HOM Lesson-1.pdf

# UNIT I ## The Development of Mathematics: Ancient Period ### Lesson 1 #### Prehistoric Times, Egyptian Mathematics, and Babylonian Mathematics #### Learning Outcomes - Identify the notable events during prehistoric times, Egyptian mathematics, and Babylonian mathematics. - Discuss the conceptio...

# UNIT I ## The Development of Mathematics: Ancient Period ### Lesson 1 #### Prehistoric Times, Egyptian Mathematics, and Babylonian Mathematics #### Learning Outcomes - Identify the notable events during prehistoric times, Egyptian mathematics, and Babylonian mathematics. - Discuss the conception and development of mathematics during prehistoric times, Egyptian mathematics, and Babylonian mathematics. ## Time to Read! ### Mathematics during Prehistoric Times - Our ancestors from the past would have possessed a broad idea of quantity and would have been able to point out, by instinct, one or two antelopes in a herd. - The conceptual transition from understanding the physical concept of two items to giving an abstract concept a symbol or term of “2,” required a long time to develop. - Some remote hunter-gatherer tribes only speak the numbers "one," "two," and "many" and some only speak the numbers from one up to five. - The early man kept track of recurring events like the moon’s phases and the seasons. - Notched bones found in Africa between 20,000 and 25,000 years ago give some of the first clues of human thought around digits. - Mathematics existed in the prehistoric period before people learned to keep written documents and records. ## The Ishango Bone ### Paleolithic and Neolithic period #### Paleolithic Period (Old Stone Age) - It is considered the longest part of the prehistoric times. - People mainly lived in caves and used plants and skin of animals for building tents and making clothing. - Primitives were seeking food to survive. - Stones, bones, and sticks were used to create weapons like daggers and spears for hunting. - Wildfires started establishing a language for communication. - At the beginning of the number system, man could distinguish between "one" and "two." #### Neolithic Age (New Stone Age) - Primitives made the transition from food hunters to food producers. - Marked with progress from obscene ways of doing things to a more civilized form. - People learned to settle and make primitive forms and dwellings as well as establish villages to live. - Development of pottery making, carpentry, weaving, and melting of copper and bronze to transform into metals. - Inventions of the wheel, wagon, hammer, tents, huts, etc. - Conception of the idea of trades, private property, and quantity. ## Ancient Egyptian Number System - The wealthy Nile Valley was home to the ancient Egyptians, who began keeping track of the seasons and lunar cycles for both agricultural and religious purposes around 5000 BC. - In ancient Egypt, surveyors for the Pharaoh used measurements based on body parts to determine the size of buildings and areas such as the width of the palm and the length from elbow to fingertips. - The decimal numerical system was invented based on the ten fingers. | Ancient Hieroglyphic Numerals | Value | | - | - | | | 1 | | | 10 | | | 100 | | | 1,000 | | | 10,000 | | | 100,000 | | | 1,000,000 | - The Moscow Papyrus is the earliest mathematical writing discovered so far in ancient Egypt and dates from the Egyptian Middle Kingdom between 2000 and 1800 BC. - It is believed that the first wholly established base-ten numeration system was created by the Egyptians at least 2700 BC (probably much early) ago. **How the Ancient Egyptians Numbered Things** - A stroke (|) was used for units, a heel-bone symbol ( ) for tens, a coil of rope for hundreds, and a lotus plant ( ) for thousands. - However, because place value was not recognized, more significant numbers were somewhat tricky. For example, a million required just one character, a million minus one required fifty-four characters. ## Egyptian Method of Multiplication - A type of arithmetic and geometry teaching manual from circa 1650 BC, the Rhind Papyrus, gives explicit illustrations of how multiplication and division were performed in those days. - It also shows other mathematical skills such as unit fractions, composite and prime numbers, arithmetic, geometric, and harmonic means, as well as the capacity to resolve first-order linear equations and arithmetic and geometric series. - According to the Berlin Papyrus, which dates to approximately 1300 BC, the ancient Egyptians were capable of solving second-order algebraic (quadratic) equations. **Example: To multiply 3 by 6.