By: Gabriela Cortese

and Lacey Inglis Babylonian Algebra The first records of Babylonian texts

appear in the Old

Babylonian Period, 1800-1600 BC. The methods of the Babylonians were some of

the very first, but their mathematicians solved

various problems in modern ways. One of the foremost civilizations in the mathematics world, the ancient Babylonians started ideas that evolved into elementary mathematics, forever etching themselves in further mathematic study and theory. Students being coached in the Babylonian methods would be given everyday

problems to expand their knowledge in algebra such as measuring creeks, weighing items, finding the number of some objects, etc. The Babylonians essentially only taught by example; with no real explanation or formula (with the exception of the quadratic formula) for why an answer was correct or not. Instead of manipulating symbols within formulas like the algebra we do the Babylonians would follow procedures to find the answer much like algorithms today. To solve quadratic equations the Babylonians would basically use the quadratic formula. They thought of quadratic equations where b and c were not inevitably integers but c was always positive. Babylonians would always use the positive roots because they believed that it made sense in the case of solving “real” problems. Quadratic Equations, like the one shown previously, would be used to find the dimension of a rectangle when you have the area and the amount that the length is greater than the width. In Babylonian mathematics, several formulas were different.

For example, a circle often had a different formula, c to the second power divided by 4 and 1/4. For Babylonians, the area of a circle was not the radius all the way around, but rather simply the area contained in the lines. Even if the diameter was known, it was calculated by the circumference. Babylonian mathematics greatly helped in advances in astronomy. The most important work of the Babylonians was their creation of the “Astronomical Diaries,” a work that graphed and calculated the positions of a great few celestial bodies. Babylonian algebra is sexagesimal, meaning that the base is 60. The Babylonians promoted this type of numeral system because 60 is a highly composite number. This aspect of Babylonian algebra is still in use today in the number of minutes in an hour and seconds in a minute, as well as the 360 degrees in a circle. The Babylonian's

ultimate algebraic goal was the computation of a number. Babylonian algebra is in fact divided into two separate types. One from the Old Babylonian Period, and the other from their predecessors, the Sumerians. Many Babylonian everyday problems consisted of area and quadratic equations because of partitioning of land. Square Roots: Babylonian's used the method of the mean to solve square roots. For example, if the square root of 2 needed to be found, per say, choose x as the first approximation and for another 2/x. The product of the two numbers is 2, and furthermore, one must be less than and the other greater than 2. Use the average to find a value closer to the square root of 2.

1. Take a = a1 as an initial approximation.

2. If a1 is smaller than the square root of 2, then 2/a1 is greater than the square root of 2. Quadratic Equations: The Babylonian method for solving quadratics was based on completing the square. These formulas were much less organized than today's, though similar, because Babylonian's only found positive solutions. Negative solutions and negative numbers in general were not put into use until several centuries later. The students problem would be to solve the equation using a prescribed method, one of the few Babylonians had. Cubics: Babylonians were also able to solve cubics, though only with positive solutions. However, the form of solving was highly restricted. For example, solving x3= a was accomplished using tables and interpolation. Mixed cubics (x3+ x2= a) were also solved using tables and interpolation. The general cubic (ax3+ bx2+ cx = d) can be reduced to the normal form y3+ ey2= g. Linear Equations: Linear Equations were solved by reducing a two variable problem to only one, in an elimination process. 4000 BC Fun Fact: Because of the amount of students in algebraic study, an enormous number of clay tablets have been found. Because of algebra, therefore, clay was a vital aspect of Babylonian economy, as some clay tablets were large enough to contain up to 200 problems. Quadratic equations, an aspect of algebra, were used incredibly often in public circumstances, such as partitioning of land. Algebra made this task significantly easier, because it found an exact size. The Sumerians inhabit

Mesopotamia. 1800-1600 BC Mathematical Texts from the Old Babylonian Period. This one contains the geometrical themes of the Old Babylonian Period.

