Numbers of Babylon By: Charles Petersen & Tony Kaufman The Babylonian civilization was one of the most advanced civilizations in the world from 3000 BC to 500 BC. Their number system was highly sophisticated and in some ways was better than even our own. They used a base 60 sexagesimal system that utilized place holders and, if thought about correctly, is very similar to our own. However, the Babylonian system had one fatal flaw. They did not develop the place holder zero until 1000 BC. This meant that the number could mean 1, 10, 100, 1000, 10,000 and so on. The number system itself consisted primarily of cuneiform wedges. The Babylonians would make these wedges using a sharp stick or rock, chiseling into a sun baked, clay tablet. A simple up and down slash like the modern � represented one unit. Once you went past the number nine you would use tens . You would keep on going like this using tens and ones until you got past the number �� or 59 . Then you would start moving into positional notation, the basis of any modern number system. The easiest way to understand positional notation is to look at our own number system and look at how it works. When you write the number 3716 what you�re really saying is this, our base taken to the third power times three + our base taken to the second power times seven + our base taken to the first power times one + our base taken to the zero power times six. Mathematically this would be written as follows: (3( )+7( )+1( )+6( ). To write the number 3716 in cuneiform you would write �� where the first stroke indicates 1( )+1( )+56( ). The Babylonian system works exactly the same way. The only reason that we would think our base is better is because it is what we were taught and to go up a power you just have to add a zero. Here is the reason that not having a zero was such a drawback to Babylonian math, requiring the mathematician to know in what context his colleague was talking about in order to figure out the actual number. The main difference between Babylonian numbers and our was that the Babylonian system was only partially abstract and ours is purely abstract (In other words where we would use the symbol 3 they would have to write out �� and where we would just write 30 they would have to write out .) In order to do addition and subtraction the Babylonians would do basically the same thing we do, except that they would have an extra carrying stage of the ones into the tens. Multiplication was a much more difficult process, but for its time was easily and efficiently done. The Babylonians used an algorithm to carry out multiplication, this algorithm was . As you can see this formula is just more multiplication, but with a squares table, which the Babylonians had to 59 , (Of course all you really need in base 60 ) it can be performed easily and efficiently. To divide the Babylonians did not have a formula. Instead they relied on the fact that so they could just use the multiplication algorithm and a reciprocal table, which the Babylonians had to several billion (Yes you read that right billions.) The thing that made the Babylonian�s number system better than most other number systems of the time was that it was capable of showing fractions and decimals without any real trouble. They did this by using negative powers. To write 1.5 in the sexagesimal you would write this�� . The first symbol represents one, then there is the decimal separator, then the two ones after that are taken to the -1st power resulting in 1.5 . This can be done with virtually any finite number. The Babylonian fractional notation would not be bested until the Renaissance, nearly 4000 years later. The Babylonians were so accurate that they had the square root of two to three sexagesimal decimal places, or six base ten decimal places, within 0.000008 of the actual number. One problem the Babylonians ran into when using fractional notation and dividing was that certain numbers reciprocals never terminated. This is one of the number one reasons for using a base sixty number system, since sixty has 10 proper divisors. Only a few numbers are non-terminating sexagesimals. In conclusion I�d like to tell you the major achievements the Babylonians made. They were some of the first to correctly do algebra. They gave us linear(ax=b,) quadratic(ax +bx=c,) three term quadratic, and indeterminate ( ) equations. They also produced the first number theory document in Plimpton 322 , a Pythagorean tablet. They gave us the square root formula used by the Greeks and the Romans. They had tables with the sexagesimal equivalents of squares to 59 , cubes to 32 and reciprocals to several billion. They had tables of logarithms and anti-logarithms for many numbers.