The Babylonians left a space for a zero, a procedure open to much interpretation. At no point in the transaction is there a need to write out a document "A repaid the loan to B and now owes B zero sheep." The need for a zero does not arise until the introduction of a place-value system. If someone takes out a loan, the amount of the loan is stamped into a clay tablet, and the tablet is destroyed when the loan is repaid. Commerce and business also get along well without a zero. Before we come to the rash conclusion that our contemporaries - and maybe we with them - are smarter than the people of ancient Babylon, let us remind ourselves that people only develop what they need for their daily lives.Ībsolute number systems have no need for a zero: In Roman notation the number 1006 is written as MVI. We are today so accustomed to using a place-value system with a zero that it appears difficult to imagine how people could have lived without the concept of "nothing". This is evident from Babylonian mathematical textbooks which, among many example calculations, contain equations such asĢ0 - 5 - 5 - 5 - 5 = We are running out of grains." -Įvery equation with a zero result receives an evasive answer, with the occasional allusion at the old system of counting by association with pebbles or grains. As strange as it may seem, the Babylonian scientists not only had no sign for zero, they had no concept of zero. The much bigger problem, however, was the lack of a pictograph for "zero". The Babylonian notation is often not very specific, and calculation errors must have been a regular occurrence. If this number is copied without accurate attention to the spaces it can appear as "1 2 5 12 2" or as "1 25 1 22" - obviously many mistakes are possible. To demonstrate the problem, imagine a number that uses four places, with the corresponding allocations "1 25 12 2". Correct spacing of the pictographs in a Babylonian number was therefore of utmost importance. The Babylonian system still required several symbols to indicate the value of each power of the base 60 in a number. To begin with, our modern notation indicates every digit in a number through a single symbol (one from the set of numerals 0 - 9). There is no doubt that science in Babylon made rapid progress after the invention of the place-value system. The admiration of the place-value system expressed by the French mathematician Pierre-Simon Laplace in the early 19th century is an acknowledgment of the genius exhibited by some unknown scientists of Babylon: "The idea is so simple that this very simplicity is the reason for our not being sufficiently aware how much admiration it deserves." It used the same principle that we use today when we write our numbers in decimal notation: 1859ġ (x 10 3) + 8 (x 10 2) + 5 (x 10) + 9 (x 1)Ĥ (x 10 3) + 8 (x 10 2) + 1 (x 10) + 8 (x 1)īabylonian scientists could therefore use the same simple rules for multiplication and long division that we use some 4,000 years later. Instead of building a number by adding as many basic number elements as necessary they began to build them by adding powers of the number base: Two examples of Sumerian and Babylonian number notation The Babylonian scientists adopted the Sumerian cuneiform script for their numbers but changed the way of writing them down drastically. (The bible mentions Babylon's 91 m tall ziggurat (temple pyramid) as the "Tower of Babel", built to defy the gods through its height.) It developed into the biggest city of the world of its time, and its temples and public buildings reached world fame. Why did it not survive? Why did quite advanced civilizations struggle on with absolute value systems until at least 200 BC, if not later? The number system of Babylonīabylon became the capital of southern Mesopotamia in about 1780 BC. We see from the list that the Babylonian place-value system preceded all others by nearly two thousand years. Central America by the Maya (around 400 - 600 AD).Mesopotamia by the Babylonians (1800 BC).Place-value systems were developed independently in All we need to know is the basic multiplication table and the rules for addition: In a position-value system the operation follows very simple rules. Multiplication of CCXII with XVI is obviously not straightforward. To see the giant leap forward let us try a simple calculation: How do you multiply 212 by 16? In Roman numerals this is written as CCXII * XVI. The principle appears so simple that it is hard to believe how long it took before a human mind came up with the idea. We all use a place-value number system every day without realizing what a magnificent invention it is. The last lecture foreshadowed the development of place-value number systems as one of the greatest achievements of the human mind. Science, civilization and society Lecture 5 Place-value number systems: The number systems of Babylon, China and the Maya civilization.
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