All of you know the history of * mathematics *is the most famous subject all over the world. It is a subject that has been studied and explored by countless individuals for centuries. But have you ever thought about where it all began? Who were the first mathematicians? What discoveries and innovations led to the complex field of mathematics we know today?

It all started in ancient civilizations such as the Egyptian, Babylonian, and Greek societies. These civilizations made significant contributions to the development of mathematics through their practical use of numbers for measuring, construction, and trade. For example, Egyptians used math to build pyramids while Greeks used it to understand astronomy.

One of the most famous mathematicians from ancient Greece was Pythagoras. He is known for his famous theorem which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is still used in geometry and has various real-world applications.

Another significant figure in ancient mathematics was Euclid, who is often referred to as the “Father of Geometry”. He developed a system of axioms and postulates that form the basis for geometric proofs. His work, Elements, is considered one of the most influential mathematical texts in history.

Moving forward in time, we come across renowned mathematicians such as Archimedes, known for his contributions to calculus and physics, and Aryabhata from India who made groundbreaking discoveries in algebra and trigonometry.

In more recent times, the field of mathematics has seen rapid growth with advancements in areas such as differential equations, topology, and number theory. These have led to innovations in fields such as engineering, computer science, and economics.

But mathematics is not just about numbers and equations. It also includes abstract concepts like geometry, logic, and set theory. These concepts help us understand the world around us and solve complex problems.

At its core, mathematics is a tool for problem-solving. It teaches us how to think critically, analyze data, and make logical deductions. This makes it an essential skill for various careers ranging from finance to technology.

So whether you are solving a simple arithmetic problem or working on cutting-edge research in theoretical mathematics, remember that every field of study is built upon a foundation of mathematical principles. And with the advances in technology today, we have access to

**Aristotle**, who was a very smart person and thought a lot about how the world works, made a mistake when he talked about how objects fall. It wasn’t until the 1590s that Galileo corrected his mistake.**Galen**was the most famous doctor in ancient times, but he wasn’t allowed to examine human bodies, so he made a lot of mistakes about what’s inside us. In 1543, Vesalius fixed some of Galen’s mistakes, and then in 1628, Harvey corrected even more.**Newton**, who was the most amazing scientist, was incorrect about some things regarding light and lenses, and he didn’t know about spectral lines. His greatest work, the rules of movement and the idea of gravity had to be changed by Einstein in 1916.

**Now we can see what makes mathematics unique.**

When the Greeks invented the method for proving things, they got it right. Everything they did stands true forever.

**Euclid**started something and it has grown a lot, but nobody had to fix his work. All his mathematical proofs still hold today.**Ptolemy**might have gotten the solar system wrong, but the trigonometry he used for his calculations is still right, and always will be.

**ANCIENT EGYPT**

**The Era and The Resources**

Around 450 BCE, a Greek man named Herodotus visited Egypt. He saw old monuments, talked to priests, and watched the Nile River and the people working along it. Herodotus thought geometry math began in Egypt.

This was because every year flooding covered the land around the Nile. So, the Egyptians needed to resurvey the land. About 100 years later, the philosopher Aristotle had the same idea. He thought Egyptian priests began geometry because they had spare time.

People often debate the origins of mathematical advancements, whether they came from practical workers like surveyors or thinkers like priests and philosophers. This ongoing debate extends beyond Egypt and continues to shape our understanding. As we’ll see, both types of contributors have played a role in the evolution of mathematics.

لوگ اکثر ریاضی کی ترقی کی ابتدا پر بحث کرتے ہیں، چاہے وہ عملی کارکنوں جیسے سرویئرز یا مفکرین جیسے پادریوں اور فلسفیوں سے آئے ہوں۔ یہ جاری بحث مصر سے باہر تک پھیلی ہوئی ہے اور ہماری سمجھ کو تشکیل دیتی ہے۔ جیسا کہ ہم دیکھیں گے، دونوں قسم کے شراکت داروں نے ریاضی کے ارتقا میں اپنا کردار ادا کیا ہے۔

Trying to understand the history of mathematics in ancient Egypt was tough for researchers until the 1800s. They faced two big issues. Firstly, they couldn’t read the existing materials because they were written in forms like hieroglyphics, hieratic, and demotic scripts, which were hard to decipher. Secondly, there weren’t many such materials available. Knowledge of hieroglyphs gradually disappeared after the third century CE when Coptic and Arabic became more common.

