ZERO

1. ZERO The story of how it finally came to be a citizen in The Land of Numbers

3. It took twelve centuries for zero to appear in western civilization. Our earliest ancestors did not need numbers either in concept or symbol.

4. They used various methods of one to one correspondence between the thing they wanted to keep track of and a simple object. Clay Balls Knots Stick Hand Illustrations taken from From One to Zero: A Universal History of Numbers by Georges Ifrah

5. Knots were used by primarily by the original inhabitants of Bolivia, Peru, and Ecuador; the Incas. The Incas used knotted ropes called quipus, to keep records for their complex accounting system. Tally sticks were made of wood or bone. They were first used by shepherds to keep track of the number of animals in their flocks. Each animal in the flock was represented by either a string tied around the stick or a notch cut in the stick. The use of tally sticks became widespread and their use persisted after the establishment of banking institutions and modern numeration. It took an act of the British Parliament to abolish their use as an official accounting and contract system. It was the burning of all the accumulated tally sticks that caused the fire which destroyed the old Houses of Parliament in 1834.

6. Intricate finger counting systems were developed. The system represented by this drawing uses each joint in the four fingers to represent an object so that twelve objects can be represented on each hand. The thumb is used to indicate the operation being performed between the objects represented by the hands. The clay envelope system was used by wealthy herd owners to ensure that shepherds arrived at their destination with all the animals belonging to the herd. The clay envelope was loaded with small clay balls, one for each animal with which the shepherd had been entrusted. The envelope was sealed and the owner or another person with writing skills indicated the contents of the herd on the envelope’s outside surface. When the shepherd arrived at his destination, the envelope was broken open and the clay balls inside and the inscription on the envelope were compared with the herd.

7. Concrete counting methods were used until about 3100 B.C. At this time Upper Egypt and Lower Egypt were united under the ruler Menes.

8. In order to govern such a large kingdom supplies of food and commodities had to be counted. Taxes had to be assessed and collected.

9. Egyptians began to use picture symbols, hieroglyphics, to represent the concrete objects that had been used for counting. A different picture was used for each power of ten. Representing concrete objects with symbols is called cipherization. Cipherization was a giant step forward in the development of numeration. The Egyptian number system was additive. This means that the symbols could be in any order. The person reading the symbols summed the values represented by each symbol to arrive at a total. The Egyptians developed mathematics along with their numeration system. They had algorithms for addition, subtraction, multiplication, division, and computing with fractions. As their civilization advanced, the Egyptians began to use symbols that were faster and easier to write. But, they continued to use mathematics for practical purposes and their numeration system remained additive. The Egyptians never had a symbol for zero because their use of numbers never required one.

10. The Egyptian Number Symbols and their decimal equivalents Take a few minutes and see if you can translate the Egyptian symbols below into their decimal equivalent.

11. 1232 1232 1232 Did you get the same answer for all three? Egyptian numerals were not positional like our decimal system. It was additive; you simply add the values of all the symbols to arrive at the total.

12. The yearly inundation of the Nile carried masses of soil from one location to another. A landowner’s holdings changed shape and size after each inundation. The Egyptians developed formulas for measurement so they could recalculate each landowner’s property tax bill after each inundation.

13. In addition to developing formulas, Egyptians took the lead in developing methods of measurement. Egyptian rope stretchers like the ones pictured above were widely recognized as the most accurate measurers in the ancient world. Many of the methods they developed are still in use today.

14. The Babylonian Empire developed at about the same time as the Egyptian Empire. While the Egyptians had few visitors due to the deserts surrounding them, the Babylonians interacted with many other cultures due to their proximity to several large bodies of water. The Babylonians developed a base sixty number system (sexagesimal). This system freed them from much of the drudgery of calculations so they were able to develop a more complex system of mathematics. The Babylonians used their mathematical system to perform astrological measurements and calculations, and to develop some theoretical mathematics such as the quadratic equation. Babylonian numeration was also superior in another way, it was positional.

15. The development of a positional system was a huge step in the advancement of mathematics. Our own base 10 numeration system is positional. • Positional Systems are important because: • There are fewer symbols to memorize; • There is less writing; • Calculations are faster and easier to perform. There was one very important difference between the Babylonian positional system and the one we use today: The Babylonians did not use a place holder to indicate empty positions in a number.

16. At first the Babylonians relied strictly on context to determine if a written number was 6 or 60. We still rely on context in some situations today. Note the real estate adds to the right. The seller is counting on the fact that most of us would realize that the price starts at 170 thousand dollars, not at 170 cents. We also rely on context when we fill up our car and the person at the counter says “ that will be 30 please”. Few of us would give the clerk thirty cents.

17. Later they began to leave a space when no symbol occupied a particular position. But, this still did not solve the problem of denoting a blank space at the end of a number. Nobody knows! There are no contextual clues and no place holders. Does this symbol stand for 1, 602, or 603 ?

18. Humans can count, measure, develop formulas, and solve quadratic equations. But, they have not developed a way to represent nothing!

19. By about 600 B.C. the government of Greece had increased its land holdings to include Egypt and Babylonia. As the Greeks conquered new lands, they assimilated the aspects of the conquered land’s culture that they admired into their own. They were then very successful in spreading this combined culture throughout the Mediterranean region. The ability of the Greeks to assimilated and then spread culture had a huge impact on the advancement of mathematics. They transformed all the rules of calculation that they had assimilated into an orderly, unified system of mathematics.

