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Evergreen Journey of Mathematics: From Stone age to AI

Evergreen Journey of Mathematics: From Stone Age to AI

I will publish this book in form of series of blogs in this website only for readers.

(....Continuation from Post - 2 )

Chapter 2

Ancient Mathematics

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As generations passed, their mathematical knowledge continued to evolve. They discovered patterns in nature and sought to understand them using numbers. For example, they observed that the moon went through phases, and the seasons changed predictably. They started recording these patterns using notches on sticks and stones. By analyzing the notches, they discovered the concept of time and developed a rudimentary calendar to track the passage of days, months, and years.

Now, let us have a look at different regions of the world to explore gradual development of mathematics.

2.1. Development in Mesopotamia (3000-600 BC):

Mesopotamia, often referred to as the cradle of civilization, witnessed significant developments in mathematics between 3000 and 600 BC. Located in the region corresponding to modern-day Iraq, Mesopotamia was home to several prominent civilizations, including the Sumerians, Akkadians, Babylonians, and Assyrians. These ancient societies made lasting contributions to various fields, including mathematics.

The Mesopotamians created impressive clay tablets containing mathematical tables, such as multiplication tables, square root approximations, and even solutions to complex equations. One of the most remarkable achievements of Mesopotamian mathematics was the development of a sexagesimal numerical system, based on the number 60. This system, still in use today for measuring time and angles, stemmed from the Sumerians' astronomical observations. They noticed that the sun completed a full rotation in the sky in approximately 360 degrees, which they divided into six equal parts, each equivalent to 60 degrees. This choice of base 60 influenced the subsequent mathematical systems of neighboring civilizations.

The Mesopotamians used a combination of symbols to represent numbers in their sexagesimal system. A single wedge-shaped mark, known as a "cuneiform" symbol, represented the number one. To indicate higher values, they added additional marks. For example, two cuneiform symbols represented the number ten, three symbols represented the number sixty, and so on. This numerical system enabled them to represent large quantities and perform calculations more efficiently than simple tallying methods.

Another notable example is the famous tablet called Plimpton 322, dating back to around 1800 BC, which contains a table of Pythagorean triples (sets of whole numbers satisfying the Pythagorean theorem). This tablet suggests that the Mesopotamians were aware of the relationships between the sides of right-angled triangles long before the Greeks.

In addition to numerical systems and tables, Mesopotamian mathematics extended into various practical applications. The need for efficient agricultural practices in a region dependent on irrigation systems led to the development of measurement techniques. Mesopotamian mathematicians devised methods to calculate the area of fields, determine the volume of containers, and measure distances. They used simple geometric shapes, such as squares and rectangles, to approximate the measurements needed for land division and construction projects.

Trade and commerce were integral to Mesopotamian society, and mathematics played a crucial role in facilitating transactions and record-keeping. Merchants and scribes developed systems of weights and measures to ensure fair exchanges of goods. The shekel, a unit of weight, became a standard for assessing the value of various commodities. Scribes meticulously recorded transactions on clay tablets, documenting quantities and prices. These early accounting practices relied heavily on mathematical calculations, fostering the development of arithmetic skills in Mesopotamian society.

Furthermore, the Mesopotamians made significant contributions to the understanding of algebraic concepts. Their mathematical texts contained solved examples of linear and quadratic equations, including methods for finding unknown quantities and calculating interest rates. They also explored practical problems involving proportions and ratios, such as determining the appropriate mixture of ingredients for various recipes.

Astronomy was intertwined with mathematics, also gained momentum in Mesopotamia. The Babylonians, in particular, were renowned for their astronomical observations and predictions. They meticulously recorded celestial events, such as lunar and solar eclipses, planetary movements, and the positions of stars. By studying these patterns, they developed sophisticated astronomical calendars, which incorporated mathematical calculations and allowed for accurate predictions of celestial phenomena.

