essay on development of number system

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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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An Historical Survey of Number Systems

• Published 2004
• History, Mathematics

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1, 2, 3, 4, 5, 6, 7, 8, 9... and 0. With just these ten symbols, we can write any rational number imaginable. But why these particular symbols? Why ten of them? And why do we arrange them the way we do? Alessandra King gives a brief history of numerical systems.

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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/

Background: Number Systems

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• Math Article

Number System

The number system or the numeral system is the system of naming or representing numbers. We know that a number is a mathematical value that helps to count or measure objects and it helps in performing various mathematical calculations. There are different types of number systems in Maths like decimal number system, binary number system, octal number system, and hexadecimal number system. In this article, we are going to learn what is a number system in Maths, different types, and conversion procedures with many number system examples in detail .  Also, check mathematics for grade 12 here.

What is Number System in Maths?

A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.

The value of any digit in a number can be determined by:

• Its position in the number
• The base of the number system

Before discussing the different types of number system examples, first, let us discuss what is a number?

What is a Number?

A number is a mathematical value used for counting or measuring or labelling objects. Numbers are used to performing arithmetic calculations.  Examples of numbers are natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents a null value. 

A number has many other variations such as even and odd numbers, prime and composite numbers. Even and odd terms are used when a number is divisible by 2 or not, whereas prime and composite differentiate between the numbers that have only two factors and more than two factors, respectively.

In a number system, these numbers are used as digits. 0 and 1 are the most common digits in the number system, that are used to represent binary numbers. On the other hand, 0 to 9 digits are also used for other number systems. Let us learn here the types of number systems.

Types of Number Systems

There are various types of number systems in mathematics. The four most common number system types are:

• Decimal number system (Base- 10)
• Binary number system (Base- 2)
• Octal number system (Base-8)
• Hexadecimal number system (Base- 16)

Now, let us discuss the different types of number systems with examples.

Decimal Number System (Base 10 Number System)

The decimal number system has a base of 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands and so on. This system is expressed in decimal numbers . Every position shows a particular power of the base (10).

Example of Decimal Number System:

The decimal number 1457 consists of the digit 7 in the units position, 5 in the tens place, 4 in the hundreds position, and 1 in the thousands place whose value can be written as:

(1×10 3 ) + (4×10 2 ) + (5×10 1 ) + (7×10 0 )

(1×1000) + (4×100) + (5×10) + (7×1)

1000 + 400 + 50 + 7

Binary Number System (Base 2 Number System)

The base 2 number system is also known as the Binary number system wherein, only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 110101 is a binary number.

We can convert any system into binary and vice versa.

Write (14) 10 as a binary number.

Base 2 Number System Example

∴ (14) 10 = 1110 2

Octal Number System (Base 8 Number System)

In the octal number system , the base is 8 and it uses numbers from 0 to 7 to represent numbers. Octal numbers are commonly used in computer applications. Converting an octal number to decimal is the same as decimal conversion and is explained below using an example.

Example: Convert 215 8 into decimal.

215 8 = 2 × 8 2 + 1 × 8 1 + 5 × 8 0

= 2 × 64 + 1 × 8 + 5 × 1

= 128 + 8 + 5

Hexadecimal Number System (Base 16 Number System)

In the hexadecimal system, numbers are written or represented with base 16. In the hexadecimal system, the numbers are first represented just like in the decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabet from A to F. The below-given table shows the representation of numbers in the hexadecimal number system .

Number System Chart

In the number system chart, the base values and the digits of different number systems can be found. Below is the chart of the numeral system.

Number System Conversion

Numbers can be represented in any of the number system categories like binary, decimal, hexadecimal, etc. Also, any number which is represented in any of the number system types can be easily converted to another. Check the detailed lesson on the conversions of number systems to learn how to convert numbers in decimal to binary and vice versa, hexadecimal to binary and vice versa, and octal to binary and vice versa using various examples.

With the help of the different conversion procedures explained above, now let us discuss in brief about the conversion of one number system to the other number system by taking a random number.

Assume the number 349. Thus, the number 349 in different number systems is as follows:

The number 349 in the binary number system is 101011101

The number 349 in the decimal number system is 349.

The number 349 in the octal number system is 535.

The number 349 in the hexadecimal number system is 15D

Number System Solved Examples

Convert (1056) 16 to an octal number.

Given, 1056 16 is a hex number.