The turning point in understanding ancient texts arrived in the early 1800s. French scholar, JeanFrancois Champollion, managed to translate hieroglyphs using multilingual tablets. He was intrigued by the Rosetta Stone, along with British physicist Thomas Young. The stone was found during Napoleon’s expedition in Egypt in 1799 and contained scripts in hieroglyphic, demotic, and Greek.

- By 1822, Champollion shared a significant part of his translations in an important letter he sent to the Academy of Sciences in Paris. By the time he died in 1832, he had published a grammar book and started a dictionary.
- We don’t add more Greek thoughts about Egyptian math or make uncertain guesses. Our story is based on real evidence.

An ancient Egyptian document, part of the Rhind collection in the British Museum, was deciphered by Eisenlohr in 1877. It turned out to be a math guide with arithmetic and geometry problems. It was written by Ahmes before 1700 B.C., and was based on an even older work, thought by Birch to go back to 3400 B.C.!

This document – the oldest math guide we know – gives us a glimpse into Egyptian math thousands of years ago. Its title is “Directions for obtaining the Knowledge of all Dark Things.” It shows us that Egyptians weren’t much interested in theory. It doesn’t have theorems. It mainly has results, possibly for a teacher to explain to students.

In 1858, Scottish collector Henry Rhind bought a papyrus roll in Luxor. It’s about a foot high and around eighteen feet long. Except for a few pieces in the Brooklyn Museum, this papyrus is now in the British Museum. It’s called the Rhind or Ahmes Papyrus, named after the scribe who copied it around 1650 BCE. The scribe says the content comes from an earlier time, around 2000 to 1800 BCE.

The papyrus, written in hieratic script, has become a key source for understanding ancient Egyptian math.

There’s another critical papyrus that’s known as the Golenishchev or Moscow Papyrus. It was bought in 1893 and you can now find it in the Pushkin Museum of Fine Arts in Moscow. This one is also about eighteen feet long, but it’s only a quarter as wide as the Ahmes Papyrus. It wasn’t written as neatly as Ahmes’s work was and we don’t know who wrote it, but it’s from around 1890 BCE. It’s got twenty-five examples in it that are mostly about everyday life. They’re not that different from the examples Ahmes wrote, apart from two that we’ll talk more about later.

#### Possible Short Questions

**Write an ancient Egypt saying about the origination of Geometry in Egypt.**

Herodotus thought geometry math began in Egypt. This was because every year flooding covered the land around the Nile. So, the Egyptians needed to resurvey the land. About 100 years later, the philosopher Aristotle had the same idea. He thought Egyptian priests began geometry because they had spare time.

**How does geometry come into the picture in Egypt?**

An ancient Egyptian document, part of the Rhind collection in the British Museum, was deciphered by Eisenlohr in 1877. It turned out to be a math guide with arithmetic and geometry problems. It was written by Ahmes before 1700 B.C., and was based on an even older work, thought by Birch to go back to 3400 B.C.!

**Write the role of mathematics between two types of contributors i.e., the Surveyor /Rope stretcher and the Priest/ the Philosophers.**

People often debate the origins of mathematical advancements, whether they came from practical workers like surveyors or thinkers like priests and philosophers. This ongoing debate extends beyond Egypt and continues to shape our understanding. As we’ll see, both types of contributors have played a role in the evolution of mathematics.

**Write the role of Jean Prancois Champollin regarding Egyptian math’s history?**

French scholar, Jean-Francois Champollion, managed to translate hieroglyphs using multilingual tablets. Champollion shared a significant part of his translations in an important letter he sent to the Academy of Sciences in Paris. By the time he died in 1832, he had published a grammar book and started a dictionary.

