history of numbers essay

The Evolution and History of Numbers and Counting

History of numbers: essay introduction, the egyptians/babylonians number history, the hindu-arabic number history, the mayan number history, history of numbers: essay conclusion, works cited.

This paper explores the evolution of number system from ancient to modern. Here, you’ll find information on the development of number system of the Egyptians/Babylonians, Romans, Hindu-Arabics, and Mayans.

The evolution of numbers developed differently with disparate versions, which include the Egyptian, Babylonians, Hindu-Arabic, Mayans, Romans, and the modern American number systems. The developmental history of counting is based on mathematical evolution, which is believed to have existed before the counting systems of numbers started (Zavlatsky 124).

The history of mathematics in counting started with the ideas of the formulation of measurement methods, which the Babylonians and Egyptians used, the introduction of pattern recognition in number counting in pre-historical times, the organization concepts of different shapes, sizes, and numbers by the pre-historical people, and the natural phenomenon observance and universe behaviors. This paper will highlight the evolution history of counting by the Egyptians/Babylonians, Romans, Hindu-Arabic, and Mayans’ counting systems. Moreover, the paper will outline the reasons why Western counting systems are widely used contemporarily.

The need for counting arose from the fact that the ancient people recognized the measurements in terms of more or less. Even though the assumption of numbers based its arguments on archeological evidence about 50,000 years ago, the counting system developed its background from the ancient recognition of more and less during routine activities (Higgins 87). Moreover, ancient people’s need for simple counting in history developed odd or even, more or less, and other forms of number systems evolved into the current counting systems. The need for counting developed from the fact that people needed a way of counting groups of individuals through population increase by birth. In addition, Menninger asserts that the daily activities of the pre-historical people, like cattle keeping and barter trade led to the need for counting and value determination (105).

For instance, in order to count cows, prehistoric people used sticks. Collecting and allocating sticks to count the animals helped determine the total number of animals present. The mathematical history evolved from marking rows on bones, tallying, and pattern recognition, which led to the introduction of numbers. The bones and wood were marked, as shown below.

Moreover, the development of numbers evolved from spoken words by pre-historical people. However, the pattern of numbers from one to ten has been difficult to trace. Fortunately, any pattern of numbers past ten is recognizable and easily traceable. For instance, eleven evolved from ein lifon, which was used to mean ‘one left’ over by the prehistoric people. Twelve developed from the lif, which meant “two leftovers” (Higgins 143). In addition, thirteen was traced from three and four from fourteen, and the pattern continued to nineteen. One hundred is derived from the word “ten times” (Ifrah and Bello 147). Furthermore, the written words used by the ancient people, like notches on wood carvings, stone carvings, and knots for counting, gave a solid base for the evolution of counting.

The Incas widely used counting boards for record-keeping. The Incas used the “quip,” which helped the pre-historical people record the items in their daily lives. The counting boards were painted with three different color levels. These were the darkest parts, representing the highest numbers; the lighter parts, representing the second-highest levels; and the white parts, representing the stone compartments (Havil 127). In addition, the quip was used to do fast mathematical computations (Zavlatsky 154). Generally, the quip used knots on cords, which were arranged in a certain way to give certain numeral information. However, the quip systems of record keeping and information have been associated with several mysteries which have not yet been established. Examples of how the knots looked are shown below.

This form is the common system of counting and numbers used in the 21st Century. In India, Al-Brahmi introduced the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 (Menninger 175). The Brahmi numerals kept changing with time. For instance, in the 4th to 6th Century, the numerals were as shown below.

Finally, the numerals were later developed to 1,2,3,4,5,6,7,8,9 with time. The earliest system of using zero was developed in Cambodia. The evolution of the decimal points emerged during the Saka era, whereby three digits and a dot in between were introduced (Hays and Schmandt-Besserat 198). The Babylonians introduced the positional system, whereby the place value of the numerical systems was established. Moreover, the positional system by the Babylonians developed the base systems to the numerical, and the Indians later developed it further. The Brahmi numerals took different incarnations to develop, which resulted in the current number system (Higgins 204).

The Gupta numerals were one of the processes passed by the Hindu-Arabic number system to become the commonly used American number version. Currently, theories about the formation and development of the Gupta numerals remain debatable by researchers.

In addition, the Europeans adopted the Hindu-Arabic system through trading, whereby the travelers used the Mediterranean Sea for trade interactions (Havil 190). The use of the abacus and the Pythagorean dominated the European number evolution. The Pythagorean used “sacred numbers” even though the two systems diminished after a short while. With time, the Europeans borrowed the Hindu-Arabic number system to establish their mathematical number systems (Ifrah and Bello, 207). However, the process through which the Europeans adopted the Hindu-Arabic system has not been proven fully. It is believed that the Europeans adopted the Hindu-Arabic number system by relying heavily on it to build their current strong numerals (Higgins 210). For instance, the scope of the positional base system is quite large, which involves the conversion of different bases using the numerical number 10.

The Mayan civilization of counting and number systems developed in Mexico through ritual systems. The rituals were calendar calculations involving two ritual systems, one for the priests and the other for the ordinary civilians (Higgins 217). For instance, priestly calendar counting used mixed base systems involving numerical number multiples. The Mayan number systems form the base of mathematical knowledge. Moreover, the Mayan system of numbers used the positioning of numbers to allocate the place value of the combined digits (Havil 223).

The Mayans used the place value of numerical numbers, which were tabled to add and multiply numbers. Ultimately, the Hindu-Arabic and the Mayan number systems contributed highly to the evolution of numbers as opposed to the Egyptian/Babylonian number systems (Menninger 199). Nevertheless, the Western number system of counting and mathematics incorporated the strong features of all the other evolutions to get a standard solid number system. For instance, the American system, commonly used in most countries, uses decimal points, place values, base values, and Roman numbers from 1 to 10 (Ifrah and Bello 225). The figure below represents a sketch of the tabled digits by the Mayans.

The American version of numbers and counting used all the development features of the Mayans, Babylonians, Incas, Egyptians, and Hindu-Arabic systems to develop a reliable and universally-accepted number system (Hays and Schmandt-Besserat 214). This aspect is outstanding as it makes the American system stand out of all the number systems and counting. Nevertheless, the commendable work of the Mayans, Babylonians, Egyptians, and Indians cannot be underrated, as the historical trace of counting and number systems would be impossible without them.

The historical trace of number systems and counting covers a wide scope of pre-historical archeological evidence. Tracing ancient times by researchers poses a significant challenge in establishing counting and number systems. The research on number systems and counting has not yet been settled on the actual source information for evidence. Ultimately, the most effective number systems that led to the current dominant Western number system are the Mayans, Hindu, and Babylonian systems relying on the Incas’ developments. The prehistoric remains left mathematical evidence as stones and wood carvings, which led to the evolution of counting. Hence mathematical methodologies evolved. The methodology of research and arguments varies on the evolution of numbers. Consequently, there are no universally-accepted research findings on the mathematical and number systems evolution.

Havil, Julian. The Irrationals : A Story of the Numbers You Cant Count on, Princeton: Princeton University Press, 2014. Print.

Hays, Michael, and Denise Schmandt-Besserat. The History of Counting , Broadway: HarperCollins, 1999. Print.

Higgins, Peter. Number Story: From Counting to Cryptography, Gottingen: Copernicus, 2008. Print.

Ifrah, Georges, and David Bello. The Universal History of Number: From Pre-history to the Invention of Computer , Hoboken: Wiley, 2000. Print.

Menninger, Karl. Number Words and Number Symbols; Cultural History of Numbers, Mineola: Dover Publications, 2011. Print.

Zavlatsky, Claudia. Africa Counts; Number and Pattern in Africa Cultures, Chicago: Chicago Review Press, 1999. Print.

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The Innovative Spirit fy17

A Smithsonian magazine special report

How Humans Invented Numbers—And How Numbers Reshaped Our World

Anthropologist Caleb Everett explores the subject in his new book, Numbers and the Making Of Us

Lorraine Boissoneault

Once you learn numbers, it’s hard to unwrap your brain from their embrace. They seem natural, innate, something all humans are born with. But when University of Miami associate professor Caleb Everett and other anthropologists worked with the indigenous Amazonian people known as the Pirahã, they realized the members of the tribe had no word used consistently to identify any quantity, not even one.

Intrigued, the researchers developed further tests for the Pirahã adults, who were all mentally and biologically healthy. The anthropologists lined up a row of batteries on a table and asked the Pirahã participants to place the same number in a parallel row on the other side. When one, two or three batteries were presented, the task was accomplished without any difficulty. But as soon as the initial line included four or more batteries, the Pirahã began to make mistakes. As the number of batteries in the line increased, so did their errors.

The researchers realized something extraordinary: the Pirahã’s lack of numbers meant they couldn’t distinguish exactly between quantities above three. As Everett writes in his new book, Numbers and the Making of Us , “Mathematical concepts are not wired into the human condition. They are learned, acquired through cultural and linguistic transmission. And if they are learned rather than inherited genetically, then it follows that they are not a component of the human mental hardware but are very much a part of our mental software—the feature of an app we ourselves have developed.”