** - Take the combination of blocks on the 1’s side, which add up to 6 (2+4), and count up the corresponding blocks on the 3’s side (6+12) to give a total of 18. **The Egyptian method of multiplication included constantly doubling the amount to be multiplied on one side and the other, which is essentially a binary factor multiplication similar to what is done today by computers. It is possible to utilize these numbers as a kind of multiplication reference table by determining the combination of powers of two that add up to the number that has to be multiplied by. This made efficient use of the idea, more than 3,000 years before Leibniz introduced the concept of binary numbers to the west, and many more years before the invention of the computer was to fully realize its potential.** **To use the Egyptian method of multiplying numbers, study the example below:** **Example 1. 12 × 15** - **Step 1.** On the left side, start by bringing down 1. - **Step 2.** Start doubling the number until it reaches the highest number before the given number, 12. At this point, we will stop at 8 because the next number is 16 which is greater than the multiplicand, 12. - **Step 3.** On the right side, bring down the multiplier (15), then start doubling until it matches the last number on the left side. - **Step 4.** Look on the left side and select the numbers that when you add will be equal to the multiplicand, and that is “8” and “4,” respectively (8+4=12). - **Step 5.** Add the numbers besides 8 and 4, that is 60+120 to give you the product, 180. ## Egyptian Method of Division - A fractional notation was created because of the market and trade-related practical issues. - The papyri discovered show us how to use the fractions in unit based on the symbol of the Eye of Horus. - The eye section was intended for a variety of values, including half, quarter, eighth, sixteenth, thirty-second, and sixty-fourth, and the total fell short of a whole by one sixty-fourth. ## Eye of Horus Papyrus **Unit fractions may also be applied to basic division sums. For instance, if there were three loaves and they needed to be divided among five individuals, the first two loaves would be cut into thirds, the third bread into fifths, and the final third from the second loaf would be cut into five pieces. Therefore, each receiver would get one-third plus one-fifth plus one-fifteenth for a total of three-fifths.** **Example: Divide 3 loaves between 5 people.** - First, divide two of two loaves into thirds… - …and the third loaf into fifths. - Each person gets one-third each, with one-third left over. - Each person gets one-fifth. - Then, divide the left-over third from the second loaf into fifths. - Each person gets a slice (one-fifteenth). - So, each person gets: 1/3 + 1/5 + 1/15 = 3/5 ## Egyptian Division ## The Egyptians - They realized that by multiplying the diameter by 8/9 and then squaring it, it was possible to determine the areas of circles with various diameters. - This offers an accurate near approximation that is within 1%. - The pyramids themselves were a significant find in Egyptian mathematics. - There is evidence that they knew the formula for a pyramid’s volume, which is 13 times the height times the length times the width, as well as the formula for a truncated or clipped pyramid. ## Abu’l-Hasan Ibn Yunus (950–1009) - A prominent Egyptian astronomer and mathematician whose works are renowned for being innovative because they were founded on tedious calculations and close attention to detail. - On the Moon, a crater bearing his name is called Ibn Yunus. ## Ahmes (1680 BCE–1620 BCE) - One of the oldest mathematical texts is the Rhind Papyrus. - It is named after Scottish Egyptologist Alexander Henry Rhind who purchased the papyrus in Egypt in 1858. - The papyrus was copied by the scribe Ahmes and according to him, the knowledge is from an ancient study from around 2000 BC. - Our primary resource for knowledge about Egyptian mathematics is the papyrus. - The Recto contains the division of 2 by the unit fractions of the odd integers 3 to 101 as well as the division of the digits 1 through 9 by 10. - The Verso has 87 problems on a variety of subjects, including the two-thirds rule, progressions, granary volumes, and the four operations. # Babylonian Mathematics - Mesopotamia, the home of the Babylonians, is frequently referred to as the Cradle of Civilization because the Sumerians developed the oldest known writing system, the pictographic cuneiform script. - Agriculture was established and practiced by the Sumerian civilization as early as the sixth millennium BC. - In the beginning, institutional requirements for measuring land plots, individual taxation, and the like, led to the development of Sumerian mathematics. - In order to construct their intricate lunar calendar and map the night sky, the Sumerians and the Babylonians had to provide explanations for extremely large numbers. # Sumerian and Babylonian Number System - A small clay cone to stand for one, a clay ball for ten, and a large cone for sixty