**What was Rosetta Stone. Who found it?**

The Rosetta Stone was a very important artifact that helped us understand ancient texts. It was discovered in Egypt during Napoleon’s expedition in 1799. The stone had writings in hieroglyphic, demotic, and Greek. It was like a key that helped us figure out how to translate hieroglyphs. Two people, Jean-Francois Champollion from France and Thomas Young from Britain were really interested in the Rosetta Stone. They worked hard and made a big discovery in understanding these ancient writings.

**Who was the Rhind (the Ahmes)?**

An ancient Egyptian document, part of the Rhind collection in the British Museum, was deciphered by Eisenlohr in 1877. It turned out to be a math guide with arithmetic and geometry problems. It was written by Ahmes before 1700 B.C., and was based on an even older work, thought by Birch to go back to 3400 B.C.!

**Note on the Rhind Papyrus (the Ahmes Papyrus)?**

The Rhind or Ahmes Papyrus is an ancient artifact named after a scribe called Ahmes. It was copied around 1650 BCE, but the content is believed to be much older, dating from around 2000 to 1800 BCE. The papyrus is about a foot tall and eighteen feet long. It is an important source for understanding the math principles of ancient Egypt and provides insight into their arithmetic and geometric problems at that time. It was purchased by a Scottish collector named Henry Rhind in 1858 and is now in the British Museum, with some parts in the Brooklyn Museum.

**Write the title of Ahmes’s Papyrus?**

Its title is “Directions for obtaining the Knowledge of all Dark Things.”

**Note on the Moscow Papyrus (the Golenishchev)?**

There’s another critical papyrus that’s known as the Golenishchev or Moscow Papyrus. It was bought in 1893 and you can now find it in the Pushkin Museum of Fine Arts in Moscow. This one is also about eighteen feet long, but it’s only a quarter as wide as the Ahmes Papyrus. It wasn’t written as neatly as Ahmes’s work was and we don’t know who wrote it, but it’s from around 1890 BCE. It’s got twenty-five examples in it that are mostly about everyday life. They’re not that different from the examples Ahmes wrote, apart from two that we’ll talk more about later.

Name few documents of Egyptian mathematics history. The Rhind or Ahmes Papyrus: This document, which Scottish collector Henry Rhind bought in 1858, has become a key source for understanding ancient Egyptian math. It is about a foot high and around eighteen feet long. The scribe, Ahmes, who copied it around 1650 BCE says the content originates from an earlier time, approximately 2000 to 1800 BCE.

The Golenishchev or Moscow Papyrus: This document is another crucial source for understanding Egyptian mathematics. It was procured in 1893 and is currently housed in the Pushkin Museum of Fine Arts in Moscow. This papyrus is also around eighteen feet long, but it’s only a quarter as wide as the Ahmes Papyrus. It contains twenty-five examples that mostly pertain to everyday life.

## NUMBERS AND FRACTIONS

Egypt has two other types of script, hieratic and demotic. Ahmes’ use of the more informal hieratic script was apt for writing with pen and ink on prepared papyrus leaves. The numbering system was still based on tens, but instead of the lengthy process of repeating hieroglyphs, they introduced special signs to stand for individual numbers and multiples of powers of 10

The ancient historian Herodotus made a significant observation about how the Egyptians did their math. Egyptians performed their calculations using small stones, with their hand moving in a right-to-left direction. Conversely, the Greeks did their calculations by moving their hand from left to right.

In this, we again see the use of the practical method of calculation often employed by ancient civilizations. Egyptians used a decimal system for their calculations. Since their calculations involved moving their hands sideways, they likely used counting boards with vertical columns. Each column probably had no more than nine stones, because ten stones would be equivalent to one stone in the column to the immediate left.

The Ahmes papyrus gives us cool information about how the ancient Egyptians used fractions. They did things very differently from us! The old Egyptians, a long time ago, had a hard time with fractions. They usually didn’t want to change both the top and bottom numbers of a fraction at the same time. The people in Babylon used the same bottom number (60) for all their fractions. The Romans did something similar but used 12 as their bottom number. But Egyptians and Greeks liked to keep the top number (numerator) the same, and they changed the bottom number (denominator) instead. Ahmes called something a “fraction” only if it had 1 as its top number, and

he wrote it by writing the bottom number with a dot above it. If a fraction couldn’t be written as just one of these “unit-fractions” (fractions with 1 on top), they would write it as the sum (addition) of two or more unit-fractions. Thus, he wrote 1/3 + 1/15 in place of 2/5.