To learn more about the invention of numbers and the enormous role they’ve played in human society, Smithsonian.com talked to Everett about his book.

How did you become interested in the invention of numbers?

It comes indirectly from my work on languages in the Amazon. Confronting languages that don’t have numbers or many numbers leads you inevitably down this track of questioning what your world would be like without numbers, and appreciating that numbers are a human invention and they’re not something we get automatically from nature. 

In the book, you talk at length about how our fascination with our hands—and five fingers on each—probably helped us invent numbers and from there we could use numbers to make other discoveries. So what came first—the numbers or the math?

I think it’s a cause for some confusion when I talk about the invention of numbers. There are obviously patterns in nature. Once we invent numbers, they allow us access to these patterns in nature that we wouldn’t have otherwise. We can see that the circumference and diameter of a circle have a consistent ratio across circles, but it’s next to impossible to realize that without numbers. There are lots of patterns in nature, like pi, that are actually there. These things are there regardless of whether or not we can consistently discriminate them. When we have numbers we can consistently discriminate them, and that allows us to find fascinating and useful patterns of nature that we would never be able to pick up on otherwise, without precision. 

Numbers are this really simple invention. These words that reify concepts are a cognitive tool. But it’s so amazing to think about what they enable as a species. Without them we seem to struggle differentiating seven from eight consistently; with them we can send someone to the moon. All that can be traced back to someone, somewhere saying, “Hey, I have a hand of things here.” Without that first step, or without similar first steps made to invent numbers, you don’t get to those other steps. A lot of people think because math is so elaborate, and there are numbers that exist, they think these things are something you come to recognize. I don’t care how smart you are, if you don’t have numbers you’re not going to make that realization. In most cases the invention probably started with this ephemeral realization [that you have five fingers on one hand], but if they don’t ascribe a word to it, that realization just passes very quickly and dies with them. It doesn’t get passed on to the next generation.

Numbers and the Making of Us: Counting and the Course of Human Cultures

Another interesting parallel is the connection between numbers and agriculture and trade. What came first there?

I think the most likely scenario is one of coevolution. You develop numbers that allow you to trade in more precise ways. As that facilitates things like trade and agriculture, that puts pressure to invent more numbers. In turn those refined number systems are going to enable new kinds of trade and more precise maps, so it all feeds back on each other. It seems like a chicken and egg situation, maybe the numbers came first but they didn’t have to be there in a very robust form to enable certain kinds of behaviors. It seems like in a lot of cultures once people get the number five, it kickstarts them. Once they realize they can build on things, like five, they can ratchet up their numerical awareness over time. This pivotal awareness of “a hand is five things,” in many cultures is a cognitive accelerant. 

How big a role did numbers play in the development of our culture and societies?

We know that they must play some huge role. They enable all kinds of material technologies. Just apart from how they help us think about quantities and change our mental lives, they allow us to do things to create agriculture. The Pirahã have slash and burn techniques, but if you’re going to have systematic agriculture, they need more. If you look at the Maya and the Inca, they were clearly really reliant on numbers and mathematics. Numbers seem to be a gateway that are crucial and necessary for these other kinds of lifestyles and material cultures that we all share now but that at some point humans didn’t have. At some point over 10,000 years ago, all humans lived in relatively small bands before we started developing chiefdoms. Chiefdoms come directly or indirectly from agriculture. Numbers are crucial for about everything that you see around you because of all the technology and medicine. All this comes from behaviors that are due directly or indirectly to numbers, including writing systems. We don’t develop writing without first developing numbers. 

How did numbers lead to writing?

Writing has only been invented in a few cases. Central America, Mesopotamia, China, then lots of writing systems evolved out of those systems. I think it’s interesting that numbers were sort of the first symbols. Those writings are highly numeric centered. We have 5,000-year-old writing tokens from Mesopotamia, and they’re centered around quantities. I have to be honest, because writing has only been invented in a few cases, [the link to numbers] could be coincidental. That’s a more contentious case. I think there are good reasons to think numbers led to writing, but I suspect some scholars would say it’s possible but we don’t know that for sure.  

Something else you touch on is whether numbers are innately human, or if other animals could share this ability. Could birds or primates create numbers, too?

It doesn’t seem like on their own they can do it. We don’t know for sure, but we don’t have any concrete evidence they can do it on their own. If you look at Alex the African grey parrot [and subject of a 30-year study by animal psychologist Irene Pepperberg], what he was capable of doing was pretty remarkable, counting consistently and adding, but he only developed that ability when it was taught over and over, those number words. In some ways this is transferrable to other species—some chimps seem able to learn some basic numbers and basic arithmetic, but they don’t do it on their own. They’re like us in that they seem capable of it if given number words. It’s an open question of how easy it is. It seems easy to us because we’ve had it from such an early age, but if you look at kids it doesn’t come really naturally. 

What further research would you like to see done on this subject?

When you look at populations that are the basis for what we know about the brain, it’s a narrow range of human cultures: a lot of American undergrads, European undergrads, some Japanese. People from a certain society and culture are well represented. It would be nice to have Amazonian and indigenous people be subject to fMRI studies to get an idea of how much this varies across cultures. Given how plastic the cortex is, culture plays a role in the development of the brain.  

What do you hope people will get out of this book?

I hope people get a fascinating read from it, and I hope they appreciate to a greater extent how much of their lives that they think is basic is actually the result of particular cultural lineages. We’ve been inheriting for thousands of years things from particular cultures: the Indo-Europeans whose number system we still have, base ten. I hope people will see that and realize this isn’t something that just happens. People over thousands of years had to refine and develop the system. We’re the benefactors of that.

I think one of the underlying things in the book is we tend to think of ourselves as a special species, and we are, but we think that we have really big brains. While there’s some truth to that, there’s a lot of truth to the idea that we’re not so special in terms of what we bring to the table genetically; culture and language are what enable us to be special. The struggles that some of those groups have with quantities is not because there’s anything genetically barren about them. That’s how we all are as people. We just have numbers.

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Lorraine Boissoneault | | READ MORE

Lorraine Boissoneault is a contributing writer to SmithsonianMag.com covering history and archaeology. She has previously written for The Atlantic, Salon, Nautilus and others. She is also the author of The Last Voyageurs: Retracing La Salle's Journey Across America. Website: http://www.lboissoneault.com/

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A brief history of numbers and counting, Part 1: Mathematics advanced with civilization

By Deseret News , Steven Law, For the Deseret News

The origins of numbers are cloaked in mystery. But, I think it’s safe to say that as civilization advanced numbers advanced with it; and it is equally safe to say that civilization could not have advanced without it.

Common intuition, and recently discovered evidence, indicates that numbers and counting began with the number one. (Even though in the beginning, they likely didn’t have a name for it.) The first solid evidence of the existence of the number one, and that someone was using it to count, appears about 20,000 years ago. It was just a unified series of unified lines cut into a bone. It’s called the Ishango Bone.

The Ishango Bone (it’s a fibula of a baboon) was found in the Congo region of Africa in 1960. The lines cut into the bone are too uniform to be accidental. Archaeologists believe the lines were tally marks to keep track of something, but what that was isn’t clear.

But numbers, and counting, didn’t truly come into being until the rise of cities. Indeed numbers and counting weren’t really needed until then. Numbers, and counting, began about 4,000 BC in Sumeria, one of the earliest civilizations. With so many people, livestock, crops and artisan goods located in the same place, cities needed a way to organize and keep track of it all, as it was used up, added to or traded.

Their method of counting began as a series of tokens. Each token a man held represented something tangible, say five chickens. If a man had five chickens he was given five tokens. When he traded or killed one of his chickens, one of his tokens was removed. This was a big step in the history of numbers and counting because with that step subtraction — and thus the invention of arithmetic — was invented.

In the beginning Sumerians kept a group of clay cones inside clay pouches. The pouches were then sealed up and secured. Then the number of cones that were inside the clay pouch was stamped on the outside of the pouch, one stamp for each cone inside. Someone soon hit upon the idea that cones weren’t needed at all. Instead of having a pouch filled with five cones with five marks written on the outside of the pouch, why not just write those five marks on a clay tablet and do away with the cones altogether? This is exactly what happened.

This development of keeping track on clay tablets had ramifications beyond arithmetic, for with it, the idea of writing was also born.

But, if you’re keeping track of your wealth with marks made on a clay tablet what’s to stop you from making your own clay tablet and stamping in 50 marks, and trading those 50 marks on a clay tablet for grain?

To prevent this from happening, the Sumerians needed an official method of keeping track, and an official group of people who kept track. A select few were allowed to enter this group. They essentially became the world’s first accountants. So a farmer may have made his own clay tablet with 50 marks on it and claimed that this proved that he was the owner of 50 chickens, but if that tablet didn’t have an official seal from the accountants it was worthless.