20. The Greeks were the first civilization to pursue knowledge for its own sake. They refined and formulated many of the mathematical principles we use today. Did you notice that these examples of Greek mathematics all deal with geometry? This is not a coincidence, the Greeks believed that all mathematics should be based on geometry. Pythagorean Theorem The first geometry book The Conics of Apollonius

21. The Greeks adapted their alphabet to use as numbers. For example : Α = 1 Β = 2 Γ = 3 Δ = 4 ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ See the pattern? The number the letter represented was determined by its position in the alphabet. So…since humans usually start at one when ordering things, there was no letter to represent zero. But, there was a more important reason as to why the Greeks never had zero. Remember that their mathematics was based totally on geometry? Well, you cannot have the side of a geometric figure with length of zero can you?

22. Although the Greeks developed the mathematics that helped us to launch rockets into space, build computers, and make purchases on the internet; they never developed a zero!

23. The Romans eventually took over most of the Greek land holdings. The Romans believed only in using mathematics for practical applications such as tallying taxes and building bridges. They already had numbers and methods for accomplishing these tasks so Greek mathematics was no longer used.

24. Remember working with Roman Numerals in elementary school?

26. Humans now have a symbol for nothing! But, it was not immediately assimilated into the formal number system. Remember, the Romans only used mathematics for measurement and counting and their number system was additive. There was no place for zero in official Rome.

27. Eventually, the western Roman Empire was overrun by barbarians . The eastern Roman empire was spared these invasions and it was here that the mathematics developed previously was preserved. It was in India and China that new mathematical concepts were being developed. India

28. Three Indian mathematicians, Brahmagupta, Mahāvīrā, and Bhāskara were the first to explore the use of zero in mathematics. They developed rules for adding and subtracting and multiplying zero, although they never agreed on a rule for division by zero. India, around 830 A.D. By treating zero as a number, the Indian mathematicians are beginning to redefine the concept of number. Numbers are no longer just symbols tied to objects, they are becoming things in and of themselves.

29. India made another important contribution to mathematics, easier to write number symbols. The symbols gradually evolved into something similar to the ones we use today.

30. These numbers were first discovered in the West in the 1100’s. They were found in a book by the Arabic mathematician , al-Khowârizmî, so they became known as Arabic numerals. We now refer to them as Hindu-Arabic numerals because we know they originated in India.

31. Fibonacci was the son of an Italian diplomat assigned to north Africa. He was educated there so he learned the mathematical systems developed in India and the Arab countries. Fibonacci saw that these systems were far superior to those being used in Europe during this time. He felt that these systems were especially suited to commerce. He returned to Europe and wrote a book, Liber Abaci, which advocated the use of the eastern mathematical systems and Hindu-Arabic numerals. He included zero in his writings, but considered it a sign. Likewise, Fibonacci did not transmit the realization reached by Brahmagupta, Mahāvīrā, and Bhāskara that zero was on equal footing with other numbers. 1175 - 1250

32. The people of Fibonacci‘s time were resistant to change and they had little need for Fibonacci’s new numbers. “Millers showed the kinds and amounts of flour in their sacks by the way they knotted the drawstrings; and everyone, merchant and banker, sophisticates and illiterates, knew how to reckon sums up to a million with their fingers.” Quote taken from: Kaplan, R (1999).The nothing that is: A natural history of zero. New York, N.Y.: Oxford University Press.

33. Which one will win? The art of calculating and Hindu-Arabic numerals must prove their superiority above the other methods. Contestant Number One Contestant Number Two

34. The art of calculating, the Hindu-Arabic numerals, and zero seemed to have no hope of winning the contest until double entry bookkeeping appeared in Italy around 1340. Italy

36. In order for an account to be balanced using the double entry method of bookkeeping, the total in both the debit and credit column must equal zero. People began to think of zero as the dividing line between positive and negative. By requiring that everything be assigned a numerical value number sense was increased.

37. Zero finally has a ticket to the land of numbers!

38. The information in this presentation came from the following sources: Kaplan, R (1999). The nothing that is: A natural history of zero. New York, N.Y.: Oxford University Press. O'Connor, J.J, Robertson,, E.F. (2000). A history of Zero. Retrieved September 18, 2006, from The MacTutor History of Mathematics archive Web site: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html Miller, J. (2006,July 4). The Math Forum at Drexel. Retrieved October 23, 2006, from Earliest Known uses of Some of the Words of Mathematics(Z) Web site: http://members.aol.com/jeff570/z.html Miller, J. (2006,July 1). The Math Forum at Drexel. Retrieved September 18, 2006, from Earliest Uses of Symbols for Constants Web site: http://members.aol.com/jeff570/constants.html Ifrah, G. (1985). From one to zero. New York, N.Y.: Viking Penguin, Inc.. Derbyshire, J (2006). Unknown quantity: A real and imaginary history of algebra. Washington, D.C.: Joseph Henry Press. . Burton, D. M. (2007). The history of mathematics an introduction. New York, N.Y.: McGraw Hill. The End