**Define “Heap”. Also write other names for the term “heap”.**

Heap: In the context of ancient Egypt, a heap refers to a missing number in algebraic puzzles. It is represented by the symbol “x” and is the value that needs to be determined. The term “heap” was used by the ancient Egyptians to describe this unknown quantity in mathematical equations. The other name of term “heap” is “aha” or “hue”.

**What Is Geometry?**

“Geometry” is a big word that comes from ancient Greek. The word is made up of two parts: “Geo” which means Earth, and “Metron” which means measurement. So, Geometry is like a tool that helps us measure the Earth and understand things like shapes, how they fit together, and how much space they take. If you love working with shapes and spaces, you might grow up to be a “geometer” – that’s just a fancy name for someone who is good at geometry!

**Shortly note on the beginning of ‘Geometry’.**

Geometry, in the beginning, was just a set of rules to help people measure things like length, width, and size. These rules were not perfect, but they were the best guesses made by trying things out. People from Babylon, Egypt, and India used geometry to build things, find their way, and measure land. They taught the Greeks what they knew. So, geometry has been around for a very long time, and ancient people used it to help with their everyday tasks.

**Aristotle says that mathematics had its birth in Egypt, on what bases?**

Aristotle believed that mathematics was first studied in Egypt because the priests there had enough free time to delve into it. Many old writers, including Herodotus, Diodorus, Diogenes Laertius, and Iamblichus, stated that Geometry specifically started in Egypt. According to a story in Herodotus’s Book II, King Sesostris split the land equally among all Egyptians and taxed them

yearly. If the river washed away some of their land, they would inform the king. Then, the king’s staff would go and measure the land to see how much was left, so they could adjust the tax. This practice, Herodotus thought, led to the creation of geometry, which then moved to Greece.

**Name few historians who said that geometry originated in Egypt?**

Herodotus, Diodorus, Diogenes Laertius, and Iamblichus, stated that Geometry specifically started in Egypt.

**What was the role of King Sesostris in the development of geometry?**

According to a story in Herodotus’s Book II, King Sesostris split the land equally among all Egyptians and taxed them yearly. He also implemented a practice where if the river washed away some of their land, they would inform the king and his staff would measure the remaining land to adjust their taxes. This practice is believed to have led to the creation and development of geometry in Egypt.

**Why did historians believe that Egypt was the birthplace of geometry?**

Historians believed that Egypt was the birthplace of geometry due to the aforementioned story about King Sesostris and his implementation of a system for measuring land. This shows an early understanding and use

**Herodotus, said the geometry originated in Egypt. On what bases? Quote its saying?**

According to a story in Herodotus’s Book II, King Sesostris split the land equally among all Egyptians and taxed them yearly. If the river washed away some of their land, they would inform the king. Then, the king’s staff would go and measure the land to see how much was left, so they could adjust the tax. This practice, Herodotus thought, led to the creation of geometry, which then moved to Greece.

**The Egyptians were familiar, 2000 years B.C., with the well-known property of the right triangle. Is it true?**

Yes, It’s true. Egyptians were familiar, 2000 years B.C., with the well-known property of the right triangle, for the special case at least when the sides are in the ratio 3 : 4 : 5

**The geometry of the Egyptians consists certain defects. Note on them.**

The Egyptians missed two important things that are needed to truly understand geometry. First, they didn’t create a proper system of geometry based on a few basic rules and assumptions.

Many of their rules, especially the ones about 3D shapes, were probably not even tested. They just thought these rules were true because they saw them work or just accepted them as facts.

The second thing they missed was their failure to see that many specific instances could be understood better under a more general perspective. This would have let them find wider and more basic principles. Instead, they saw some of the easy truths about geometry as many different special cases, each needing its own unique way to be solved.