It was the Egyptians who transformed the number one from a unit of counting things to a unit of measuring things. In Egypt, around 3,000 BC, the number one became used as a unit of measurement to measure length. If you’re going to build pyramids, temples, canals and obelisks you’re going to need a standard unit of measurement — and an accurate method of applying it to real objects. What they invented was the cubit, which they considered to be a sacred measurement. A cubit is the length of a man’s forearm, from elbow to fingertips, plus the width of his palm. Considered sacred as they were, they had officially ordained sticks which they kept in the temples. If copy cubits were needed they were made from one of the original cubits kept in the temple. Thanks to this very official, very guarded and very precise unit of measurement the Egyptians were able to create colossal buildings and monuments with wondrous accuracy.

The Egyptians were the first civilization to invent different symbols for different numbers. They had a symbol for one, which was just a line. The symbol for ten was a rope. The symbol for a hundred was a coil of rope. They also had numbers for a thousand and ten thousand. The Egyptians were the first to dream up the number one million, and its symbol was a prisoner begging for forgiveness, which was a person on its knees, hands upraised in the air, in a posture of humility.

Greece made further contributions to the world of numbers and counting, much of it under the guidance of Pythagoras. He studied in Egypt and upon returning to Greece established a school of math, introducing Greece to mathematical concepts already prevalent in Egypt. Pythagoras was the first man to come up with the idea of odd and even numbers. To him, the odd numbers were male; the evens were female. He is most famous for his Pythagorean theorem, but perhaps his greatest contribution to math was laying the groundwork for Greek mathematicians who would follow him.

Pythagoras was one of the world’s first theoretical mathematicians, but it was another famous Greek mathematician, Archimedes, who took theoretical mathematics to a level no one had ever taken it to before. Archimedes is considered to the greatest mathematician of antiquity and one of the greatest of all time. Archimedes enjoyed doing experiments with numbers and playing games with numbers.

But as trivial as his math games may have seemed to outsiders they often led to results that proved practical in the real world, some of which we still benefit from today. One example: Archimedes wondered if you could turn the surface of a sphere into a cylinder, and if you did, what would be the difference in area covered? Archimedes successfully worked this problem out, and to him that was the end of it. But thanks to the formulas he left behind, later mapmakers were able to turn the surface of the globe into a flat map.

Archimedes is also famous for his Archimede’s screw, which is a circular inclined plane (a screw) inside a tube that pumps water from one level to a higher level. He is equally famous for inventing a method of determining the volume of an object with an irregular shape. The answer came to him while he was bathing. He was so excited he leapt from his tub and ran naked through the streets screaming “Eureka!,” which is Greek for “I have found it.”

Archimedes made many, many other mathematical contributions, but they are too numerous to mention here during a brief history of numbers.

The Greek’s role in mathematics ended, quite literally, with Archimedes. He was killed by a Roman soldier during the Siege of Syracuse in 212 BC. And thus ended the golden age of mathematics in the classical world. Under the rule of Rome, mathematics entered a dark age, and for a couple different reasons.

In Part 2 we’ll look at numbers from the mathematical dark ages of the Romans to the modern digital age.

If you'd like to read Steven Law's previous science articles go to www.curiosity101.com .

A Brief History of Numbers: How 0-9 Were Invented

Have you ever wondered how numbers first came about?

Using only ten symbols (0 – 9), we can write and rational number imaginable. But why do we use these ten symbols? And why is there 10 of them?

Strange as it seems to us now, there was a time when numbers, as we know them, simply weren’t invented.

How early humans kept count

Early humans in the Paleolithic age likely counted animals and other everyday objects by carving tally marks into cave walls, bones, wood or stone. Each tally mark stood for one and each fifth mark was scored through to help keep track.

This system is fine for small numbers, but it doesn’t really work with large numbers – try writing 27,890 using tally marks.

Symbols for numbers developed with early civilizations

As early civilizations developed, they came up with different ways of writing down numbers. Many of these systems, including Greek, Egyptian and Hebrew numerals, were essentially extensions of tally marks. The used a range of different symbols to represent larger values. For example, in the Ancient Egyptian system, a coiled rope represented 100 and a water lily represented 1000.

Each symbol was repeated as many times as necessary and all were added together, so under the Ancient Egyptian system, 300 would be shown as three coiled ropes.

But even with this system, it was still a cumbersome method for writing large numbers.

Positional notation: An easier way to write down large numbers

Early number systems all have one thing in common. They require someone to write down many symbols to record a single number and create new symbols for each larger number.

A positional system allows you to reuse the same symbols, by assigning the symbols different values based on their position in the sequence.

Several civilisations developed positional notation independently, including the Babylonians, the Chinese and the Aztecs.  

By the 7th Century, Indian mathematicians had perfected a decimal (or base ten) positional system, which could represent any number with only ten unique symbols. Over the next few centuries, Arab merchants, scholars and conquerors began to spread it into Europe.

A key breakthrough of this particular system (which was also independently developed by the Mayans) was the number 0. Older positional notation systems, which didn’t have 0, would leave a blank in its place, making it hard to distinguish between 63 and 603 or 12 and 120. Having and using 0 helps make writing down numbers clearer and easier for everyone to understand.

Positional notation doesn’t have to be based around a decimal or base 10 system. The Babylonians invented a base 60 system, which is still the foundation of the way we now tell time: each day is made up of 60 minute hours and 60 second minutes.

Modern ways of managing numbers and complex calculations

Today, we mostly take our number system for granted. 

Modern students are no longer worrying about the best way to record numbers. Instead, they build skills to check the reasonableness of answers and must be familiar with a wide range of mathematical knowledge to know that the answer is correct.

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5.1: The Evolution of Numeration Systems

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Introduction

As we begin our journey through the history of mathematics, one question to be asked is “Where do we start?” Depending on how you view mathematics or numbers, you could choose any of a number of launching points from which to begin. Howard Eves suggests the following list of possibilities.[i]

Where to start the study of the history of mathematics…

• At the first logical geometric “proofs” traditionally credited to Thales of Miletus (600 BCE).
• With the formulation of methods of measurement made by the Egyptians and Mesopotamians/Babylonians.
• Where prehistoric peoples made efforts to organize the concepts of size, shape, and number.
• In pre-human times in the very simple number sense and pattern recognition that can be displayed by certain animals, birds, etc.
• Even before that in the amazing relationships of numbers and shapes found in plants.
• With the spiral nebulae, the natural course of planets, and other universe phenomena.

We can choose no starting point at all and instead agree that mathematics has always existed and has simply been waiting in the wings for humans to discover. Each of these positions can be defended to some degree and which one you adopt (if any) largely depends on your philosophical ideas about mathematics and numbers.

Nevertheless, we need a starting point. Without passing judgment on the validity of any of these particular possibilities, we will choose as our starting point the emergence of the idea of number and the process of counting as our launching pad. This is done primarily as a practical matter given the nature of this course. In the following chapter, we will try to focus on two main ideas. The first will be an examination of basic number and counting systems and the symbols that we use for numbers. We will look at our own modern (Western) number system as well those of a couple of selected civilizations to see the differences and diversity that is possible when humans start counting. The second idea we will look at will be base systems. By comparing our own base-ten (decimal) system with other bases, we will quickly become aware that the system that we are so used to, when slightly changed, will challenge our notions about numbers and what symbols for those numbers actually mean.

Recognition of More vs. Less

The idea of numbers and the process of counting goes back far beyond when history began to be recorded. There is some archeological evidence that suggests that humans were counting as far back as 50,000 years ago.[ii] However, we do not really know how this process started or developed over time. The best we can do is to make a good guess as to how things progressed. It is probably not hard to believe that even the earliest humans had some sense of more and less . Even some small animals have been shown to have such a sense. For example, one naturalist tells of how he would secretly remove one egg each day from a plover’s nest. The mother was diligent in laying an extra egg every day to make up for the missing egg. Some research has shown that hens can be trained to distinguish between even and odd numbers of pieces of food.[iii] With these sorts of findings in mind, it is not hard to conceive that early humans had (at least) a similar sense of more and less. However, our conjectures about how and when these ideas emerged among humans are simply that; educated guesses based on our own assumptions of what might or could have been.

The Need for Simple Counting

As societies and humankind evolved, simply having a sense of more or less, even or odd, etc., would prove to be insufficient to meet the needs of everyday living. As tribes and groups formed, it became important to be able to know how many members were in the group, and perhaps how many were in the enemy’s camp. Certainly it was important for them to know if the flock of sheep or other possessed animals were increasing or decreasing in size. “Just how many of them do we have, anyway?” is a question that we do not have a hard time imagining them asking themselves (or each other).

In order to count items such as animals, it is often conjectured that one of the earliest methods of doing so would be with “tally sticks.” These are objects used to track the numbers of items to be counted. With this method, each “stick” (or pebble, or whatever counting device being used) represents one animal or object.

Of course, in our modern system, we have replaced the sticks with more abstract objects. In particular, the top stick is replaced with our symbol “1,” the second stick gets replaced by a “2” and the third stick is represented by the symbol “3,” but we are getting ahead of ourselves here. These modern symbols took many centuries to emerge.

