Hello, students! Welcome to today's mathematics lesson. I am so happy to be with you as we explore one of the most fascinating chapters in your Class 8 textbook — Chapter 3, titled "A Story of Numbers." Now, students, you might be wondering why this chapter is called "A Story of Numbers." Well, that's exactly what we are going to discover together. Today, we are going to travel back in time, thousands and thousands of years ago, to understand how our modern number system came into existence. This is not just about learning numbers — this is about understanding one of the greatest inventions in human history. Are you ready for this exciting journey? I am sure you are!
Let us begin with a little story. Imagine a lazy afternoon, and a young girl named Reema is flipping through an old book in her house. Suddenly, whoosh! A piece of paper slips out and floats to the floor. She picks it up and stares at strange symbols all over it. "What is this?" she wonders. She runs to her father, holding the paper as if it were a secret treasure. He looks at it and smiles. "Around 4000 years ago," he tells her, "there flourished a civilisation in a region called Mesopotamia, in the western part of Asia, containing a major part of present-day Iraq and a few other neighbouring countries. This is one of the ways they wrote their numbers!" Reema's eyes light up. "Seriously? These strange symbols were numbers?" Her curiosity was sparked, and questions started swirling in her head.
Students, this is exactly where our journey begins. Sensing Reema's curiosity, her father started telling her how the idea of number and number representation evolved over the course of time, across geographies, to finally reach its modern efficient form. So let us get ready to travel back in time with them!
Now, students, let us think about why humans even needed numbers in the first place. Humans had the need to count even as early as the Stone Age — around ten thousand years ago. They were counting to determine the quantity of food they had, the number of animals in their livestock, details regarding trades of goods, the number of offerings given in rituals, and so on. They also wanted to keep track of the passing days, for example, to know and predict when important events such as the new moon, full moon, or onset of a season would occur. However, when they said or wrote down such numbers, they didn't make use of the numbers that we use today. Can you imagine that? The numbers we use so casually today — 1, 2, 3, 4, 5 — these took thousands of years to develop!
Now, students, here is something wonderful about our own country, India. The structure of the modern oral and written numbers that we use today had its origin thousands of years ago in India. Ancient Indian texts, such as the Yajurveda Samhita, mentioned names of numbers based on powers of 10, almost as we say them orally today. For example, they listed names for the numbers one, which is eka in Sanskrit; ten, which is dasha; hundred, which is shata; thousand, which is sahasra; ten thousand, which is ayuta, and so on, all the way up to 10¹² and beyond. Isn't that amazing, students? Our ancestors in India were using number names similar to what we use today, thousands of years ago!
The way we write our numbers today — using the digits 0 through 9 — also originated and were developed in India around 2000 years ago. The first known instance of numbers being written using ten digits, including the digit 0 (which was then notated as a dot), occurs in the Bakhshali manuscript, which dates back to around the 3rd century CE. Then, around 499 CE, Aryabhata was the first mathematician to fully explain and do elaborate scientific computations with the Indian system of 10 symbols. This is truly a proud moment for all of us, students!
Now, how did these numbers travel from India to the rest of the world? The Indian number system was transmitted to the Arab world by around 800 CE. It was popularised in the Arab world by the great Persian mathematician Al-Khwārizmī — after whom the word "algorithm" is named — through his book "On the Calculation with Hindu Numerals" written around 825 CE, and by the noted philosopher Al-Kindi through his work "On the Use of the Hindu Numerals" around 830 CE. From the Arab world, the Hindu numerals were transmitted to Europe and to parts of Africa by around 1100 CE. Though Al-Khwārizmī's work on calculation with Hindu numerals was translated into Latin, it was the Italian mathematician Fibonacci who, around the year 1200, really made the case to Europe to adopt the Indian numerals. However, students, here is an interesting historical fact: the Roman numerals were so ingrained in European thinking and writing at the time that the Indian numerals did not gain widespread use for several more centuries. But eventually, during the European Renaissance and by the 17th century, not adopting them became impossible because it would impede scientific progress. Their use then spread to every continent, and are now used in every corner of the world.
