To what Extent did Babylonian Mathematics Relate to the Egyptians? Thoughts European Algebraists

Introduction

The contribution of Babylonian and Egyptian mathematics to today’s mathematics. Throughout history, an innumerable amount of mathematical theories was thoroughly developed, and though not all were proved, these concepts have been considered the outline for the mathematics practiced today. Millenniums before Christ, each existing tribe distinctly contributed their thoughts and ideas to the massive measure of calculations currently in use. Most of these have been carefully modified, however, for us to comprehend the mathematical procedures taken by ancient groups. Two of the most famous tribes were the Babylonians and the Egyptians and both Mesopotamian civilizations have been compared to one another. Although one was better and more advanced than the other in many aspects, they both provided necessary information that led to the introduction of mathematics.

Number System

Each empire significantly contributed to the mathematics concept, but Babylonian math is considered “more ‘serious’ than Egyptian” (Hodgkin, 2005) since they were more advanced in the problems they solved. Their mathematics, “written in a dead language” (Hodgkin, 2005) known as “Sumerian or Akkadian or a mixture” (Hodgkin, 2005) of both, were displayed on a multitude of clay tablets that have been found throughout history. Thus, not everything experts have translated is correct, since these translations are merely human attempts to interpret a complex language unknown today. Because they were “both difficult and in some sense useless” (Hodgkin, 2005), all mathematical processes Babylonians created and helped solve “have been filtered through particular preoccupations of Neugebauer” (Hodgkin, 2005). Despite this, a miniscule amount of what “has ever been created has been discarded” (Kline, 1962), meaning that everything in existence today was created using the adjusted theories and formulas the Egyptians, the Babylonians, and several other empires created.

During the time these existed, Babylonians occupied what is currently known as “modern Iraq” (Kline, 1962). Here they endeavored to create the beginning steps of the miscellaneous mathematics that successfully became a part of the subject. One of their most significant contributions was their practice with numbers and the different operations they managed to do with them. In modern days, numbers are written “in a ‘place-value’ system” (Hodgkin, 2005) and, similar to the Babylonians’, the symbols in use are 1, 2, 3… 9, the use of “a ‘zero’ sign” (Hodgkin, 2005) is the only exception. Babylonians, “the inventors of the notation of numbers” (Cajori, 1919), needed but did not use a figure for 0. Instead of a symbol, they used an “o” to “indicate the absence of a figure” (Cajori, 1919) in rare instances. “Depending on where it is placed” (Hodgkin, 2005), the value of a number undeniably alters if another digit is added in front or following it. For instance, (‘2’ has a distinct value from .2, .02, .002, etc.) (Hodgkin, 2005). This is called the “decimal notation” (Cajori, 1919) system, meaning that a base 10 is incorporated in all numbers, allowing decimals and fractions to be created. Before this base emerged, however, base 60 was invented by the inhabitants of the Mesopotamian region, Babylonia. Unalike us, they used a “sexagesimal scale” (Cajori, 1919) instead, a system which integrated a base 60 in each integer. For example, “1.4= 60+4” (Cajori, 1919) is equivalent in value to “1.4= 82” (Cajori, 1919), where each digit’s “1” symbol was used to represent the base number, 60 (Cajori, 1919). This base was not only utilized to find angle measurements, but also geographical coordinates and the time of the day as well. Every time a clock or a watch is read today, base 60 facilitates the understanding of it. Babylonian fractions also used their sexagesimal system. In “A History of Mathematics” by Florian Cajori, he uses ½ and 1/3 as an example. Both fractions are “sixtieths”, meaning they are only a simplified version of fractions in base 60 since, 1/2= 30/60 just like 1/3= 20/60 (Cajori, 1919).

Various theories exist with the attempt of explicating the uncertainty of the origination of the sexagesimal scale and its progression into decimal notation. Its purposes are unknown as well but there are many suppositions as to why and how it was created. In the years before Christ, “it was supposed that the early Babylonians reckoned the year at 360 days” (Cajori, 1919), therefore, they developed the assumption that a circle, undivided, was a total of 360 degrees, a degree for each day of the year. It is also presumed that Babylonians had discovered that the circumference of a circle was equal to three times its diameter or six times its radius, this led to them “dividing it into 6 segments of 60 degrees each” (Cajori, 1919). With these calculations, it is believed that, at one point, the sexagesimal notation would have had been developed.

