Reading — Test 12

Reading passage

For most of human history, the concept of nothing was not considered a number at all, let alone a number worth writing down. Yet the invention of zero—both as a placeholder in a system of notation and as an abstract quantity subject to its own rules of arithmetic—ranks among the most consequential intellectual achievements in the history of mathematics. Its story is not one of a single discovery but of several independent inventions, separated by centuries and continents, that were eventually woven together into the numeral system used almost universally today. The earliest known use of a zero-like symbol appeared in Babylonian mathematics around the third century BCE. Babylonian scribes wrote numbers in a positional, base-60 system, meaning that the value of a digit depended on its place within a sequence, much as the digit "3" means something different in "30" than in "300" in our own base-10 system. The trouble with any positional system is that it requires some way of marking an empty place; without such a marker, the numbers 216 and 2106 would look identical. Babylonian mathematicians solved this problem by inserting a special wedge-shaped mark wherever a place was empty. Crucially, however, this mark functioned only as a placeholder within a numeral. It was never used to represent the value of nothingness on its own, nor was it added, subtracted, or otherwise manipulated the way an ordinary number would be. A separate, and apparently unconnected, placeholder notation later arose among the Maya of Mesoamerica, who used a shell-shaped glyph in their vigesimal, or base-20, calendrical calculations. Like its Babylonian counterpart, the Maya symbol marked absence within a numeral but was not treated as a number in the fullest sense. The transformation of zero from a mere placeholder into a genuine number—one that could be added, subtracted, and reasoned about—is credited to mathematicians on the Indian subcontinent. By the fifth century CE, Indian astronomers were using a dot or small circle, called shunya, meaning "void" or "empty," within a decimal place-value system. The decisive conceptual leap came in 628 CE, when the mathematician Brahmagupta composed a treatise titled Brahmasphutasiddhanta, in which he formally defined zero as the result of subtracting a number from itself and laid out rules for how it behaved when combined with positive and negative quantities. Brahmagupta correctly stated, for instance, that a positive or negative number multiplied by zero yields zero, though his attempt to define division by zero produced results that later mathematicians would revise. Nonetheless, his willingness to treat zero as a legitimate object of arithmetic, deserving the same systematic treatment as any other quantity, distinguished the Indian approach from earlier placeholder notations and set the stage for zero's wider adoption. From India, knowledge of zero and the accompanying decimal place-value system spread westward through the Islamic world, where scholars preserved, refined, and transmitted it further. In ninth-century Baghdad, the mathematician Muhammad ibn Musa al-Khwarizmi, working at the House of Wisdom, wrote influential texts explaining Indian numerals and arithmetic techniques, including the use of zero, or sifr in Arabic—a term from which both "zero" and "cipher" ultimately derive. Al-Khwarizmi's works, along with those of other scholars in the region, carried this numeral system across the Islamic world and eventually into contact with European traders and scholars, particularly in Spain and North Africa. The figure most responsible for popularizing these numerals in Europe was the Italian mathematician Leonardo of Pisa, better known as Fibonacci, who had studied under Arab teachers in North Africa. In 1202, he published Liber Abaci, a text that demonstrated the practical advantages of the Hindu-Arabic numeral system, including zero, for commerce and calculation. Despite these advantages, adoption in Europe was neither immediate nor uncontested; some merchants and civic authorities distrusted the new symbols, and a few Italian cities briefly restricted their use in official record-keeping, fearing that figures such as zero could be altered or forged more easily than Roman numerals. Only over the following centuries did the system now taken for granted throughout the world gradually displace its rivals, carrying with it a concept once dismissed as nothing at all.