Another possible way of employing the “tally stick” counting method is by making marks or cutting notches into pieces of wood, or even tying knots in string (as we shall see later). In 1937, Karl Absolom discovered a wolf bone that goes back possibly 30,000 years. It is believed to be a counting device.[iv] Another example of this kind of tool is the Ishango Bone, discovered in 1960 at Ishango, and shown below.[v] It is reported to be between six and nine thousand years old and shows what appear to be markings used to do counting of some sort.

Spoken Words

As methods for counting developed, and as language progressed as well, it is natural to expect that spoken words for numbers would appear. Unfortunately, the developments of these words, especially those corresponding to the numbers from one through ten, are not easy to trace. Past ten, however, we do see some patterns:

Eleven comes from “ein lifon,” meaning “one left over.”

Twelve comes from “twe lif,” meaning “two left over.”

Thirteen comes from “Three and ten” as do fourteen through nineteen.

Twenty appears to come from “twe-tig” which means “two tens.”

Hundred probably comes from a term meaning “ten times.”

Written Numbers

When we speak of “written” numbers, we have to be careful because this could mean a variety of things. It is important to keep in mind that modern paper is only a little more than 100 years old, so “writing” in times past often took on forms that might look quite unfamiliar to us today.

As we saw earlier, some might consider wooden sticks with notches carved in them as writing as these are means of recording information on a medium that can be “read” by others. Of course, the symbols used (simple notches) certainly did not leave a lot of flexibility for communicating a wide variety of ideas or information.

Other mediums on which “writing” may have taken place include carvings in stone or clay tablets, rag paper made by hand (12 th century in Europe, but earlier in China), papyrus (invented by the Egyptians and used up until the Greeks), and parchments from animal skins. And these are just a few of the many possibilities.

These are just a few examples of early methods of counting and simple symbols for representing numbers. Extensive books, articles and research have been done on this topic and could provide enough information to fill this entire course if we allowed it to. The range and diversity of creative thought that has been used in the past to describe numbers and to count objects and people is staggering. Unfortunately, we don’t have time to examine them all, but it is fun and interesting to look at some systems in more detail.

Egyptian Hieroglyphs

The system of ancient Egyptian numerals was used in Ancient Egypt from around 3000 BCE until the early first millennium CE. It was a system of numeration based on multiples of ten written in hieroglyphs. The Egyptians had no concept of a place-valued system such as the decimal system. The hieratic form of numerals stressed an exact finite series notation, ciphered one-to-one onto the Egyptian alphabet.

These symbols represent the powers of ten. From left to right: 1 (stroke), 10 (heel bone), 100 (scroll), 1000 (lotus flower), 10,000 (finger), 100,000 (tadpole), 1,000,000 (astonished person).

They used a simple grouping system by combining multiple symbols of the same value to add them up.

Convert to our decimal system:

Since there are 3 scrolls, 4 heel bones, and 7 strokes, this represents the number 347.

The Egyptians also had a method for multiplication. This method is based on the fact that every positive integer can be written as the sum of different powers of 2.

Write the numbers 35 and 79 as a sum of powers of 2.

\(35=2^5+2^1+2^0=32+2+1\)

\(79=2^6+2^3+2^2+2^1+2^0=64+8+4+2+1\)

Multiply \(35 \times 79\)

\(35 \times 79\ = 35 \times (64+8+4+2+1)\)

\(35 \times 79\ = 35 \times 64 + 35 \times 8 + 35 \times 4 + 35 \times 2 + 35 \times 1\)

Now make a table of Powers of 2 times 35 just by doubling 35 many times.

Thus \(35 \times 79\ =2240+280+140+70+35=2765\).

Roman Numerals

Even though Roman Numerals are rare in today’s society, they are still used and expected to be understood. They are taught in grades three through five, depending on the district. They can be seen in clocks, the Super Bowl, Film Credits for the copyright date like MCMLXII, preface of textbooks and others like Star Wars Episode VI and WWII.

An line over a numeral means to multiply by 1000. You'll see in this next table.

For all numbers except 4 and 9, we ADD the Roman Numerals together, in order from left to right, greatest value to lowest value.

\[\begin{aligned} 11 &=10+1=X I \\ 8 &=5+1+1+1=V I I I \\ 123 &=100+10+10+1+1+1=C X X I I I \\ 3816 &=1000+1000+1000+500+100+100+100+10+5+1=M M M D C C C X V I \\ 7002 &=5000+1000+1000+1+1=\overline{V I I }I I \end{aligned} \nonumber \]

For any number that includes a 4 or a 9, we subtract. When we are looking at a Roman Number expression and we see a Roman character OUT OF ORDER , which is the clue to SUBTRACT !

\[\begin{aligned} 4 &=5-1=I V \\ 9 &=10-1=I X \\ 40 &=50-10=XL \\ 99 &=(100-10)+(10-1)=X C I X \\ 400 &=500-100=C D \end{aligned} \nonumber \]

Notice \(99 \neq 100-1\) because you can only subtract numerals from the next two higher numerals.

[i] Eves, Howard; An Introduction to the History of Mathematics, p. 9.

[ii] Eves, p. 9.

[iii] McLeish, John; The Story of Numbers - How Mathematics Has Shaped Civilization, p. 7.

[iv] Bunt, Lucas; Jones, Phillip; Bedient, Jack; The Historical Roots of Elementary Mathematics, p. 2.

[v] http://www.math.buffalo.edu/mad/Ancient-Africa/mad_zaire-uganda.html

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A brief history of numbers.

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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

• Table of Contents

This fine book gives what its title promises. It was written for students and teachers of mathematics and, of course, anyone else who would appreciate a well-written treatment of the subject. It is a popularization in that it is not burdened with footnotes or other scholarly apparatus but it is not a popularization because the author is not writing down to his readers. He is writing right at them.

The history follows the conventional course, from the ancient Egyptians and Greeks through the Arabs and medieval and renaissance Europe to Descartes, Newton, and on through the twentieth century. The author says that he left out most of Asia, Africa, and all of the Americas because there wasn’t space for everything.

One thing, among many, that the book does well is bring out the tremendous difficulty we have had in arriving at our present understanding of numbers. For the ancient Greeks, 1 was not a number. It was the unit, out of which numbers were made. It wasn’t too long, in historical terms, before it became a number, but I’m sure that struggle was involved, the dinosaurs against the revolutionaries. The struggle was even greater with negative numbers. They were first rejected, and then sort of accepted without being understood. Wallis (1616–1703), while doing things like writing an infinite product for \(\pi\), was unable to get his head around them, and even in the first half of the nineteenth century they had their opponents.

Another is to point out the very different ways our predecessors thought. For at least a millennium and a half, a ratio like 2:3 was not a number. It was a ratio, an entirely different thing. A proportion like 2:3 :: 4:6 had nothing to do with numbers. It was a proportion. Equalities were comparisons, not equations that could be operated on. Numbers were like labels. You can’t multiply labels. 2:3 = .666… made no sense at all.

“What was wrong with those people?” is what too many of us think, consciously or not. The author shows that there was nothing wrong with them. They were doing difficult work and making progress. Try doing algebra without symbols, using nothing but words, and you’ll see what they were up against.

The book contains much that will be new, even to experienced readers. For example, I never thought about how Euclid was careful to put geometry on what he thought was a firm foundation with his axioms and postulates, but when it came to proving that there are infinitely many primes he didn’t include any similar foundation for numbers.

In his last chapters the author doesn’t hesitate to lay out material that some mathematics undergraduates don’t encounter. There are algebraic numbers, with a proof of their countablilty, Kummer’s ideal numbers, and a development of the number system, starting with Peano’s postulates for the positive integers, to integers as pairs of positive numbers, rationals as pairs of integers, reals as Dedekind cuts, and complexes as pairs of reals. There are transfinite cardinals and ordinals. There is the axiom of choice and well-ordering. Good stuff.

The copy editing leaves something to be desired. Displayed lines have inconsistent punctuation, there are symbols not in italics that should be, and in the space of sixteen pages we find

p. 148: Bombelli writing “ R . q .21” for the square root of ‒121

p. 152: congruences “module” 9

p. 153: a parenthesis closed that was never opened

p. 158: 6 times 216 is 393216

p. 163: Viète writing “it is not the reckoning’s fault bu the reckoner’s”.

The meaning is not obscured, though you may have to think a second or two about the fourth, but one expects better. If the OUP needs to save money I would rather it use cheaper paper instead of cheaper copy editors.

Woody Dudley was born so long ago that his days are numbered. They are now positive and rational, but may become irrational and then in any case, inevitably, imaginary.