**How did Egyptians find the area of isosceles triangle?**

The Egyptians had a strength in geometry, which revolved around creating shapes and finding out their areas. They made a mistake in calculating the area of an isosceles triangle. If the triangle had sides of 10 ruths each and a base of 4 ruths, they incorrectly stated its area as 20 square ruths. This is actually half of the result you get when you multiply the base by one side.

**How did Egyptians find the area of isosceles trapezoid?**

The area calculation for an isosceles trapezoid follows a similar approach. It’s computed by multiplying half the sum of the parallel sides by the length of one non-parallel side.

**How did Egyptians find the area of circle?**

The area of a circle is found by deducting from the diameter of its length and squaring the remainder.

**It is often said that the ancient Egyptians were familiar with the Pythagorean theorem. Is it true?**

Some people believe that the ancient Egyptians knew about the Pythagorean theorem, but we don’t have any evidence of this in the old papers we’ve found so far. However, the Ahmes Papyrus, one of these old papers, does contain some problems related to geometry.

**Write edfu corollary and its importance?**

The ancient Egyptians found a way to figure out the area of a triangle. Here’s how they did it: if you take two sides of the triangle and add them together, then take half of this amount, and multiply it by half of the third side, that gives you the area of the triangle.

**About relationships, what was the concept of Egyptians?**

The Egyptians were quite ahead of their time when it came to understanding the connection between different geometric shapes, almost like the Greeks who came after them. They didn’t have any formal theorems or proofs like we do today. But, they were among the first to make accurate observations about the outer edges and the inside space of circles and squares.

**Write the rule for finding the area of the general quadrilateral in Egypt era?**

The rule for figuring out the area of a typical four-sided shape, also known as a quadrilateral, is a bit different. What you need to do is find the average of the lengths of two opposite sides, then multiply those averages together. While this method might not be completely accurate, it’s the approach that was used in ancient times.

**What were the concept of Egyptians about Degree of accuracy or about Angles?**

Degree of accuracy is not a good measure of success in math or architecture. We shouldn’t overemphasize this aspect of Egyptian work.

**Write the Ahmes’s value for π?**

The number 22/7 we commonly use today for π is approximately 3.14. But it’s important to

remember that Ahmes, an ancient Egyptian, used a simpler value for π, which was just about. Other Egyptians also used Ahmes’s value, which is confirmed in a historic document called the Kahun Papyrus from the twelfth dynasty. Here, the volume of a cylinder is figured out by multiplying its height by the area of its base. The area of the base is calculated using Ahmes’s method.

**SLOPE PROBLEMS**

**How did Egyptian introduce about cotangent of an angle or Slope?**

When the Egyptians built the pyramids, they had to make sure all sides sloped at the same angle. This was very important for the pyramid’s structure. This might be the reason why they came up with a concept related to what we now call the “cotangent of an angle“.

**In modern age we measure the steepness of a straight line through the ratio of the “rise” to the “run.” How did Egyptian measure?**

The Ancient Egyptians used the reciprocal of this ratio. They used a term called “seqt”, representing the horizontal departure of an oblique line from the vertical axis per unit change in height. The units of measurement were “hands” for horizontal distance and “cubits” for vertical length, making the seqt equivalent to today’s concept of the cotangent of an angle.

**What is meant by the word seqt?**

The Ancient Egyptians used the term “seqt” to describe how a slanted line moves away from the vertical axis as its height changes. In simpler terms, it is like the cotangent of an angle.

## BABYLONIAN (MESOPOTAMIAN)

**Write the time period of Babylonian Era?**

The Babylonian era, often referred to as the period associated with the ancient Mesopotamian civilizations, spanned from around 2000 BCE to approximately 600 BCE. The Babylonian Empire formally ended in 538 BCE with the conquest by Cyrus of Persia.

**How did Babylonian save their records?**

The Babylonians saved their records by impressing them on soft clay tablets using styluses. These clay tablets were then baked in the hot sun or in ovens, making them more durable and less vulnerable to the ravages of time compared to Egyptian papyri.