First we need to convert the given hexadecimal number into decimal number

= 1 × 16 3 + 0 × 16 2 + 5 × 16 1 + 6 × 16 0

= 4096 + 0 + 80 + 6

= (4182) 10

Now we will convert this decimal number to the required octal number by repetitively dividing by 8.

Therefore, taking the value of the remainder from bottom to top, we get;

(4182) 10 = (10126) 8

Therefore, 

(1056) 16 = (10126) 8

Convert (1001001100) 2 to a decimal number.

(1001001100) 2

= 1 × 2 9 + 0 × 2 8 + 0 × 2 7 + 1 × 2 6 + 0 × 2 5 + 0 × 2 4 + 1 × 2 3 + 1 × 2 2 + 0 × 2 1 + 0 × 2 0

= 512 + 64 + 8 + 4

Convert 10101 2 into an octal number.

 10101 2 is the binary number

We can write the given binary number as,

Now as we know, in the octal number system,

Therefore, the required octal number is (25) 8

Convert hexadecimal 2C to decimal number.

We need to convert 2C 16  into binary numbers first.

2C → 00101100

Now convert 00101100 2 into a decimal number.

101100 = 1 × 2 5  + 0 × 2 4 + 1 × 2 3  + 1 × 2 2  + 0 × 2 1 + 0 × 2 0

= 32 + 8 + 4

Video Lesson on Numeral System

Number System Questions

• Convert (242) 10 into hexadecimal. [ Answer: (F2) 16 ]
• Convert 0.52 into an octal number. [ Answer: 4121]
• Subtract 1101 2 and 1010 2 . [ Answer: 0010]
• Represent 5C6 in decimal. [ Answer:  1478]
• Represent binary number 1.1 in decimal. [ Answer: 1.5]

Also Check: Binary Operations

Computer Numeral System (Number System in Computers)

When we type any letter or word, the computer translates them into numbers since computers can understand only numbers. A computer can understand only a few symbols called digits and these symbols describe different values depending on the position they hold in the number. In general, the binary number system is used in computers. However, the octal, decimal and hexadecimal systems are also used sometimes.

More Topics Related to Number Systems

Frequently Asked Questions on Number System

What is number system and its types.

The number system is simply a system to represent or express numbers. There are various types of number systems and the most commonly used ones are decimal number system, binary number system, octal number system, and hexadecimal number system.

Why is the Number System Important?

The number system helps to represent numbers in a small symbol set. Computers, in general, use binary numbers 0 and 1 to keep the calculations simple and to keep the amount of necessary circuitry less, which results in the least amount of space, energy consumption and cost.

What is Base 1 Number System Called?

The base 1 number system is called the unary numeral system and is the simplest numeral system to represent natural numbers.

What is the equivalent binary number for the decimal number 43?

How to convert 30 8 into a decimal number.

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• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Linear Algebra
• CBSE Class 8 Maths Formulas
• CBSE Class 9 Maths Formulas
• CBSE Class 10 Maths Formulas
• CBSE Class 11 Maths Formulas

What is the importance of the number system?

The Number System is a way of representing numbers on the number line with the help of a set of Symbols and rules. All the Mathematical concepts and formulas are based on Number system. We have curated the importance of number system in this article below.

A number refers to a word or symbol which represents a particular quantity. It is with the help of numbers only that multiple arithmetic operations are performed and we have been able to develop so much in the field of physics and mathematics. One cannot live their life without the usage of numbers, even for the most basic chores or tasks. Even the money exchanged for commodities is a certain value represented by numbers.

A bunch of numbers grouped together is used to assign a person as their contact number. such is the prominence of numerals in our lives. Hence, it is imperative to know more about the numbers and number systems , as has been discussed below.

Counting Numbers

Such a set of numbers as is used to count certain objects is called counting numbers. Such a set of numbers starts with 1(one) and goes on till infinity. One here represents a single object. For example, Mr. A had one pencil and one pen in his hand, or I ate one banana today. Adding up two counting numbers yields another counting number. These are used in real life for basic exchange, calculations, and operations.

The Number Zero 

Zero is denoted by 0. It is used to depict nothing. In other words, if something has no value at all, it is assigned the number zero as a quantity. The number zero comes before all the counting numbers, and forms the set of ‘whole numbers’.

The different types of numbers in mathematics,

• Natural Numbers A set of numbers as is used to count certain objects are called natural numbers. Such a set of numbers starts with 1(one) and goes on till infinity. It is to be noted that natural numbers include only positive integers.
• Whole Numbers A set of numbers that includes all the positive integers and zero.
• Integers An integer is defined as such a whole number that can assume either positive, negative or no value at all.  
• Real Numbers Such numbers which include both rational numbers and their irrational counterparts.
• Rational Numbers Such numbers can be expressed in the form of a fraction.
• Irrational Numbers Such numbers cannot be expressed as a fraction.