1. The System of Numbers: An Overview 2. Writing Numbers: Now and Back Then 3. Numbers and Magnitudes in the Greek Mathematical Tradition 4. Construction Problems and Numerical Problems in the Greek Mathematical Tradition 5. Numbers in the Tradition of Medieval Islam 6. Numbers in Europe from the 12th to the 16th Centuries 7. Number and Equations at the Beginning of the Scientific Revolution 8. Number and Equations in theWorks of Descartes, Newton, and their Contemporaries 9. New Definitions of Complex Numbers in the Early 19th Century 10. "What are numbers and what should they be?" Understanding Numbers in the Late 19th Century 11. Exact Definitions for the Natural Numbers: Dedekind, Peano and Frege 12. Numbers, Sets and Infinity. A Conceptual Breakthrough at the Turn of the Twentieth Century 13. Epilogue: Numbers in Historical Perspective

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The number e.

Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.

References ( show )

• E Maor, e : the story of a number ( Princeton, 1994) .
• J L Coolidge, The number e, Amer. Math. Monthly 57 (1950) , 591 - 602 .

Intellectual Mathematics

How to write a history of mathematics essay

This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.

The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.

So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.

Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.

The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.

Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.

For many examples of how one might write a critique, see my book reviews categorised as “critical.”

Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.

Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?

Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.

When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?

Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.

The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.

Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.

Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn ? Or does mathematics work according to the model of Kitcher ? Or that of Lakatos or Crowe ? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska ? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?

Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.

Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.

In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book . It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper ) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis , 1621, or many other works listed in a bibliography by Schüling .

Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.

Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.

Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:

The table shows the state of the literature before and after your research project has been carried out.

A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.

You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.

An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.

Many areas of the history of mathematics have been thoroughly Described but never Explained.

What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.

Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.

See also History of mathematics literature guide .

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In This Article Expand or collapse the "in this article" section Book of Numbers

Introduction.

• General Introductions to the Pentateuch
• Ancient Jewish Midrashim
• Ancient Christian Interpretation
• Modern Commentaries, 1900–1990
• Theological Commentaries
• Surveys of Recent Critical Scholarship
• Structure of the Book
• Studies on Nonpriestly Texts
• Studies on Priestly Texts
• Redactional Composition
• Numbers 6:1–21
• Numbers 6:22, 23–27
• Numbers 16–17
• Numbers 28–30

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Book of Numbers by Reinhard Achenbach LAST REVIEWED: 22 November 2022 LAST MODIFIED: 15 December 2011 DOI: 10.1093/obo/9780195393361-0068

The book of Numbers is the fourth part of the Torah (or, in Greek, the Pentateuch). In the Mishnah and in the Talmud it is named Homesh ha-Pequdîm (i.e., the Fifth of the Mustered) because of the censuses recorded at the beginning of the book and in chapter 26. The Septuagint names the part Arithmoi (Censuses or Numbers), while the Vulgate calls it Numeri (Numbers). Rashi (Rabbi Shelomo ben Yitzhak) referred to the book with its first words, “wa-yedabber” (i.e., “and he said”), but the traditional Hebrew name is Be-midbar (i.e., “in the wilderness” [of Sinai]). In Christian Bibles it is sometimes called the Fourth Book of Moses, presuming Moses’s authorship of the Pentateuch. As the first and last verses of the final composition of the book indicate, the content was thought to contain the revelations Moses received from God during the wandering of the Israelites from the wilderness of Sinai to the “plains of Moab” at the border of the promised land. The reports of these revelations are connected to a series of narratives from scribal epical and historical tradition on the wilderness wandering and the conquest of Transjordan. The book also contains material from the priestly tradition, such as narratives and regulations about proper procedures in the sanctuary, about the institutions of Israel, and about ritual obligations concerning purity and with regard to land conquest and land inheritance. The frequent change between the narratives and legal material from the priestly tradition is characteristic of the book’s final composition. The core of the book also contains parts from different narrative cycles on a confrontation with the Moabites in the Balaam story, which take up the preexilic epic tradition (9th–6th centuries BCE ) and were rewritten and expanded by priestly scribes in the postexilic period of the Second Temple in Jerusalem (5th–4th centuries BCE ). This was rewritten and expanded by Pentateuch redactors, who stressed the authority of Moses and Aaron as mediators of God’s will and the Torah. For general information, see the articles on Numbers in encyclopedias and lexicas about the Bible.

The biblical text of the book of Numbers has a long history of tradition. Scholars have tried to reconstruct the original text by comparing all manuscripts available, including ancient translations. The most acknowledged Hebrew Masoretic version from the St. Petersburg manuscript, which goes back to the family of Ben Asher (Tiberias, 1008 CE ), is the basic text of the critical edition Elliger and Rudolph 1977 . Among the Samaritan manuscripts from the congregation of the Samaritans, Tal 1994 is one of the most important editions. The most significant translations are those in Aramaic; their roots go back to the reading of the holy scripture in the ancient synagogue already at rabbinic times. Among the many editions of the Targum Onkelos, the best place to begin is Rosenbaum and Silbermann 1972 or Sperber 2004 . Greek translations, which often rely on very old variants of the Hebrew text, are available in several critical editions, an older one being Rahlfs 1935 . More recent editions are Wevers 1982 and Wevers 1998 , which comes with annotations. For further ancient versions of the biblical text there is an online directory, Marlowe 2001–2011 .

Elliger, Karl, and Wilhelm Rudolph, eds. Biblia Hebraica Stuttgartensia . Rev. ed. Stuttgart: Deutsche Bibelgesellschaft, 1977.

Presents the text of the Codex Petropolitanus B 19, from 1008 CE , with critical annotations on variants. The edition is used by all scholars as a basic text for research and has been reprinted several times. A new edition is under work, examining all text variants of the manuscript versions from the Qumran caves.

Marlowe, Michael D. Bible Research: Internet Resources for Students of Scripture . 2001–2011.

This is a wonderful collection of links to ancient versions of the Bible for students and scholars who are looking for detailed information on texts, versions, and history of the canon.

Rahlfs, Alfred, ed. Septuaginta . Stuttgart: Württembergische Bibelanstalt, 1935.

Classical Septuagint edition.

Rosenbaum, M., and A. M. Silbermann, eds. Pentateuch with Targum Onkelos, Haphtaroth, and Rashi’s Commentary, Numbers . Jerusalem: Silbermann Family, 1972.

A traditional edition of the Chumash for rabbinic use, including the Aramaic Targum text of Numbers.

Sperber, Alexander, ed. The Bible in Aramaic: Based on the Old Manuscripts and Printed Texts . Leiden, The Netherlands: Brill, 2004.

A reprint of the 1959–1962 classical critical edition of the Aramaic Targum text.

Tal, Avraham, ed. The Samaritan Pentateuch: Edited according to MS 6 ( C ) of the Shekhem Synagoge . Texts and Studies in the Hebrew Language and Related Subjects 8. Tel Aviv: Tel Aviv University, 1994.

The editor has provided the oldest complete version of the Samaritan Pentateuch.

Wevers, John William, ed. Septuaginta: Vetus Testamentum Graecum . Vol. 3. Göttingen, Germany: Vandenhoeck & Ruprecht, 1982.

Advanced critical edition of the Göttingen Septuagint with references to many manuscripts in the notes.

Wevers, John William. Notes on the Greek Text of Numbers . Society of Biblical Literature Septuagint and Cognate Studies Series 46. Atlanta: Scholars Press, 1998.

Gives valuable explanations about text-critical decisions.

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History of numbers

Introduction

Numbers were first invented in 35,000 B.C. during which whole numbers and rational numbers were the most applicable variables in their commercial calculations. The history of numbers can, therefore, be traced back to the discovery era as artifacts and bones which formed tally marks. People preferred numbers which could be constructed mechanically. However, the Isaac Newton’s inventions introduced the idea of continuous variable numbers in1600 B.C. Later on in the 1800s, the discontinuous functions were introduced in order to clear the confusion brought about by the continuous variable. The geometry and other more technical problems, however, led to the introduction of square roots and development of algebraic numerals. Innovations and technological advancements have thus led to the invention of different types of numbers classified into sets. This paper seeks to analyze the History of Numbers and find out what a “Big Number” is and whether there is a zillion numbers.

Types of numbers

Natural numbers.

The natural numbers was first invented in 1500B.C by the Egyptians through the use of distinct hieroglyphs. Initially, numbers 1-10 was first invented and later they introduced all numbers with powers of 10 and their highest limit was a million. During this period zero was not considered as a number. In the 1 st century B.C the Mayan civilization started using zero as a number a practice which only revolved within the Mesoamerica. The first universal study of numbers was credited due to the works of Archimedes and Pythagoras famous philosophers in the Greek empire. The two emphasized on a more advanced natural numbers. The natural numbers are the most commonly used form of numbers which seeks to enhance the arithmetic operations. The natural numbers are mainly based on tens number system where the expressions are mainly done in ten digits. In this case, the rightmost digit mostly assumes the value of ones while any other digit placed after the rightmost digit assume the value of ten times the value of the digit to its right. A letter N is also commonly used when referring to a set of natural numbers. It is also important to note that natural numbers are also referred to as positive integers (Joshi, 1989). The numbers are therefore said to comprise a set of non-negative integers. Among the notable properties of the natural numbers includes the ability to implement some additions, subtractions, multiplication and divisions. All these properties make it easy to use natural numbers in mathematical calculations.