Babylonians contributed greatly in the discovery of the Pythagorean Theorem. Babylonians are credited with properly implementing the principle of the Pythagorean Triples. The Akkadians invaded Babylonia

around 2300 BC. This invasion was relevant

in the world of mathematics because the

Akkadians created an abacus for counting,

as well as developing forms of arithmetic. Babylonians, whose numerical system was sexagesimal, established the modern increments of time. This system was put in place in the Old Babylonian Period, although its roots came from the earlier Sumerians. Kidinnu, also spelled Kiden, Kidenas, or Cidenas, was a Babylonian mathematician and astronomer, and he was possibly the only name known from the Babylonian period associated with math. He was one of the key contributors of his time period, as he helped to clarify ideas of solar motion and the moon, as well as graphing and calculations in astronomy. During the second half of Babylonian algebra's development period, around 4500 to 4000 B.C., the more geometric works of the Babylonian mathematics were reflected in the artistic yet mathematical patterns on the faces of buildings. Pottery, masonry, and architectural construction was flooded with rich designs, carefully measured and calculated by Babylonian mathematicians. Algebra was used in Babylonian lifestyle in commerce, interest, tax dividends, harvest, surplus accommodations, calenders, and canals. 3000 BC Modern day division is a Babylonian concept. The Babylonians did not have a logical procedure for solving long division so instead they used this equation with a table of reciprocals. a/b = a x 1/b Pi was estimated by the Babylonians. Their resulting number was 3.125, a number at the forefront of Babylon's mathematical intelligence. These tablets explain fractions, algebraic, quadratic, and cubic equations, and some Pythagorean theorem. Fun Fact: Babylonian mathematics relates to any math used by the people in Mesopotamia from the early Sumerians to around 539 BC when Babylon fell. The Babylonians divided the day into 12 hours, each with 60 minutes. Therefore, two of our minutes would equal one for the Babylonians. The Babylonians were somewhat more advanced than the Egyptians in their mathematics, they used no zero, the could solve linear systems, they could extract square roots, and they studied things such as circular measurement. A clay tablet known as Plimpton 322 which was believed to date back to around 1800 BC shows that the Babylonians may have discovered that the square of the hypotenuse equals the sum of the square of the other two sides. Many years before the Greek Pythagoras. Fun Fact: Since the Babylonian civilization was rediscovered many astronomers and mathematicians from other societies borrowed their strategies.

Only two symbols were used by the Babylonians to show the numbers up to 50. One is the numeral that looks like a slingshot, while the other looks as if you were looking at the slingshot from the top, while it has something in it. Sumer (what is now modern-day Iraq) was where writing, the wheel, agriculture, the arch, and many other innovations started. It is also referred to as “the Cradle of Civilization.” Fun Fact: There is one Babylonian tablet that gives an approximation of two squared to five decimal places. Because of the Babylonian’s sexagesimal numerical system they figured that a circle contains 360 degrees. A large fragment from a tablet was found and it contained the question: “The side of the square equals one. I have drawn four triangles in it. What is the surface area?” Babylonian schools would teach young scribes geometry so they were able to write deeds and calculate agricultural yields. Fun Fact: The Babylonians disregarded zero as just a place-value holder than more of a value itself. The Sumerians adopted a form of writing that was based on cuneiform symbols. These symbols were written on tablets made of clay that baked in the sun. The Babylonians later adopted the same type of writing. The Babylonian calendar was a lunisolar calendar that contained twelve months. Each of these months would begin when a new crescent moon was found on the western horizon at sunset. There are very few well known Babylonian mathematicians but two were known as Nabu-rimanni and Kidinu. They were both believed to live around 490-480 BC. It was likely that these mathematicians worked together to develop new ideas and formulas.

Clay tablets dating from around 2100 BC showing a problem using the area of an irregular shape. These were the sorts of tablets that students would do school work on. Modern Day Discoveries This tablet was used for astronomy. It was used for mapping the stars. A tablet that has been thoroughly examined. It contains carvings that indicate a mathematical problem that was used on young scribes. Babylonian mathematics were until recently characterized as a collection of theories that appeared to be measured by trial and error. Babylonian algebraic texts are the only ones that can actually be analyzed, as the rest(geometric, some astronomic) are too short. The End

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# Babylonian Algebra

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