Grade 6 | Informative | Text-Dependent
Read the following excerpts from two books about the history of the number zero. Write an essay in which you explain how each excerpt focuses on a different aspect of the history of the number zero. Then explain how these aspects are connected. Use evidence from the excerpts to support your points.
by Denise Schmandt-Besserat
The most universal way of counting, the one the majority of people use today, is known as abstract counting, using abstract numbers. We separate, or abstract, the idea of “one,” “two,” “three,” and so on, from the thing we are counting. This system is very convenient because:
• Abstract numbers count anything.
• Each abstract number is expressed by a word that remains the same no matter what is being counted. (This is not so with concrete counting. In that system, number words are limited counting small amounts of only certain types of common things in daily life.)
Another advantage is that abstract numbers are infinite. For example, our largest numbers are the googol (a 1 followed by one hundred zeros) and the googolplex (a 1 followed by a googol of zeros). But if ever needed to count beyond these numbers we could keep adding zeros like this: 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...
Mathematicians think that abstract counting developed over a long period of time. Some suggest that the evolution of counting may have happened in three steps: 1. counting without numbers, 2. concrete counting, and 3. abstract counting. Ancient objects used for counting found in the Middle East support this idea.
The earliest counting devices are notched bones that were found among the remains of hunters and gatherers who lived about ﬁfteen thousand years ago in what is now the Middle East. Although we do not know what these ancient people counted with the notched bones, these counting devices may tell how they counted. Because each notch is similar to the next one and because there never seems to be a total indicated on the bones, it is likely that the hunters and gatherers had not yet developed numbers. Each notch probably stood for “and one more.”
The counters found in the villages and towns built by farmers between ﬁve thousand and ten thousand years ago were small tokens of many shapes. Each token shape was used to count only one type of thing. For example, sheep were counted with disks, but jars of oil were counted with egg-shaped tokens. (We know this because the signs for sheep and oil in early Sumerian writing pictures a disk and an egg shape.)
The fact that each different type of item was counted with a different-shaped token suggests that the early farmers had different sets of numbers to count various things. They counted concretely. They used the tokens by matching them with the number of things counted: One sheep was shown by one disk, two sheep by disks, and so on.
We owe the invention of abstract numbers to the Sumerians who lived in the first cities, in the region of present-day Iraq, about ﬁve thousand years ago. The Sumerian tablet in the man’s hand shows an account of thirty-three jars of oil. The sign on the right stands for “jar of oil.” The other signs represent numbers. Each circle is 10, and each long sign is 1.
Why is this counting system different from the others? For the first time, number and things counted were separated, or abstracted. Sheep and jars of oil were finally counted with the same numbers!
Why did it take thousands and thousands of years to invent abstract numbers? Why weren’t they invented sooner? It was not a question of intelligence: The size of your brain is the same as that of a child who lived fifty thousand years ago. Probably it was a matter of need. The simple life of hunters and gatherers required little counting, since these people lived on the animals they caught or that plants and fruits they gathered daily. The fact that it was the first farmers who invented tokens suggests that domesticating animals and plants made counting necessary. It makes a lot of sense that counting became important when the life of a community depended on knowing how many bags of grain to keep for planting the next harvest and how many animals would feed the village during the winter season.
There can also be no doubt that abstract counting was invented to cope with the development of business, trade, and taxes in the ﬁrst cities. A more precise method of counting became necessary once workshops produced quantities of pottery and tools. But it was the tax system that had the biggest impact on counting. Every month, each Sumerian had to deliver to the ruler specific amounts of fish, oil, grain, or animals. Because of this, the palace accountants had to come up with a way to keep track of large amounts of goods.
The three steps in counting, therefore, were responses to new demands brought about by the increased complexity of life.
Once abstract numbers were invented, they were used more and more widely in trade and in calculations needed for everyday life.