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This can be defined as numbers which have a value less than zero. Integers comprise of the negative numbers and are usually noted by the negative sign which is put in front of the number. Integers and natural numbers can be combined to form a set of integers denoted by letter Z. It is important to understand that integers consist of the natural numbers but a negative sign is added onto them. Integers also comprise of the smallest group of numbers. Similarly, integers have the associative, distributive and communicative properties which make them easier to manipulate (Flegg, 2002). The properties also make the integers user-friendly while computing.

Rational numbers

This is a fraction which has a non-zero denominator. A rational number, therefore, represents a fraction of the whole. During the initial stages, integers were mainly reciprocals of the positive integers. But with the mathematics advancements, fractions which consist of both numerators and denominators have been improvised. The continued advancement has enhanced rational numbers to be expressed in decimal forms (Kline, 1990). In such case, the denominator is mainly powered by tens, hundreds or so considering the number of digits to the right of the decimal. It is possible to do some additions, subtractions, multiplications and divisions to the rational numbers. In some instances the fraction numbers are rationalized, this mostly occurs when the denominator consist of irrational number or complex numbers (Nahin, 1998).

Real number

This are all measuring numbers, either whole or fraction, positive or negative. Real numbers assist in improving the accuracy in the measurement parameters as they tend to be more precise. It sometimes becomes more difficult to compute real numbers unless there is an algorithm. The expression of the real number in decimal form seeks to establish least error margin (Robert, 2000).

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Complex numbers

This is the more advanced group of numbers which exhibits abstract formulas. Some of the examples of the complex numbers contained in the polynomials and advanced roots. Emergence of more advanced formulas has therefore led to the emergence of more complex numbers (Ahlfors, 1979).

The biggest number

Numerically the biggest natural number is a billion. A billion comprise of a ten digit numbers. The number can be written as 1,000,000,000. This is the universally recognized big number which exists throughout the world.

It is important to note that numbers has continued to gain popularity in the day-to-day human life. The reason being that it becomes more important as it help people to combine, divide, multiply and subtract their daily transactions and identify whether they are worthy or not. As the Dwivedi argues, combinations of numbers are always meaningful to the lives of human beings and that they are not like the combinations of words and alphabets which sometimes become meaningless. According to his book the serial and natural quality of numbers enables them to remain succinct. Numbers continues to play a big role with the introduction of the money economy in the current world. It has always people to determine whether or not they are operating under a profit. It also enables people to put take precautions when a continued loss is observed within a venture. Numbers also eradicates the much confusion which used to exist before. It has also answered most of the life ambiguities and abstracts thus assisting people to understand the current environmental conditions. Numbers also enables people to fix time frame in their operations. Through it people have been able to maximize their returns (Kline, 1972).

According to the research already done identical numbers do attract one another. There is a mutual affinity of numbers which are identical. Our lives are therefore governed and categorized with some specific numbers. For instance people tend to formulate some social affinities if they share some identical numbers. The numerology has also made it easier for people to understand their past, present and future. The precision, interest and the definite nature of numbers have also assisted people to understand one another (Nicolas, 1998). People are able to identify their date of birth the year, month and the day which enables him or her to have a well synchronized program of events in life. Through such numbers the individuals are able to identify the spiritual numbers which guides and regulates the individual operations. Through numbers someone can successfully derive expressions such as squares by multiplying the number with itself. The presence of numbers assists in the function interpretation and thus helps to prove some theories mathematically. We can therefore say that presence of numbers assist in eradication confusions which mostly occupies our day-to-day life (Dantzig, 1930). It is therefore appropriate to argue that numbers have greatly contributed to the education advancement as it assist scholars and other elites to prove their written theory using some mathematic formulae.

The history of numbers have come from far, from simple to complex and its still evolving as people innovate more. Having being first invented by the Egyptians, the number has continued to gain universality as more people seek to use it dress their daily issues. The different types of numbers have therefore been developed in order to address the daily issues. The real and complex numbers on the other hand assist people to reduce the error margin as they are more accurate than the natural numbers. Originally people only used natural numbers but as the field diversified there was need to improvise rational numbers. Additionally the presence of integers also assisted people to address the negative numbers. The precision, interest and the definite nature of numbers assisted people to understand one another. People are able to identify their date of birth the year, month and the day which enables them to have a well synchronized program of events in life. Through such numbers the individuals are able to identify the spiritual numbers which guides and regulates their operations. The serial and natural quality of numbers enables them to remain succinct.

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Writing numbers When to use words and when to use numerals

It can be difficult to know how to write numbers in academic writing (e.g. five or 5 , 1 million or 1,000,000 ). This section gives some guidelines on when to use words to write numbers, and when to use numerals . There are also some exceptions to the rule which are considered, i.e. times when you might expect to use words but should instead use numerals. There is also a checklist at the end, that you can use to check the use of numbers in your own writing.

When to use words

In general, words should be used for zero to ten , and numerals used from 11 onwards. The same rule should be applied to ordinal numbers, i.e. use words for first, second up to tenth, and numbers plus 'th' (or 'st') from 11th onwards. However, it is always best to check what the accepted practice is at your university (or in your department/on your course), and remember that some common referencing systems have their own, different requirements, as follows.

• MLA . Use words if the number can be written using one or two words (e.g. three , twenty-seven ).
• APA . Use words for numbers zero to nine.
• Chicago . Use words for numbers zero to one hundred.

Before looking at when to use numerals (which is almost all other situations, see next), it is useful to look at important exceptions.

(1) When the number begins a sentence , you should use words , whatever the size of the number (though if possible, rewrite the sentence so the number is not at the beginning).

• Fifty respondents agreed with the statement.
• There were 50 respondents who agreed with the statement. [ rewritten sentence ]
• 50 respondents agreed with the statement.

(2) When expressing part of a very large round number , e.g. million, billion, you should use words for that large number part (it is common to use abbreviations m for million and bn billion ).

• The population of the earth is now in excess of 7 billion people.
• The population of the earth is now in excess of 7bn people.
• The population of the earth is now in excess of 7,000,000,000 people.
• The population of the UK is approximately 70 million.
• The population of the UK is approximately 70,000,000.

Conversely, numerals should be used rather than words, whatever the size of the number, when large and small numbers are combined , since this makes comparisons easier.

• There were 2 respondents in the first category, and 22 in the second.
• There were two respondents in the first category, and 22 in the second.

When to use numerals

Numerals are used for almost all other situations. These include the following.

• Measurements (e.g. 6 kg, 3 cm, 10 min, 2 hr, 3 days, 6 years, 5 decades )
• Currency (e.g. $10, £50, £60 billion )
• Statistical data , including survey data (e.g. A survey of participants revealed that 4 out of 5 students worked. )
• Mathematical functions (e.g. v 2 = u 2 + 2as )
• Decimals (e.g. 2.5, 4.54 )
• Percentages (e.g. 75% )
• Ratios (e.g. 3:1 )
• Percentiles/quartiles (e.g. the 95th percentile, the 1st quartile )
• Times (e.g. 12.30 a.m., 6 p.m., 16:00 )
• Dates (e.g. Wednesday 25 December 2019 )
• Scores/points on a scale (e.g. This item scored 5 on a 9-point scale )

Other important points

The following are a few other points to remember when using numbers.

• Consistency. You should be consistent in how you write numbers; for example, if write a figure like 7bn in one place, do not write a figure like 5 billion in another.
• Use of commas. When giving numerals of 1,000 or larger, use commas for each thousand, e.g. 5,500, 8,326,500 .
• Use of hyphens. When displaying a range, use a hyphen, with no space, e.g. 30%–50%
• Expressing fractions. Fractions can be written either as numerals e.g. 2/3 or words e.g. two-thirds . If using words, use a hyphen.

American Psychological Association (2019a) Numbers Expressed in Words . Available at: https://apastyle.apa.org/style-grammar-guidelines/numbers/words (Accessed: 26 December, 2019).

American Psychological Association (2019b) Numbers Expressed in Numerals . Available at: https://apastyle.apa.org/style-grammar-guidelines/numbers/numerals (Accessed: 26 December, 2019).

Harvard Wiki (2019) Numbers . Available at: https://wiki.harvard.edu/confluence/display/HSG/Numbers . (Accessed: 26 December, 2019).

University of Bristol (2015) Using numbers . Available at: http://www.bristol.ac.uk/arts/exercises/grammar/grammar_tutorial/page_33.htm (Accessed: 26 December, 2019).

University of New England (nd) Numbers in academic writing . Available at: https://aso-resources.une.edu.au/academic-writing/miscellaneous/numbers/ (Accessed: 26 December, 2019).

University of Oxford (2015) Style Guide . Available at: https://www.ox.ac.uk/sites/files/oxford/media_wysiwyg/University%20of%20Oxford%20Style%20Guide.pdf (Accessed: 26 December, 2019).

