## Mathematics ( 4000 BC – 539 BC )

### Babylonian Mathematics refers to mathematics developed in Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC and is especially known for the development of the Babylonian Numeral System

#### What was math used for?

Sumerian mathematics initially developed largely as a response to bureaucratic needs when their civilization settled and developed agriculture (possibly as early as the 6th millennium BCE) for the measurement of plots of land, the taxation of individuals, etc. In addition, the Sumerians and Babylonians needed to describe quite large numbers as they attempted to chart the course of the night sky and develop their sophisticated lunar calendar.

They were perhaps the first people to assign symbols to groups of objects in an attempt to make the description of larger numbers easier. They moved from using separate tokens or symbols to represent sheaves of wheat, jars of oil, etc, to the more abstract use of a symbol for specific numbers of anything. Starting as early as the 4th millennium BCE, they began using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. Over the course of the third millennium, these objects were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text. A rudimentary model of the abacus was probably in use in Sumeria from as early as 2700 – 2300 BCE.

image source: http://www.florentintise.com/abacus/

#### the numeric system

Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand the five fingers on the other hand. Unlike those of the Egyptians, Greeks and Romans, Babylonian numbers used a true place-value system, where digits written in the left column represented larger values, much as in the modern decimal system, although of course using base 60 not base 10. Thus, 1 1 1 in the Babylonian system represented 3,600 plus 60 plus 1, or 3,661. Also, to represent the numbers 1 – 59 within each place value, two distinct symbols were used, a unit symbol (1) and a ten symbol (10) which were combined in a similar way to the familiar system of Roman numerals (e.g. 23 would be shown as 23). Thus, 1 23 represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context.

It has been conjectured that Babylonian advances in mathematics were probably facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 – in fact, 60 is the smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12 pence, 2 x 12 hours, etc).

The Babylonians also developed another revolutionary mathematical concept, something else that the Egyptians, Greeks and Romans did not have, a circle character for zero, although its symbol was really still more of a placeholder than a number in its own right.

The accepted reason for the use of a sexagesimal system is that it was based in astronomy and the desire of the Babylonians to develop accurate calendars to chart the turning of the seasons and predict the best times for planting, extremely importantly in a culture with a strong agricultural base. Initially, the Babylonians believed that there were 360 days in a year, and this formed the basis of their numerical system; they divided this into degrees and this represented the daily movement of the sun around the sky. They then transferred this into measuring circles by dividing degrees into minutes. Our entire system of astronomy, geometry, and dividing the day into hours, minutes and seconds hails from this period of history.

#### fractions

In the sexagesimal system, any fraction in which the denominator is a regular number (having only 2, 3, and 5 in its prime factorization) may be expressed exactly. The table below shows the sexagesimal representation of all fractions of this type in which the denominator is less than 60. The sexagesimal values in this table may be interpreted as giving the number of minutes and seconds in a given fraction of an hour; for instance, 1/9 of an hour is 6 minutes and 40 seconds.

Fraction: Sexagesimal: Fraction: Sexagesimal: Fraction: Sexagesimal: 1/2 1/3 1/4 1/5 1/6 1/8 1/9 1/10 30 20 15 12 10 7,30 6,40 6 1/12 1/15 1/16 1/18 1/20 1/24 1/25 1/27 5 4 3,45 3,20 3 2,30 2,24 2,13,20 1/30 1/32 1/36 1/40 1/45 1/48 1/50 1/54 2 1,52,30 1,40 1,30 1,20 1,15 1,12 1,6,40

However numbers that are not regular form more complicated repeating fractions. For example:

1/7 = 0;8,34,17,8,34,17 … (with the sequence of sexagesimal digits 8,34,17 repeating infinitely many times) = 0;8,34,17
1/11 = 0;5,27,16,21,49
1/13 = 0;4,36,55,23
1/14 = 0;4,17,8,34
1/17 = 0;3,31,45,52,56,28,14,7
1/19 = 0;3,9,28,25,15,47,22,6,18,56,50,31,34,44,12,37,53,41

The fact in arithmetic that the two numbers that are adjacent to sixty, namely 59 and 61, are both prime numbers implies that simple repeating fractions that repeat with a period of one or two sexagesimal digits can only have 59 or 61 as their denominators (1/59 = 0;1; 1/61 = 0;0,59), and that other non-regular primes have fractions that repeat with a longer period.

#### From numbers to geometry

The idea of square numbers and quadratic equations (where the unknown quantity is multiplied by itself, e.g. x2) naturally arose in the context of the meaurement of land, and Babylonian mathematical tablets give us the first ever evidence of the solution of quadratic equations. The Babylonian approach to solving them usually revolved around a kind of geometric game of slicing up and rearranging shapes, although the use of algebra and quadratic equations also appears. At least some of the examples we have appear to indicate problem-solving for its own sake rather than in order to resolve a concrete practical problem.

The Babylonians used geometric shapes in their buildings and design and in dice for the leisure games which were so popular in their society, such as the ancient game of backgammon. Their geometry extended to the calculation of the areas of rectangles, triangles and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders (although not pyramids).

The famous and controversial Plimpton 322 clay tablet, believed to date from around 1800 BCE, suggests that the Babylonians may well have known the secret of right-angled triangles (that the square of the hypotenuse equals the sum of the square of the other two sides) many centuries before the Greek Pythagoras. The tablet appears to list 15 perfect Pythagorean triangles with whole number sides, although some claim that they were merely academic exercises, and not deliberate manifestations of Pythagorean triples.

#### who inherited and developed this knowledge?

The Sumerians, Babylonians and other inhabitants of the Euphrates valley certainly made some sophisticated mathematical advances, developing the basis of arithmetic, numerical notation and using fractions. Their work was adopted by the Greeks, and it is likely that the Greeks learned mathematical techniques from the Babylonian culture, as ideas traveled along the Silk Route from Anatolia (Turkey) to China. Alexander the Great is known to have sent astronomical records from Babylonia to Aristotle after he conquered the area. Their knowledge passed to the Greeks and formed the basis of pure mathematics as the master manipulators of numbers, the Greeks, took this knowledge and began to explore the relationships between numbers.

info sources: https://en.wikipedia.org/wiki/Sexagesimal

http://www.storyofmathematics.com/sumerian.html

https://explorable.com/babylonian-mathematics