Mesopotamian mathematical inerpretation

The mathematical legacy of Mesopotamia has endured through the ages, thanks to the survival of clay tablets and the subsequent decipherment of cuneiform script. The knowledge and techniques developed in this ancient civilization influenced later mathematical traditions in neighbouring regions, including Egypt, Greece, and India.

The achievements of Mesopotamian mathematics showcase the ingenuity and practicality of ancient civilizations. Their advancements in numerical systems, tables, measurement techniques, algebra, and astronomy laid the groundwork for future mathematical developments and contributed to the rich tapestry of mathematical knowledge that has shaped human history.

2.2.  Advancement in Egypt (3000-500 BC):

The ancient Egyptians made remarkable contributions to mathematics, especially in the fields of geometry and arithmetic. They devised practical methods for measuring land, building pyramids, and constructing irrigation systems. The Egyptians used a decimal numerical system and had a deep understanding of fractions, enabling them to perform complex calculations related to trade and architecture. They also developed the Rhind Mathematical Papyrus, which contained a wealth of mathematical problems and solutions. Their mathematical knowledge was essential for various practical purposes, including architecture, surveying, trade, and record-keeping.

Egyptian numeral system

One of the most remarkable achievements of Egyptian mathematics was their mastery of geometry. The Egyptians' practical need to measure and survey the land for agricultural purposes, as well as their fascination with the monumental architecture of the pyramids, motivated them to develop geometric techniques. They observed that the Nile River flooded predictably each year, leaving behind fertile soil, and they used geometry to divide and allocate the land for farming. By employing simple geometric principles, such as the Pythagorean theorem and the properties of right angles, the Egyptians accurately surveyed and established boundaries for their fields.

The Egyptians developed a numerical system based on hieroglyphic symbols, which they used for recording numbers. They used a decimal system, utilizing various symbols for the powers of ten, including symbols for one, ten, one hundred, one thousand, and so on. This numerical system allowed for the representation of both large and small numbers, enabling complex calculations.

Arithmetic played a vital role in the Egyptian civilization. The Egyptians developed sophisticated methods for performing addition, subtraction, multiplication, and division. They used a combination of hieroglyphic symbols and verbal instructions to carry out these operations. One interesting feature of their arithmetic system was the use of unit fractions. They expressed fractions as the sum of unit fractions, such as 1/2 + 1/4 + 1/8, rather than using a single fraction. This method made calculations involving fractions more manageable and aligned with their practical needs in trade and commerce.

Measurement was a crucial aspect of Egyptian mathematics, primarily driven by their practical requirements in construction and trade. The Egyptians developed an elaborate system of weights and measures to ensure fairness and consistency in transactions. They used standardized units, such as the cubit for measuring lengths, the hekat for measuring volume, and the deben for measuring weight. These units were based on body parts or commonly used objects and provided a standardized framework for construction projects, trade, and taxation.

The Egyptians developed a deep understanding of fractions and their applications. They recognized the importance of fractions in dividing and distributing resources. For example, when sharing food or other commodities, they used fraction-based calculations to ensure equitable distribution. Their familiarity with fractions extended to more complex operations, such as multiplication and division of fractions.

Egyptian mathematics also had a significant impact on the practical aspects of their daily lives. They developed a calendar system based on astronomical observations, which consisted of 12 months of 30 days each, with an additional five or six epagomenal days. This calendar facilitated agricultural planning and the organization of religious festivals. The Egyptians calculated the length of the year with impressive accuracy, coming close to the current accepted value of 365.25 days.

Egyptian pictograph of practical life

Record-keeping was a critical application of Egyptian mathematics. Scribes meticulously recorded transactions, tax payments, and other administrative matters on papyrus scrolls or stone inscriptions. The skillful manipulation of numbers and calculations allowed for accurate bookkeeping, ensuring the smooth functioning of the Egyptian economy and administration.

The mathematical legacy of ancient Egypt influenced subsequent civilizations, including the Greeks and Romans. Greek mathematicians, such as Euclid, studied and incorporated Egyptian geometric principles into their own works. The Greeks recognized the value of Egyptian mathematics and regarded it as a foundation for their own mathematical developments.


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