were later replaced by cuneiform equivalents during the third millennium so that numerals could be written with the same stylus as the words in the text. - As early as 2700–2300 BC, Sumer presumably used a primitive version of the abacus. | | | | ----------- | ----------- | | 1 | | | 2 | | | 3 | | | 4 | | | 5 | | | 6 | | | 7 | | | 8 | | | 9 | | | 10 | | | 11 | | | 12 | | | 13 | | | 14 | | | 15 | | | 16 | | | 17 | | | 18 | | | 19 | | | 20 | | | 21 | | | 22 | | | 23 | | | 24 | | | 25 | | | 26 | | | 27 | | | 28 | | | 29 | | | 30 | | | 31 | | | 32 | | | 33 | | | 34 | | | 35 | | | 36 | | | 37 | | | 38 | | | 39 | | | 40 | | | 41 | | | 42 | | | 43 | | | 44 | | | 45 | | | 46 | | | 47 | | | 48 | | | 49 | | | 50 | | | 51 | | | 52 | | | 53 | | | 54 | | | 55 | | | 56 | | | 57 | | | 58 | | | 59 | | **Babylonian Numerals** - Sumerian and Babylonian mathematics used the sexagesimal, or base 60, mathematical system, and could be physically numbered using the twelve knuckles on one hand and the five fingers on the other. - The Babylonians used a comprehensive place-value system for their numerals, unlike the Egyptians, Greeks, and Romans. - More important numbers were written in the left column, just like in the modern decimal system, but the numbers were written in base 60 instead of base 10. - For example, 3,600 + 60 plus 1 in the Babylonian numerical system, or 3,661. - Additionally, two separate symbols – a unit symbol (T) and a ten sign ( ) – were used to indicate the numbers 1 through 59 within each place value. - These symbols were joined similarly to the well-known Roman numeral system. - For example, 23 would be shown as «TTT. - However, the sign for 60 was the same as that for 1, and as they are lacking a symbol for the decimal point, it was frequently necessary to determine a symbol’s real place value from its context. - To use this base system, we must learn the fundamental place value of its symbol (the diagram below uses a triangle for convenience): - The first place value is equivalent to 1-59 using its symbols. - The second place value is equivalent to 60 times its symbol given. - The third place value is equivalent to 60^2 times its symbol given. - The fourth place value is equivalent to 60^3 times its symbol given. - Looking at the symbol, the inverted triangle is equivalent to 1. - Now looking at its place value, this will give: - (602 x 1) = third place value - (60 × 1) = second place value - (1) = first place value - Now, add these values to have (3,600) + (60) + (1) = 3,661. - The important thing here is the spacing of writing the symbols because this will tell us the place value to get the correct answer. **Another example:** - _a_. = (60 × 1) + (14) = 74. - _b_. = (60^2 x 10) + (60 × 1) + (5) = 36,065 - Many divisors of 60 have been proposed, including 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 60 (in fact, 60 is the smallest integer divisible by all numbers from 1 to 6). - In addition, 60 seconds continue to be used in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle in the present day. - All of them are evidence of the ancient Babylonian system that helped the Babylonians expand their mathematical expertise. - The same explanations apply to why the number 12, which is composed of the digits 1, 2, 3, 4, and 6, has historically been one of the most prevalent multiples that we are using such as 12 months, 12 inches, 12 pence, 2 x 12 hours, and so on. - The Babylonians developed the circle character for zero, a mathematical concept that was unknown to the Egyptians, Greeks, and Romans. However, this circle character was still functioning more as a substitute for a number than a real one. ## Babylonian Tablets - Around 2100 BC, the Babylonians created clay tablets that detailed their methods for calculating the area of irregular shapes. - Backgammon and other dice games were immensely popular among the ancient Babylonians, and they used geometric shapes in their structures, patterns, and dice for these games. - They used geometry to compute the volumes of common things like bricks and cylinders as well as the areas of triangles, trapezoids, and rectangles (although not pyramids). ## Babylonian Clay Tablets - It is possible that the right-angled triangle's secret was known to the Babylonians before Pythagoras of Greece, thanks to the renowned and contentious Plimpton 322 clay tablet, which is believed to have been made around 1800 BC. - According to this little-known fact, the square of the hypotenuse is equivalent to the sum of the squares of the other two sides. - Some people think that the tablet was only an intellectual exercise, and that the 15 perfect Pythagorean triangles with whole number sides were not supposed to be Pythagorean triples. - They base their belief on the fact that the sides of the triangles were all whole numbers.