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Below is a checklist for using numbers in academic writing. Use it to check your writing, or as a peer to help.

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Compare & contrast essays examine the similarities of two or more objects, and the differences.

Cause & effect essays consider the reasons (or causes) for something, then discuss the results (or effects).

Discussion essays require you to examine both sides of a situation and to conclude by saying which side you favour.

Problem-solution essays are a sub-type of SPSE essays (Situation, Problem, Solution, Evaluation).

Transition signals are useful in achieving good cohesion and coherence in your writing.

Reporting verbs are used to link your in-text citations to the information cited.

The Science of Magic

Mikko hypponen: who owns the internet, openmind books, scientific anniversaries, can technology create a ‘hit', featured author, latest book, the history of the number pi.

On February 5 or 6, 1897, the House of Representatives of the State of Indiana (USA) passed one of the most absurd laws in history by a vote of 67 to 0. By introducing as a “new mathematical truth” a supposed method for squaring the circle —defining with compass and straightedge a square with the same area as a circle— invented by the physician and amateur mathematician Edward Goodwin, the law established de facto a value of 3.2 for the number pi. Fortunately, the text was never voted on in the Senate, enduring only as one of the more bizarre episodes in the history of the world’s most popular irrational number, a mathematical constant whose endless quest has captivated human beings for centuries.

Although today we know pi ( π ) as the ratio between the length of a circumference and its diameter, the first historical approximations arose when analysing the relationship between polygons and circles. In ancient Babylon, a value of 3/8, or 3.125, was calculated by relating the length of a circumference to the perimeter of an inscribed hexagon, as deduced from a clay tablet dated around 1900 B.C. Another estimated value appears in the Rhind Papyrus, an Egyptian mathematical document from 1650 B.C. that yields a calculation of 256/81, around 3.1604. Interestingly, before the proposal from the State of Indiana, perhaps the last integer value of pi appears in the Bible: the First Book of Kings, written about the 6th century BC, speaks of a sea of molten metal with a circumference of 30 cubits and a diameter of 10 cubits, which would give a value of pi equal to three.

Archimedes algorithm 

Around 250 BC, the Greek polymath Archimedes created an algorithm, based on the Pythagorean theorem, which allowed a better approximation; by inscribing and circumscribing a circle with polygons he calculated its upper and lower limits, 3/7 and 310/71, which predicted an average value of 3.1418…. Archimedes also observed that this same number related the area of a circle to its radius. However, and although in classical Greece the letter π (“p”) was used in the notation of geometric calculations because it was the first letter of “periphery” or “perimeter”, it was not until the eighteenth century when its use began to be standardised; it was Leonhard Euler who in 1736 established the definition of “ π ” as half the circumference of a circle whose radius is equal to one, or 3.14….

Decimals began to flourish in the first millennium of our era at the hands of Chinese, Indian and Arab mathematicians, who undertook cumbersome calculations to reach the seventh or ninth digits of pi. With the development of calculus in the 17th century by Isaac Newton and Gottfried Leibniz, the English mathematician and physicist published up to the fifteenth digit. The race progressed slowly: by the end of the 19th century, the figure stood at 527 digits. The exception, of course, was the curious case of Indiana, fortunately frustrated. Although legislators saw in Goodwin’s proposal an opportunity to collect royalties for the state, the intervention of mathematician Clarence Waldo succeeded in getting the Senate to simply shelve a proposal that had already been ridiculed in American newspapers.

Ramanujan and the leap from hundreds of decimal places 

It was the development of computing in the 20th century that brought about the leap from hundreds of decimal places of pi —the record for hand calculation is 620 digits, set in 1946— to thousands and then millions, making this quest a more affordable endeavour. The revolution in computing pi relied on formulas developed in the early 20th century by the Indian genius Srinivasa Ramanujan , who filled hundreds of pages of his notebooks with methods that were not rediscovered until decades later and are still in use today. In 1985, one of the formulas created by Ramanujan made it possible to surpass 17 million digits of pi.

Today the decimals recorded are counted in the tens of trillions. After Emma Haruka Iwao, a Japanese computer scientist at Google, reached more than 31.4 trillion digits on March 14, 2019, since January 29, 2020 the record has been set at 50 trillion , a mark achieved by the American cybersecurity analyst Timothy Mullican, who used an old computer expanded with second-hand hardware purchased on eBay. 

However, although pi has a finger in every pie in mathematics and is an essential element in fields such as wave physics, for its practical applications scientists make do with much less. According to NASA engineer Marc Rayman , only 15 digits are used for space mission calculations, and 40 would suffice to calculate the circumference of the visible universe with an accuracy the size of a hydrogen atom. Nevertheless, there is no doubt that the race to increase this infinite numerical string will continue, because if there is one thing that knows no limits, like pi itself, it is human curiosity.

Javier Yanes

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History of Numbers Essay

Introduction.

Numbers were first invented in 35,000 B.C. during which whole numbers and rational numbers were the most applicable variables in their commercial calculations. The history of numbers can therefore be traced back to the discovery era as artifacts and bones which formed tally marks. People preferred numbers which could be constructed mechanically. However, the Isaac Newton’s inventions introduced the idea of continuous variable numbers in1600 B.C. Later on in the 1800s the discontinuous functions were introduced in order to clear the confusion brought about by the continuous variable.

The geometry and other more technical problems however led to the introduction of square roots and development of algebraic numerals. Innovations and technological advancements have thus led to the invention of different types of numbers classified into sets. This paper seeks to analyze the History of Numbers, and find out what a “Big Number” is and whether there is a zillion numbers.

Types of numbers

Natural numbers.

The natural numbers was first invented in 1500B.C by the Egyptians through the use of distinct hieroglyphs. Initially numbers 1-10 was first invented and later they introduced all numbers with powers of 10 and their highest limit was a million. During this period zero was not considered as a number. In the 1st century B.C the Mayan civilization started using zero as a number a practice which only revolved within the Mesoamerica. The first universal study of numbers was credited due to the works of Archimedes and Pythagoras famous philosophers in the Greek empire. The two emphasized on a more advanced natural numbers. The natural numbers are the most commonly used form of numbers which seeks to enhance the arithmetic operations. The natural numbers are mainly based on tens number system where the expressions are mainly done in ten digits. In this case the rightmost digit mostly assumes the value of ones while any other digit placed after the right most digit assume the value of ten times the value of the digit to its right. A letter N is also commonly used when referring to a set of natural numbers. It is also important to note that natural numbers are also referred to as positive integers (Joshi, 1989). The numbers are therefore said to comprise a set of non-negative integers. Among the notable properties of the natural numbers includes the ability to implement some additions, subtractions, multiplication and divisions. All this properties make it easy to use natural numbers in mathematical calculations.

This can be defined as numbers which have a value less than zero. Integers comprise of the negative numbers and are usually noted by the negative sign which is put in front of the number. Integers and natural numbers can be combined to form a set of integers denoted by letter Z. It is important to understand that integers consist of the natural numbers but a negative sign is added onto them. Integers also comprises of the smallest group of numbers. Similarly, integers have the associative, distributive and communicative properties which make them easier to manipulate (Flegg, 2002).  The properties also make the integers user friendly while computing.

Rational numbers

This is a fraction which has a non-zero denominator.  A rational number therefore represents a fraction of the whole.  During the initial stages integers were mainly reciprocals of the positive integers. But with the mathematics advancements, fractions which consist of both numerators and denominators have been improvised. The continued advancement has enhanced rational numbers to be expressed in decimal forms (Kline, 1990). In such case the denominator is mainly powered by tens, hundreds or so considering the number of digits to the right of the decimal. It is possible to do some additions, subtractions, multiplications and divisions to the rational numbers.  In some instances the fraction numbers are rationalized, this mostly occurs when the denominator consist of irrational number or complex numbers (Nahin, 1998).

Real number

This are all measuring numbers, either whole or fraction, positive or negative. Real numbers assist in improving the accuracy in the measurement parameters as they tend to be more precise. It sometimes becomes more difficult to compute real numbers unless there is an algorithm.  The expression of the real number in decimal form seeks to establish least error margin (Robert, 2000).

Complex numbers

This is the more advanced group of numbers which exhibits abstract formulas. Some of the examples of the complex numbers contained in the polynomials and advanced roots. Emergence of more advanced formulas has therefore led to the emergence of more complex numbers (Ahlfors, 1979).

The biggest number

Numerically the biggest natural number is a billion. A billion comprise of a ten digit numbers. The number can be written as 1,000,000,000. This is the universally recognized big number which exists throughout the world. 