The following diagram depicts all kinds of numbers discussed so far,

Apart from the above-mentioned types, we have the following categories of numbers as well,

• Even Numbers Such numbers as can be divided by 2 are called even numbers. Example: 2, 4, 6, 8, …, 1024, etc.
• Odd Numbers Such numbers as are not divisible by 2 are called odd numbers. Example: 3, 5, 7, 10, …, 1345, etc.
• Prime Numbers Such a number that can be divided exactly by itself or 1. Example: 5, 7, 13, 23, etc.
• Composite Numbers Such numbers as having multiple factors other than 1 and the number itself. Example: 16, 20, 50, etc.

Number System

It is clear that numbers are used to represent a certain quantity. When certain symbols or digits are used to represent the numbers themselves, it forms a number system. Hence, a number system is such a system as can be used to define a set of values, which are further used to represent a quantity. 

• Number System in Math

Types of Number Systems

There are various types of number systems in mathematics. The four most common number system types are:

• Decimal number system
• Binary number system
• Octal number system
• Hexadecimal number system

Decimal Number System 

Such a number system as has a base value of 10 is termed a decimal number system . It uses the digits between 0 – 9 for the creation of numbers. In this system, each digit is depicted as its product with different powers of 10.

Another feature to be noted is that the place value keeps on increasing from right to left, with the extreme right termed as ones, then tens, hundreds, thousands, and so on. The units (ones) place would be depicted as 10 0 , tens would be 10 1 , hundreds 10 2 , and so on.

For example: 548 has place values as

(5 x 10 2 ) + ( 4 x 10 1 ) + (8 x 10 0 ) = 5 x 100 + 4 x 10 + 8 x 1 = 500 + 40 + 8 = 548

Binary Number System 

As the name suggests, this type of number system has a base value of 2 (binary). This system uses only two digits, i.e., 0 and 1 for creating numbers. Extensively used in computer applications, this system is really east to utilize. For example: 

14 can be written as 1110 50 can be written as 110010

• Binary Number System

Octal Number System 

As the name suggests, this system has a base value of 8(octal). Hence it uses 8 digits to create numbers. For example:

(112) 10 can be expressed as (287) 8 . (287) 10 can be expressed as (372) 8 .

• Octal Number System

Hexadecimal Number System 

This system has a base value of 16 and hence uses 16 digits for the creation of numbers. For example:

(255) 10 can be written as (FF) 16 (1096) 10 can be written as (448) 16 (4090) 10 can be written as (FFA) 16

• Hexadecimal Number System

It is safe and wise to agree that number system holds its importance for everything which includes proportion and percentage. Number system plays a crucial role, both in our everyday lives and the technological world. With its myriad qualities, it simplifies our lives a lot, which has been discussed as follows:

• It enables to keep count of all the things around people. Like how many apples are in the basket, or the number of milk cartons to be purchased, etc.
• It enables the unique and accurate representation of different types of numbers.
• Making a phone call is possible only because we have a proper and efficient number system.
• Elevators used in public places also depend upon number systems for their functioning.
• Computation of any kind of interest on amounts deposited in banks.
• Creation of passwords on computers, security purposes.
• Encrypting important data, by converting figures into another number system to avoid hacking and misuse of data.
• It enables easy conversion of numbers for technical purposes.
• The entirety of computer architecture depends upon number systems (octal, hexadecimal). Every fiber of data gets stored in the computer as a number.

Conceptual Questions

Question 1: Convert 128 10 to an octal number.

In case of octal conversion, we have to divide the numbers by 8. Operation Output Remainder 128/8 16 0 16/8 2 0 2/8 0 2 Therefore, the equivalent octal number = 200 8

Question 2: Convert 128 10 to hexadecimal.

In case of hexadecimal conversion, we divide the numbers by 16. Operation Output Remainder 128/16 8 0 8/16 0 8 Therefore, the equivalent hexadecimal number is 80 16

Question 3: Convert (1101) 2 into a decimal number.

Now, multiplying each digit from MSB to LSB with reducing the power of the base number 2. 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 8 + 4 + 0 + 1 = 13 Therefore, (1101) 2 = (13) 10

Question 4: Convert (214) 8 into a binary number.

We know, 2 → 010 1 → 001 4 → 100 Therefore,(214) 8 = (010001100) 2

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