And with the greater use of numbers also came the need for larger and larger numbers. In the country of Sumer, the most common large number that was used in everyday life was 60. It was called “the big one,” which suggests that, at some time, it had been the highest number. But by 2500 B.C., the Sumerians’ largest number had grown up to 36,000. It was probably used very rarely and then only by palace accountants to calculate tax collections.
It was much the same for our large numbers today. The googol and the googolplex were invented in the 1950s by mathematicians who needed to do very large calculations. But we never use these large numbers in daily life. The largest numbers we read about in newspapers are in the trillions. One trillion is a 1 followed by twelve zeros.
“The History of Counting,” by Denise Schmadnt-Besserat. Copyright © 1999.
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by Charles Seife
Life without Zero
The point about zero is that we do not need to use it in the
operations of daily life. No one goes out to buy zero ﬁsh. It is in
a way the most civilized of all the cardinals, and its use is only
forced on us by the needs of cultivated modes of thought.
-Alfred North Whitehead
It’s difficult for a modern person to imagine a life without zero, just as it’s hard to imagine life without the number seven or the number 31. However, there was a time where there was no zero – just as there was no seven and 31. It was before the beginning of history, so paleontologists have had to piece together the tale of the birth of mathematics from bits of stone and bone. From these fragments, researchers discovered that Stone Age mathematicians were a bit more rugged than modern ones. Instead of blackboards, they used wolves.
A key clue to the nature of Stone Age mathematics was unearthed in the late 1930s when archeologist Karl Absolom, sifting through Czechoslovakian dirt, uncovered a 30,000-year-old wolf bone with a series of notches carved into it. Nobody knows whether Gog the caveman had used the bone to count the deer he killed, the paintings he drew, or the days he had gone without a bath, but it is pretty clear that early humans were counting something.
A wolf bone was the Stone Age equivalent of a supercomputer. Gog’s ancestors couldn’t even count up to two, and they certainly did not need zero. In the very beginning of mathematics, it seems that people could only distinguish between one and many. A caveman owned one spearhead or many spearheads; he had eaten one crushed lizard or many crushed lizards. There was no way to express any quantities other than one and many. Over time, primitive languages evolved to distinguish between one, two, and many, and eventually one, two, three, many, but didn’t have terms for higher numbers. Some languages still have this shortcoming. The Siriona Indians of Bolivia and the Brazilian Yanoama people don’t have words for anything larger than three; instead, these two tribes use the words for “many” or “much.”
Thanks to the very nature of numbers – they can be added together to create new ones – the number system didn’t stop at three. After a while, clever tribesmen began to string number-words in a row to yield more numbers. The languages currently used by the Bacairi and the Bororo peoples of Brazil show this process in action; they have number systems that go “one,” “two,” “two and one,” “two and two,” “two and two and one,” and so forth. These people count by twos. Mathematicians call this a binary system.
Few people count by twos like the Bacairi and Bororo. The old wolf bone seems to be more typical of ancient counting systems. Gog’s wolf bone had 55 little notches in it, arranged into groups of five; there was a second notch after the first 25 marks. It looks suspiciously as if Gog was counting by fives, and then tallied groups in bunches of five. This makes a lot of sense. It is a lot faster to tally the number of marks in groups than it is to count them one by one. Modern mathematicians would say that Gog, the wolf carver, used a ﬁve-based or quinary counting system.
But why ﬁve? Deep down, it’s an arbitrary decision. If Gog put his tallies in groups of four, and counted in groups of four and 16, his number system would have worked just as well, as would groups of six and 36. The groupings don’t affect the number of marks on the bone; they only affect the way that Gog tallies them up in the end – and he will always get the same answer no matter how he counts them. However, Gog preferred to count in groups of ﬁve rather than four, and people all over the world shared Gog’s preference. It was an accident of nature that gave humans ﬁve ﬁngers on each hand, and because of this accident, ﬁve seemed to be a favorite base system across many cultures. The early Greeks, for instance, used the word “ﬁving” to describe the process of tallying.
“Zero: The Biography of a Dangerous Idea,” by Charles Seife. Copyright © 2000.