**In which institutes mostly material (mathematical) of Babylonian era saved?**

The university libraries at Columbia, Pennsylvania, and Yale have large collections of ancient tablets from Mesopotamia, including some from the Babylonian era.

**In 9th century who tried to read Babylonian Script?**

F.W. Grotefend, a German philologist, attempted to read Babylonian Script in the 9th century.

**CUNEIFORM WRITING **

**What is Cuneiform Writing?**

Cuneiform writing was used by ancient Mesopotamian civilizations. It consisted of wedge-shaped marks on clay tablets, providing valuable insights into history, culture, and language.

Babylonian used special symbol for 10. Write it. Write Grotefend belief about it. Babylonian use special symbol for 10 and it’s < like this. Grotefend believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust out.

**In the Babylonian notation two principals were employed. Name them.**

In the Babylonian notation two principals were employed _ the additive and multiplicative. The additive system used wedge-shaped symbols to represent numbers, with the value of each symbol increasing from right to left. For example, a single wedge represented 1, two wedges represented 2, and so on. Meanwhile, the multiplicative system used different symbols to represent powers of 10. This allowed for larger numbers to be easily expressed using fewer symbols.

**What is cuneiform writing?**

Cuneiform writing was one of the earliest forms of writing in human history. It originated in ancient Mesopotamia around the 4th millennium BCE and was widely used by civilizations such as Sumerians, Akkadians, Babylonians and Assyrians. The term “cuneiform” comes from the Latin word “cuneus”,

**Write the Babylonian notation for 1,2,3,4,23 and 30 etc.**

The answer of the following question on the page no 30 and 31.

The usage of symbols and notations has been an integral part of human history, serving as a way to record and communicate information. The Babylonian civilization is known for their advanced numerical system, which was based on a combination of additive and multiplicative principles. This system provided valuable insights into their culture, language, and even their religious beliefs.

One interesting aspect of the Babylonian notation is their use of a special symbol for 10. This symbol, <, was believed by German philologist Georg Friedrich Grotefend to have originated from the picture of two hands held in prayer. The fingers were close together with the thumbs thrust outwards, representing the number 10.

Aside from this special symbol for 10, the Babylonians also used.

**How did Babylonian represent intermediate numbers?**

The Babylonians represented intermediate numbers by using a combination of vertical and oblique impressions with different stylus sizes. Vertical impressions represented 10 units, oblique impressions with the smaller stylus represented a unit, oblique impressions with the larger stylus represented 60 units.

## SEXAGESIMAL

**What is Sexagesimal system?**

The sexagesimal system is a numerical system that uses a base of 60. It originated in ancient Mesopotamia and was adopted for various measurements, including time and angle. It allows for easy subdivisions and has been used for centuries, despite the predominance of the decimal system in modern mathematics.

**What led to the invention of the sexagesimal system? This question have no exact answer but regarding this Cantor offered a theory. Write that.**

According to Cantor’s theory, the invention of the sexagesimal system may have been influenced by the Babylonians’ division of the circle into 360 degrees. They noticed that the radius can be applied to the circumference as a chord six times, with each chord subtending an arc of 60 degrees. This observation may have led them to divide the circle and other measurements into 60 parts, giving rise to the sexagesimal notation.

## POSITIONAL NUMERATION

**Is the zero symbol presented in Babylonian mathematics?**

They did not have a zero symbol, although they sometimes left a space where a zero was intended. Instead, they used a placeholder in their positional notation system. This was achieved by repeating the same symbol multiple times to indicate the absence of a value. For example, 60 would be written as two symbols for 10 placed side by side.

This positional notation system was revolutionary at the time and allowed for more complex calculations to be performed with ease. However, it did not have a true zero like we use today in modern mathematics.

The concept of zero was later developed by the ancient Indian mathematician Brahmagupta in the 7th century AD. He introduced the idea of negative numbers and defined them as being “an amount less than nothing.” This paved the way for zero to be used as an actual number rather than just a placeholder.

**Since Babyloinians had no symbiol for zero then what did they use for value having zero magnitude?**

They sometimes left a space where a zero was intended. This was a common practice in other ancient civilizations as well, such as the Mayans and the Romans. It wasn’t until the 9th century that the concept of zero was introduced to Europe by the Persian mathematician Al-Khwarizmi.