Everything Babylonians created in the field of mathematics or even began to discover was notated in base 60. They “had employed positional notation for fractions, but they used base 60 rather than base 10” (Kline, 1962). “Their empire survived until the 7th century BC, when it in turn was engulfed by the Assyrians” (McLeish, 1994) and, after that, base 10 was introduced. Since “it was built on the base 60, the Babylonian system itself was cumbersome to use” (Devlin, 2000), which did not allow it to gain popularity within other civilizations. And because the level of perplexity it spread amongst the people was in an excessive measure, base 10 was developed in the sixteenth century by “European algebraists” (Kline, 1962). Although “it still survives in our practice of dividing hours and angles into 60 minutes and 60 seconds” (Kline, 1962), base 10 started emerging “by the use of fingers” (Struik, 1948). This base helped introduce fractions and allowed fractions to be converted to decimals. For instance, “twelve was conceived as 10+2, or 9 as 10-1” (Struik, 1948). Despite the number, 10 was always considered the base and from there, another integer was either added to it or subtracted from it. However, “the disappointing feature of the decimal representation” (Kline, 1962) was that not the entire group of fractions could be converted into a decimal, examples of those could be “1/3” (Kline, 1962), 2/3, 1/7, 2/7, etc. A multitude of other fractions are ones that do not have a firm number of digits after its decimal point and, therefore, an infinite number of fractions will have to be added to a whole number to obtain the specific fraction since the “more and more decimal digits” (Kline, 1962) added, the “closer and closer to the fraction” (Kline, 1962).

On the other hand, the Egyptian number system had never consisted of the sexagesimal base. Instead, they calculated mathematics with a “decimal system without positional notation” (Imhausen, 2016). This means that instead of using the current numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, where the “absolute value is determined by its position within the number” (Imhausen, 2016), their system used “individual symbols for each power of 10” (Imhausen, 2016). They “indicated each higher unit by a new symbol” (Struik, 1948) and used hieroglyphic signs to represent these different units of tens they needed for their calculations. The “1” symbol was, most certainly, invented “from marks on a tally stick” (Imhausen, 2016), or it could also be a form of drawing one single finger when it was held up, since the decimal notation emerged because it consisted of using one’s fingers. Egyptians’ “systematic use of fractions of the form ” (Heaton, 2017) is shown in the “Ahmes Papyrus” (Heaton, 2017) just like their impressive abilities to “solve problems that involve unknown quantities” (Heaton, 2017). Egyptians made possible the division of a number into another digit when they “invented new numbers” (Félix, 1960), for that, they have been recognized.

Quadratic and Cubic Equations

“Babylonians used ‘tables’ for a great number of procedures: multiplication, division, fractions, square and cube roots, and much more” (McLeish, 1994). Apart from the sexagesimal scale, Babylonians’ “resourcefulness in algebra was astounding” (Sarton, 1957), they were able to solve quadratic and cubic equations during the time they were alive as well. They discovered some of the sequence for squared numbers; (12, 22, 32, 42) (Kline, 1962), etc. could be written as (1, 4, 9, 16, etc.) (Kline, 1962) and it was then that what today we call “Pythagorean triples” was developed. It is assumed that “Babylonians understood the Pythagorean theorem as early as 1800-1650 BC” (Mankiewicz, 2004). Inhabitants of this region noticed “32+42=52; 52+122=132” (Kline, 1962), in which case, along with countless other set of 3s, became a Pythagorean triple since they made a right triangle’s sides with each number being the measurement of one of its sides. In any right triangle, “the sum of the squares on the two shorter sides equals the square on the longer side” (Mankiewicz, 2004) which is what the Babylonians noticed with 3,4,5 and with 5,12,13. Centuries later, it was discovered that, depending on what type of right triangle we were dealing with, whether it was a 30-60-90 or a 45-45-90, its sides could be found by solving three small equations for each one. for the first and for the latter, . As Babylonians taught themselves to solve quadratic and cubic equations, they did so “in two variables” (Struik, 1948), meaning, that they followed the first coefficient of the equation with an , and added an to the one after it, then , , etc. until the last coefficient did not have a variable. Though only certain coefficients were used by the Babylonians, “their method leaves no doubt that they knew the general rule” (Struik, 1948).

Babylonians and Egyptians used as the Pythagorean theorem equation and has been one of the few that were not modified to fit our understanding throughout history. The process of the Egyptians, however, was very distinct to find and develop the Pythagorean theorem. Instead, they used a “rope marked into twelve equal lengths” (Heaton, 2017). Mathematicians and other Egyptians then realized that, if a triangle was made with “one side of 3 units” (Heaton, 2017) and another one of 4, then the last one would have to be 5, making a right triangle. A mathematician has stated that “‘the rope of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together’” (Mankiewicz, 2004). They discovered how the theorem was related to right triangles but weren’t the only ones since “‘hypotenuse’ is derived from the Greek for ‘stretched against’” (Heaton, 2017). It is understood that Pythagorean theorem “has been applied to a huge range of both practical and poetic problems” (Heaton, 2017), and it does not matter “whether we are talking about the length of fields, spears, shadows or buildings” (Heaton, 2017), the same connection could be said made in each existent triangle.

