Fibonacci Quarterly, is published and an asteroid, 6765 Fibonacci orbits.

Fibonacci’s signature work, Liber Abaci, was dedicated to a Scottish scholar, Michael Scot, who assisted with the translation of Islamic mathematics into Latin. While not featured in Jimbo’s Assumption, Fibonacci emerges in the sequel and Scot will appear in the third episode. - RNP

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