Fioravante Ascolese’s essay traces the birth and evolution of zero—from a philosophical notion of “nothingness” to a cornerstone of modern mathematics and technology. From its Babylonian and Indian origins to its transmission through the Islamic world and its subsequent diffusion in Europe with Fibonacci, zero emerges as a symbol of knowledge and progress. The author follows its trajectory up to the binary system of the digital age, illustrating how “nothingness” has become the very principle of infinity and the foundation of the scientific and harmonic order of the world.
Rosa Bianco
“Make room,” protested Zero, “I exist too.” (*): The Rise of Nothingness
In ancient times, civilizations were unaware of the concept of zero because their world was firmly rooted in the concreteness of things. It was logical to count people, sheep, and objects—but not what was not there. With intellectual progress came the dawning idea of nothingness—the absence of something, of anything—and the desire to represent it symbolically. Thus emerged zero, though not yet understood as a number in its own right.
In the final centuries of Babylonian civilization (7th–6th century BCE), there arose a need to introduce into their positional numeral system—one that, despite its sophistication, lacked a zero—a new symbolic sign that could function as a placeholder. Their system was sexagesimal (base 60), beginning with a symbol for one and ending with another representing sixty. The innovation served merely to indicate continuity beyond sixty, without yet possessing the numerical value of our modern zero.
In another part of the world, the Maya civilization also developed a positional numeral system—base-twenty—that incorporated a symbol for zero, which not only functioned as a placeholder but also as a number in its own right; however, until the arrival of the Spaniards (from 1517 onward) the Maya remained unknown to Europe, which by that date had already long been familiar with and using the decimal system.
In ancient Greece, too, zero denoted “nothingness”—both in philosophy and in mathematics—but it did not assume numerical value.
One must wait until the 5th and 6th centuries CE—during the period when the Indian civilization achieved its highest flowering in the arts, in the sciences (especially astronomy), and in mathematics—when zero finally acquired its full numerical meaning, thereby enabling the representation of arbitrarily large numbers and inclusive numbers (i.e., without repetition, unlike non-positional systems such as the Roman numerals). In the Indian culture the idea was already present that multiplication and division are inverse operations; Indian mathematicians also engaged in division in which zero appears as divisor, though they reached conclusions different from those held today. Indeed, today the operation is considered impossible when the dividend is ≠ 0, because no number multiplied by zero can yield a result different from zero; and if one faces 0 ÷ 0 the division is defined as indeterminate because it has infinitely many solutions, since any number multiplied by zero gives zero (the zero-product law).
With the progressive expansion of Islamic civilization, cultural and scientific exchanges intensified across Asia, North Africa, and Europe. The Islamic world’s territorial reach extended beyond the former empire of Alexander the Great, encompassing regions of the Indian subcontinent and, in Europe, parts of Spain and Sicily. Such extensive contact inevitably fostered profound intellectual and cultural interactions between the Islamic and Indian worlds.
In mathematics, this cross-fertilization gave rise to the term ṣifr, from which derive the Italian cifra and the Latin zephirum, later zevero, and ultimately zero. Around 780 CE, the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī became the director of the House of Wisdom (Bayt al-Ḥikma) in Baghdad. He translated into Arabic several Indian and Persian mathematical treatises. From him we inherit the term algebra—that is, a system of relations among quantities expressed through letters and numbers within equations, enabling the generalization of operations. The term algorithm also originates from the Latinized form of his name (Algoritmi), signifying a set of sequential instructions designed to achieve a specific goal—a concept now fundamental in event planning and, above all, in computer science.
Around 1170, in Pisa, Leonardo, son of Guglielmo dei Bonacci, was born during the flourishing age of the Maritime Republics. Guglielmo, a wealthy merchant and customs officer for the Republic of Pisa, stationed in Béjaïa (Bougie), Algeria, introduced his son to arithmetic and commercial practices. Leonardo—later known to history as Fibonacci—displayed remarkable intelligence and an aptitude for study. He mastered mathematical concepts that were then unknown in the Western world and embarked on extensive travels for both study and commerce. From Algeria and North Africa he journeyed to Syria and Greece, eventually reaching Constantinople, where he continued to study and critically reformulate the works of Indian, Persian, and Arab mathematicians.