Algebra and Geometry

Algebraic notation was nonexistent in the Babylonian civilization, yet “their algorithms were evaluating” (Rudman, 2008) several different equations. It was then assumed that they were “geometric diagrams” (Rudman, 2008) instead, which was simply referred to as “geometric algebra” (Rudman, 2008). Neither Egyptian nor Babylonian mathematics relating to algebra were considered superior to the other, but “from the time of the Babylonians and Egyptians to about 1550, all the equations solved had numerical coefficients” (Kline, 1962). Babylonians did focus more on it than Egyptians, however, “because of their unique problem of multiplication with a large base” (Rudman, 2008), meaning that Babylonians’ process for this mathematical operation was very complicated due to the sexagesimal scale they used. Babylonians “were in full technique of handling quadratic equations” (Struik, 1948) ever since “King Hammurabi reigned in Babylon” (Struik, 1948), but Egyptians were “only able to solve simple linear equations” (Struik, 1948). On the other hand, multiplying was “not a problem for the Egyptians” (Rudman, 2008). Despite it not being “named geometry until around 460 B.C.” (Brumbaugh, Ashe, Rock & Ashe, 1997), they are considered the creators of geometry because it is what they worked with when they wanted to “restore land boundaries associated with landmarks” (Brumbaugh, Ashe, Rock & Ashe, 1997).

A problem relating to the Babylonian algebra “asks for a number which, added to its reciprocal, yields a given number” (Kline, 1990). They turned the equations

Babylonians were able to obtain the quadratic formula after much thinking and any problem in which it was required to find “two numbers whose sum and product are given” (Kline, 1990) was reduced to the same quadratic formula. The more complex algebraic problems “were reduced by transformations to simpler ones” (Kline, 1990). Babylonians found a way to solve “special problems involving five unknowns in five equations” (Kline, 1990), and the unknowns were usually length, breadth, and area. To use special symbols for each, the three unknowns were represented by a word in the Sumerian language, which was no longer spoken. They were very commonly used symbols and were easily detected by Babylonians despite not “knowing how they were pronounced in Akkadian” (Kline, 1990). Babylonians solved algebraic problems “by describing only the steps required to execute the solution” (Kline, 1990). Neither Egyptians nor Babylonians included units in their calculations, but “the cubit was one unit for measuring length” (Brumbaugh, Ashe, Rock & Ashe, 1997) used by both civilizations. This unit of length was measured “from the point of the elbow to the tip of the middle finger on the same arm” (Brumbaugh, Ashe, Rock & Ashe, 1997).

As for geometry, its role “in Babylonia was insignificant” (Kline, 1990). Any problems that were associated with the “division of a field” (Kline, 1990) or with the physical construction of something were automatically “converted into algebraic problems” (Kline, 1990). Also, the “figures” (Kline, 1990) that show the problems of geometry were very undefined drawings whose “formulas may have been incorrect” (Kline, 1990) since, for example in their computations to find area, it is unknown whether the triangles the Babylonians worked with were right or equilateral, or “whether their quadrilaterals were squares” (Kline, 1990). Many pieces of information are missing in this area of mathematics to know exactly how Babylonians proceeded with their geometric calculations. Therefore, no useful proof and information has been found.

Area and Volume

Babylonians used 3 as “their value of pi” (Kline, 1962), meaning it was less precise than that of the Egyptians who used 3.16 (Kline, 1962). Egyptians, two millenniums before Christ, held an adequate number of digits on their “system of numeration,” (Sarton, 1957) including “large numbers of the order of millions, and with a decimal system” (Sarton, 1957). As time went on, they discovered ways to obtain the volume and area of different geometric shapes and prisms with formulas still used today. To find a circle’s area, Egyptians “squared eight-ninths of its diameter” (Sarton, 1957), giving an almost accurate answer. However, Egyptians and Babylonians both used the formula “where is a number which we can approximate” (Hodgkin, 2005) to solve for the area of a circle, which is one of the formulas in use today as well. Egyptians were able to find the volume of a truncated pyramid during the time of their existence. It was a complicated process, but they did the following for a pyramid whose , , and has a height of 6: 56 (Friberg, 2005)

Egyptians had also figured out how to “calculate the volume of a certain cylinder” (Cuomo, 2001) and were able to prove why their method had the correct steps as they solved the formula (Cuomo, 2001). Mathematicians have tried to clarify “how it was possible for Egyptian mathematicians to find the correct expression for the volume of a truncated pyramid” (Friberg, 2005) for many years. As Egyptians were in the process of trying to understand the intricate mathematics they had started to develop, Babylonians had already created the “methods for finding areas and volumes” (Heaton, 2017) when “Hammurabi became king of Babylon (c. 1750 BC)” (Heaton, 2017). Meaning that in this aspect of mathematics, Babylonians were more advanced than Egyptians since they had already discovered the process and formulas needed to find the area and volume of a figure while Egyptians had just begun their exploration of it.