It is important to note that numbers has continued to gain popularity in the day-to-day human life. The reason being because it becomes more important as it help people to combine, divide, multiply and subtract their daily transactions and identify whether the are worthy or not. As the Dwivedi argues, combinations of numbers are always meaningful to the lives of human beings and that they are not like the combinations of words and alphabets which sometimes become meaningless. According to his book the serial and natural quality of numbers enables them to remain succinct. Numbers continues to play a big role with introduction of money economy in the current world. It has always people to determine whether or not they are operating under a profit. It also enables people to put take precautions when a continued loss is observed within a venture. Numbers also eradicates the much confusion which used to exist before. It has also answered most of the life ambiguities and abstracts thus assisting people to understand the current environmental conditions. Numbers also enables people to fix time frame in their operations. Through it people have been able to maximize their returns (Kline, 1972).  

According to the research already done identical numbers do attract one another. There is a mutual affinity of numbers which are identical. Our lives are therefore governed and categorized with some specific numbers. For instance people tend to formulate some social affinities if they share some identical numbers. The numerology has also made it easier for people to understand their past, present and future. The precision, interest and the definite nature of numbers have also assisted people to understand one another (Nicolas, 1998).  People are able to identify their date of birth the year, month and the day which enables him or her to have a well synchronized program of events in life. Through such numbers the individuals are able to identify the spiritual numbers which guides and regulates the individual operations. Through numbers someone can successfully derive expressions such as squares by multiplying the number with itself. The presence of numbers assists in the function interpretation and thus helps to prove some theories mathematically. We can therefore say that presence of numbers assist in eradication confusions which mostly occupies our day-to-day life (Dantzig, 1930). It is therefore appropriate to argue that numbers have greatly contributed to the education advancement as it assist scholars and other elites to prove their written theory using some mathematic formulae. 

The history of numbers have come from far, from simple to complex and its still evolving as people innovate more. Having being first invented by the Egyptians, the number has continued to gain universality as more people seek to use it dress their daily issues. The different types of numbers have therefore been developed in order to address the daily issues. The real and complex numbers on the other hand assist people to reduce the error margin as they are more accurate than the natural numbers. Originally people only used natural numbers but as the field diversified there was need to improvise rational numbers. Additionally the presence of integers also assisted people to address the negative numbers. The precision, interest and the definite nature of numbers assisted people to understand one another. People are able to identify their date of birth the year, month and the day which enables them to have a well synchronized program of events in life. Through such numbers the individuals are able to identify the spiritual numbers which guides and regulates their operations. The serial and natural quality of numbers enables them to remain succinct.

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History of Numbers and Counting Essay Example

History of numbers: essay introduction.

Different versions of the evolution of numbers have led to different outcomes. These include the Egyptian, Babylonians and Hindu-Arabic versions as well as the Mayans, Romans and modern American number systems. The mathematical evolution is the basis of the development history of counting. It is believed that this was before the beginning of the counting systems. (Zavlatsky, 124).

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The history of mathematics and counting began with the idea of formulating measurement methods. This was used by the Babylonians/Egyptians. This paper will highlight the evolution history of counting by the Egyptians/Babylonians, the Romans, Hindu-Arabic, and the Mayans' counting systems. The paper will also explain why Western counting systems are so popular today.

The Egyptians/Babylonians Number History

The ancients recognized the measurement in terms of more and less, which led to the need for counting. Although the assumptions of numbers were based on archeological evidence from around 50,000 years ago (Higgins, 87), the background of the counting system is the ancient recognition of more or less in routine activities (Higgins, 87). The need for simple counting was also recognized by ancient people. These ancient people developed other types of number systems, such as more or less and odd numbers. Because people need a way to count groups of people through the increase in population by birth, counting became a necessity. Menninger also claims that pre-historical activities like barter and cattle trading led to the need to count and determine value (105).

Prehistoric people used sticks to count cows. The total number of animals in the world was determined by the allocation and collection of sticks. The mathematical history was formed from the marking of rows of bones, tallying and pattern recognition. This led to the introduction numbers. As shown below, the bones and woods were marked.

The spoken words of prehistoric people are what influenced the evolution of numbers. It has been hard to track down the pattern of numbers between one and ten. Any pattern beyond ten can be easily identified and traced. One example is ein lifon. This prehistoric term meant "one left" and led to eleven. Twelve were derived from the lif which was "two leftovers". (Higgins143). The pattern was extended to nineteen by tracing thirteen from the combination of three and four, which were taken from fourteen. The word "ten times," which is the number of hundred, can be deduced from Ifrah Bello (147). The written words of ancient people, such as notches on stone carvings and knots for count, provided a solid foundation for the evolution in counting.

The Incas used the counting of boards for their record-keeping. The Incas used "quip" to help pre-historic people record their items. Three different colors were used to paint the counting boards. These represented the highest number, the darkest part represented the second-highest level, and the lighter parts represented the stone compartments (Havil 277) The quip could also be used to perform fast mathematical computations (Zavlatsky, 154). The quip generally used knots on cords that were arranged in certain ways to provide certain numerical information. The quip system of information and record keeping has been linked to many mysteries that have yet to be solved. Below are examples of how knots look.

History of the Hindu-Arabic Number History

This is the 21st century's common system for counting and numbers. Al-Brahmi introduced in India the numbers 1, 2, 4, 5, 6, 7, 8, 8 and 9 (Menninger 175). With time, the Brahmi numerals changed. The numerals in the 4th through 6th centuries were, for example, as shown below.

The numerals were eventually extended to 1,2,3,4,5 and 6,7,7,8,9 over time. From Cambodia came the earliest system for using zero. The Saka era saw the introduction of three digits with a dot between (Hays & Schmandt-Besserat, 198). This was the beginning of the evolution in decimal points. Babylonians introduced the positional systems, which established the place value for numerical systems. The Babylonians created the base systems for the numerical using the positional system, which was later further developed by the Indians. Brahmi numerals went through many incarnations before they were able to create the current number system (Higgins, 204).

Gupta numerals are one of the ways that the Hindu-Arabic numbers system evolved to become the most widely used American number version. Researchers are still unsure of the origins and evolution of the Gupta numbers.

The Europeans also adopted the Hindu-Arabic trading system, which meant that travelers could use the Mediterranean Sea to trade (Havil 191). European number evolution was dominated by the Pythagorean and the abacus. Even though both systems declined after a while, the Pythagorean still used "sacred number" The Europeans borrowed the Hindu–Arabic number system over time to create their mathematical number systems (Ifrah 207 and Bello 207). The exact process by which the Europeans adopted Hindu-Arabic systems has yet to be established. The Europeans may have adopted the Hindu-Arabic numbering system because they rely heavily on it to build their strong numerals (Higgins, 210). The positional base system's scope is extensive, and involved conversion of various bases using numerical numbers 10.

History of the Mayan Number

Mexico was home to the Mayan civilization, which developed number and counting systems through rituals. Two ritual systems were used to calculate calendar dates. One for priests, the other for common civilians (Higgins 217). The priestly calendar counting utilized mixed base systems that involved multiples of numerical numbers. The Mayan number system forms the basis of mathematical knowledge. The Mayan system of number used the position of numbers to assign the place value of the combined numbers (Havil 223).

The Mayans used place values of numerical numbers to multiply and add numbers. Ultimately, the Hindu-Arabic and the Mayan number systems contributed highly to the evolution of numbers as opposed to the Egyptians/Babylonians number systems (Menninger 199). The strong features of all other evolutions were incorporated into the Western number system for counting and mathematics to create a solid number system. The American system, which is used in most countries, uses decimal points, place values, and Roman numbers 1-10 (Ifrah 225 and Bello 225). Below is a sketch of what the Mayans called the tabled numbers.

To create a reliable, universally accepted number system, the American version of numbers and count used all the development features from the Mayans, Babylonians and Incas. This is a unique aspect that makes the American system stand apart from all other number systems and counting. The merits of the Mayans Babylonians, Egyptians and Indians must not be underestimated as without them, history would not be complete.

Essay Conclusion on History of Numbers

There is a lot of pre-historical archeological evidence that shows the historical history of counting and number systems. Researchers face a huge challenge when trying to trace the ancient times of number systems and counting. Research on the topic number systems and counting is still ongoing. The Mayans, Hindus, and Babylonians, which rely on Incas development, are the most successful number systems. Prehistoric evidence, such as wood carvings and stones, left mathematical evidence that led to the evolution in counting. There are many methods and arguments for the evolution of numbers. There are therefore no accepted research findings regarding the evolution of numbers and mathematical systems.

Works Cited

Havil, Julian. The Irrationals: A Story of the Numbers You Cant Count on, Princeton: Princeton University Press, 2014. Print.

Hays, Michael, and Denise Schmandt-Besserat. The History of Counting, Broadway: HarperCollins, 1999. Print.

Higgins, Peter. Number Story: From Counting to Cryptography, Gottingen: Copernicus, 2008. Print.

Ifrah, Georges, and David Bello. The Universal History of Number: From Pre-history to the Invention of Computer, Hoboken: Wiley, 2000. Print.

Menninger, Karl. Number Words and Number Symbols; Cultural History of Numbers, Mineola: Dover Publications, 2011. Print.

Zavlatsky, Claudia. Africa Counts; Number and Pattern in Africa Cultures, Chicago: Chicago Review Press, 1999. Print.

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