Upon returning to Pisa, Fibonacci was welcomed with high honors, his international fame preceding him. The Republic granted him a stipend so that he could devote himself entirely to his studies, free from financial concerns. In 1202 he published the Liber Abbaci, a monumental treatise introducing the Indo-Arabic numerals of the decimal system—therefore including zero—which made possible the construction of an infinite set of natural numbers, as well as the execution of fundamental operations, the use of fractions (numeri rotti), and the extraction of roots.
Particularly noteworthy is his solution to the problem of rabbit reproduction, which generated the now-famous Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, …, in which each number is obtained by summing the two preceding ones.
This sequence reveals extraordinary secrets of nature. Indeed, phyllotaxis—the arrangement of leaves and flowers on a stem—follows the Fibonacci sequence as a means of maximizing exposure to sunlight. For example, lilies have three petals, dog roses have five, certain daisies have thirteen, others twenty-one, thirty-four, fifty-five, or even eighty-nine. If one observes the arrangement of sunflower seeds, one can discern a double spiral: one clockwise and the other counter-clockwise.
The word spiral is crucial to understanding this mysterious order. If, using the initial numbers of the sequence, one constructs squares whose sides correspond to these numbers and then draws an arc of a circle within each square, the resulting figure is a spiral—the so-called golden spiral. Astronomical observations reveal that some galaxies also exhibit this same spiral form.
This proportional relationship can be formalized geometrically:
“The golden ratio is the relationship between two homogeneous magnitudes in which the greater is the mean proportional between the smaller and the sum of both.” (cf. op. cit., p. 841)
The golden ratio (φ) is an irrational number, whose approximate value is 1.618, and it has become a symbol of mathematical harmony.
Philosophers, mathematicians, architects, and artists have long regarded the golden ratio as the embodiment of beauty, harmony, and perfection. This concept inspired Leonardo da Vinci’s Vitruvian Man, the Parthenon, numerous cathedrals, and other monumental works. To reflect upon the golden ratio is to sense an underlying order—what some perceive as the imprint of the Divine. It is no coincidence that, a couple of centuries after Fibonacci, the mathematician and friar Luca Pacioli wrote his Summa de Arithmetica, Geometria, Proportioni et Proportionalità (1494) and later his De Divina Proportione (1509), in which the notion of divine harmony and proportion found its most elevated human feelings expression.
Returning to Fibonacci’s era, it is useful to briefly contextualize the political landscape of Italy. In the central-northern regions, city-states (Comuni) flourished, while in the south, Frederick II, Holy Roman Emperor, ruled as an enlightened sovereign. He promulgated the Constitution of Melfi, which introduced a series of rights alongside obligations, founded a university in Naples bearing his name, engaged in cultural exchanges with the Islamic world, and sought to meet Fibonacci, even posing him mathematical questions.
In response, between 1225 and 1226, Fibonacci published the Liber Quadratorum, in which he addressed these problems and described so-called square numbers: numbers raised to the exponent 2, which can be represented geometrically as squares—1, 4, 9, 16, 25, 36…
My essay about the uprising of zero continues with the introduction of integers (ℤ), a concept absent from Indo-Arabic mathematics and from Fibonacci’s own work. Integers include, besides positive numbers (the equivalent of natural numbers), negative numbers, which are the symmetrical counterparts of natural numbers and are preceded by the minus sign (−) to distinguish them from positive numbers. Practical examples include thermometer readings or floor levels in a building. In these cases, zero no longer represents an empty set but a numerical same identical value to any other number.
Finally, zero forms the foundation of the binary numeral system, which underpins computer science and digital technology. This system uses only two digits—0 and 1—called bits (binary digits). These digits correspond, in the connected electrical circuits, to “on” (1, information passes) and “off” (0, information does not pass). Like the decimal system, it is positional: each shift to the left increases the power of two corresponding to the column. For example, while 0 and 1 are straightforward, the number 2 requires two bits: it is written as 10 (read as “one zero”), where 0 counts units and 1 counts pairs. Applying this principle, decimal numbers 0, 1, 2, 3, … are represented in binary as 0, 1, 10, 11, …
Even supercomputers, which operate with parallel circuits accessing enormous amounts of information, rely fundamentally on the binary system—using only 0 and 1.
Thus, over centuries and millennia, zero has acquired extraordinary significance, evolving from a philosophical concept of “nothingness” to a stonecorner of mathematics, science, and technology.
Zero has finally became a true real number meaning.
Fioravante Ascolese.
References
Le Garzantine. Matematica. Milano: Garzanti, pp. 1329 ff.
F. Op. cit., p. 841.
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