**Al-Khwarizmi’s** work on algebra and his use of Hindu-Arabic numerals greatly influenced European mathematics and helped pave the way for modern mathematics as we know it today. The introduction of zero allowed for more advanced mathematical concepts to be developed, such as calculus and complex equations.

Today, zero is an essential part of our everyday lives. From counting money to measuring time, zero plays a crucial role in many fields including science, technology, and finance. It may seem like a

**Shortly note on Posintional system of Babylonians. As they have no absolute positional system.**

The Babylonians had a positional system for representing numbers, but it was not absolute. They used symbols for units and tens, and these symbols could be assigned different values depending on their position in the number. However, the Babylonians did not have a zero symbol, which

made distinguishing between certain numbers challenging. They introduced a special sign, consisting of two oblique wedges, as a placeholder for missing numerals. This helped to alleviate some ambiguity, but their positional system remained relative rather than absolute.

## SEXAGESIMAL FRACTIONS

**Write the superiority of Babylonian mathematics over Egyptian mathematics.**

The superiority of Babylonian mathematics over Egyptian mathematics lies in the Babylonians’ extension of positional notation to cover fractions as well as whole numbers. By incorporating fractions into their notation system, the Babylonians gained computational power comparable to the modern decimal fractional notation. This allowed them to perform operations with fractions as effortlessly as with whole numbers, a crucial advantage that the Mesopotamians fully utilized.

**On what mathematical concept, Babylonian had at their command?**

The Babylonians had the concept of positional notation, which allowed them to represent both whole numbers and fractions using a consistent system. This greatly expanded their ability to perform calculations and solve complex problems, as they no longer had to rely on cumbersome methods for working with fractions. With positional notation, the Babylonians could easily add, subtract, multiply, and divide any combination of whole numbers and fractions.

The concept of positional notation is based on the idea that the value of a number depends on its place or position in a written representation. In this system, each digit has a specific value determined by its position in relation to other digits. For example, in our modern decimal system, the numeral “123” represents one hundred plus two tens plus three ones. The same concept applies to positional notation used by the Babylonians.

However, what made the Babylonian’s use of positional notation unique

**APPROXIMATIONS**

**In modern mathematics Which symbol is used to separate the integral and fractional parts? Show with example.**

In modern characters, this number can be appropriately written as 1;24,51,10, where a semicolon is used to separate the integral and fractional parts, and a comma is used as a separatrix for the sexagesimal positions.

**In modern mathematics Which symbol is used as a separatrix for the sexagesimal positions? Show with example.**

a comma is used as a separatrix for the sexagesimal positions. Here is an example for this 1;24,51,10,

**Explain Newton’s algorithm with example in Babylonian mathematics history.**

Newton’s algorithm, also known as the Babylonian square root algorithm, was a method used in Babylonian mathematics to approximate square roots. The algorithm involved iteratively refining an initial approximation to the desired square root. Here is an example of how it was used:

Let’s say we want to find the square root of 23 in Babylonian mathematics. First, we would start with an initial guess, let’s say 4. We then divide our number (23) by our guess (4), which gives us 5.75 as our new approximation.

Next, to refine this approximation, we take the average of our current approximation and the original number divided by our current approximation: (5.75 + (23/5.75))/2 = 5.28587…

We repeat this process until we have reached a desired level of accuracy or until the calculated average no longer changes significantly.

In this case, after a few more iterations, we get a final approximation of the square root of 23 as 4.79583152331, which is quite close to the actual value of 4.7958315233… (rounded to 10 decimal places).

This method may seem tedious compared to modern methods such as using a calculator, but it was a significant advancement in mathematics during its time and laid the foundation for more sophisticated techniques. It also allows us to understand and appreciate the development of mathematical concepts and how they have evolved over time.

Linearc.xyz offers various tools and resources for exploring different mathematical methods, including Babylonian mathematics and its root-finding algorithm. Our user-friendly interface makes it easy for anyone to input their desired numbers and see step-by-step calculations for finding square roots