Multiplication

Although nothing has been found to reveal “how the Egyptians did addition” (Rudman, 2008), there has been proof of how they progressed with multiplication. Egyptians who daily calculated for the product of two numbers had to memorize many of the smaller products of other numbers to be able to get to the bigger products. To facilitate their mathematical operations, they invented what is “now called Egyptian multiplication” (Rudman, 2008), which was the approach of “two-number additions a multiplication algorithm” (Rudman, 2008). Previous to the developing of writing, “pebble-counting arithmetic and abacus arithmetic” (Rudman, 2008) had been in use by Egyptians and by Babylonians. An example of the method used earlier is given in “How Mathematics Happened” by Peter Strom with the multiplication of “” (Rudman, 2008). To find the product of this, they would do “” (Rudman, 2008) and then would add 173 to the result for “twelve successive additions” (Rudman, 2008). However, multiplication calculations were found this way until a new method was introduced, one that didn’t require as many steps. Using the same numbers, this one, instead of adding 173 twelve times, Egyptians added each new sum plus itself. Meaning: (Rudman, 2008)

This new method helped Egyptians move much faster in their calculations but is not very similar to what we use presently. The Babylonians, on the other hand, contributed greatly to what is currently in use. Archeologists discovered that they “employed various multiplication techniques” (Rudman, 2008) by thoroughly analyzing “more than five hundred thousand Babylonian clay tablets” (Rudman, 2008). A vast amount of their “multiplication tables” (Rudman, 2008) continue to exist just like some of their calculations still do as well.

Studies of the Universe

The mathematical findings of Mesopotamian regions were complexly joined “with a desire to study the night sky” (Heaton, 2017) as well as with the “practical tasks of taxation, trade and measurement” (Heaton, 2017). It is said that without the mathematics of Babylonians, it would have been impossible to “record accurately the passage of stars” (Heaton, 2017). Babylonians noticed the sky’s movements and used them to help themselves in their calculations. One example of this were “the four phases of the Moon” (McLeish, 1994), they “gave the lunar month, which could be either 29 or 30 days long” (McLeish, 1994). They proceeded with this and “developed a lunar calendar” (Kline, 1962), a twelve-month calendar that followed every year but “did not coincide with the year of the seasons” (Kline, 1962). As a result, every 19 years, Babylonians added 7 months to their time in order to stabilize it as much as possible. They noticed and understood that the moon regularly moved around the Earth just like the Earth moved around the sun. What did not make sense to them was the cycle the rest of the planets followed. Babylonians only knew that they “moved with varying speeds at different times of the year” (Kline, 1962), which made it difficult for them to comprehend. Several other civilizations, except for Egyptians, then followed this style of calendar (Kline, 1962). On the other hand, Egyptians constructed a secret calendar only they knew of and understood. Historians and mathematicians assume that Egyptians were aware of the amount of time the Earth took in its revolution around the sun.

Egyptians and Babylonians both thought that the universe was made up of “gods who would most likely behave in a gentlemanly and beneficent manner” (Kline, 1962). This was the most reasonable theory they had for the universe, and because it was logical to them, they did not conceive any others. They believed “crops grew under the influence of the sun” (Kline, 1962) and that “animals mated and rivers overflowed at special times of the year” (Kline, 1962). It is still unknown how Babylonians and Egyptians were able to proceed with their mathematical findings and their curiosity of our universe as well. However, because they were interested in everything of their surroundings is why “the science of astrology” (Kline, 1962) was comprehended and emphasized in their societies and throughout the rest of Europe too.

Both groups of Mesopotamian regions assumed that to be able to foresee “the future one had to know the courses of planets and stars” (Kline, 1962). In a way, this was a battle between religion versus science and mathematics because the idea of “pseudoscience was widely accepted” (Kline, 1962) and it was sometime “after the belief that heavenly bodies were gods disappeared” (Kline, 1962).

Conclusion

Both groups of people had a very different opinion on what was necessary in calculations. Despite this, Babylonians and Egyptians both created an impressive amount of mathematics, most of which was very similar to the other. Their theories may have been conceived in a distinct way, but these Mesopotamian regions developed and expanded the idea of what today is taught around the world. Babylonians contributed with only the sketch and the outline of some aspects of the subject while giving complete and correct processes on others. Their sexagesimal scale as well as their way of working with Pythagorean theorem are examples that continue to exist just like Egyptians findings of the volume of a pyramid and their decimal notation are still in use as well. Some of the Babylonians’ mathematics were more accurate than the Egyptians’ and some of the Egyptians’ works were more mathematically correct than the Babylonians’, but they both wrote down their thoughts and their calculations, something that has significantly helped us discover more about their culture and their intelligence.

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