Finding Zero Read Online Free Page A

this is merely a hypothesis, but it is a likely one.
    The Ishango bone is certainly the best early example of pre-counting. But there is some evidence that European humans may also have used some kind of pre-counting: In addition to the Ishango bone, several animal bones with markings that are likelytally counts have been discovered in Europe and also dated to the Paleolithic. 1
    A later piece of evidence for something resembling counting comes from the mysterious Neolithic stone arrangements, believed to be about 6,000 years old, at Carnac on the coast of Brittany in France. Interestingly, the megaliths are found in groups, and the groups are often comprised of a prime number of megaliths: 7, 11, 13, and 17. Was this by chance, or does it represent a form of counting? Does it, perhaps, represent an even deeper understanding of numbers? We don’t know. Carnac is a mystery that archaeology has never been able to explain, despite attempts over many decades. Nobody knows why so many very heavy stones were placed in rows. Perhaps a connection exists with Stonehenge, where similar stones were placed in circles around the same time.
    But Ishango and Carnac do not present what we consider numbers. In everyday life, numbers began with simple representations of a quantity by people using their fingers. Aristotle wrote two-and-a-half millennia ago, “Or is it because men were born with ten fingers and so, because they possess the equivalent of pebbles to the number of their fingers, come to use this number for counting everything else as well?” 2 And since we also have ten toes, early societies used these as well to count beyond ten. Remnants of such a base-20 number system that existed long ago can still be seen in French, through words such as quatre-vingt (four twenties) for 80.
    Clearly, the form of counting we use evolved from an accident of nature: our having five fingers on each hand and five toes on each foot. Indeed, in some languages, such as Old Khmer, five was used as a point to anchor the other numbers: After four comes five,then five-and-one, five-and-two, and so on to ten, which becomes the next anchor. When we look at the early European numbers, the Roman numerals, we see the same trend: The Roman numbers IV, V, VI, VII, VIII, are all anchored at five (V), and only after eight do we come to measure them in relation to ten: IX, X, XI, XII, XIII. So the use of five and ten as key numbers evolved in different parts of the world.
    In ancient India, the number 10 was the anchor. Decimal numbers were evident very early on there, as early as the sixth century BCE as observed in some inscriptions. And once they learned the powers of ten—simply by using ten fingers, and then using ten fingers ten times, as in ten people, one for each finger, each holding up his or her ten fingers—the Indians of antiquity understood that this process can go on forever. Ten times ten people holding their fingers up was ten to the third power, and ten times ten times ten people with ten fingers each was ten to the fourth power, and so on without a limit.
    During the Han Dynasty in China (206 BCE to 220 CE), a mathematical work titled Nine Chapters on the Mathematical Art appeared, employing both positive numbers, colored red, and negative ones, colored black. And in Egypt of the third century, a leading Greek mathematician named Diophantus obtained negative answers to some of his equations but immediately dismissed them as unrealistic. So the idea for negative numbers is quite old, but people did not understand such numbers. The double-entry bookkeeping system used in accounting today was developed in Europe in the thirteenth century in part to avoid using negative numbers. To define negative numbers requires the concept of zero.
    Negative numbers are, in a sense, a reflection across zero of the positive numbers. You can see this if you draw the number line, starting at zero and going to the right to 1, 2, 3, and so on; and