Now, students, here is something that might surprise you. Because European scholars learned the Indian numerals from the Arab world, they called them "Arabic numerals" to reflect their European perspective. On the other hand, as noted above, Arab scholars such as Al-Khwārizmī and Al-Kindi called them "Hindu numerals." During the period of European colonisation, the European term "Arabic numbers" became widely used. However, in recent years, this mistake is being corrected in many textbooks and documents around the world, including in Europe. The most commonly used terminologies for the numbers we use today are "Hindu numerals," "Indian numerals," and the transitional "Hindu-Arabic numerals." It is worth noting that the word "Hindu" here does not refer to a religion, but rather a geography or people from whom these numbers came. So, students, whenever you write numbers today, remember that you are using a system that originated in India!
Now, let us understand some foundational ideas needed to count and to determine the number of objects in a given collection. Imagine that we are living in the Stone Age, say, around ten thousand years ago. Suppose we have a herd of cows. Here are some natural questions that we might ask about our herd. First, how do we ensure that all cows have returned safely after grazing? Second, do we have fewer cows than our neighbour? Third, if there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour? We need to tackle these questions without the use of the number names or written numbers of the Hindu number system. How do we do it? Let us explore some possible methods.
Students, here is the first method we can use. We could tackle the questions by using pebbles, sticks, or any object that is available in abundance. Let us choose sticks. For every cow in the herd, we could keep a stick. The final collection of sticks tells us the number of cows, which can be used to check if any cows have gone missing. This way of associating each cow with a stick, such that no two cows are associated or mapped to the same stick, is called a one-to-one mapping. This mapping can then be used to come up with a way to represent numbers. So, if we have one cow, we keep one stick. If we have two cows, we keep two sticks, and so on. This is a very natural and intuitive way of counting, students!
Now, how will you use such sticks to answer the other two questions — whether we have fewer cows than our neighbour, and if so, how many more we need? Well, students, if we have a collection of sticks for our herd and our neighbour has a collection of sticks for their herd, we can compare the two collections directly. If our collection has fewer sticks, then we know we have fewer cows. And to find out how many more cows we need, we can keep adding sticks to our collection until it has the same number of sticks as our neighbour's collection. The number of sticks we added tells us how many more cows we need. Simple, isn't it?
Now, students, here is the second method we could use. Instead of objects, we could use a standard sequence of sounds or names. For example, we could use the sounds of the letters of any language. While counting, we could make a one-to-one mapping between the objects and the letters — that is, associate each object to be counted with a letter, following the letter order. This mapping can then be used to come up with a way of verbally representing numbers. For example, we get the following number representation if we use English letters 'a' to 'z'. The number 1 would be represented as 'a', the number 2 as 'b', the number 3 as 'c', and so on, until the number 26, which is represented as 'z'. An obvious limitation of using only the letters of the English alphabet in this form is that it cannot be used to count collections having more than 26 objects. So, students, if you were using this method, how many numbers could you represent in this way using the sounds of the letters of your own language? Think about it!
Now, students, here is the third method we could use. We could use a sequence of written symbols as follows. Let me write these symbols for you. For the number 1, we use the symbol I. For 2, we use II. For 3, we use III. For 4, we use IV. For 5, we use V. For 6, we use VI. For 7, we use VII. For 8, we use VIII. For 9, we use IX. For 10, we use X. Similarly, 11 is XI, 12 is XII, 13 is XIII, 14 is XIV, 15 is XV, 16 is XVI, 17 is XVII, 18 is XVIII, 19 is XIX, and 20 is XX. Students, do you see a way of extending this method to represent bigger numbers as well? How would you represent, say, 25 or 30 or 50 using this system? This is something we will explore shortly.
From the discussion above, we see that for counting and finding the size of a collection, we need a standard sequence of objects, or names, or written symbols, that has a fixed order. Let us call this standard sequence a number system. A collection of objects can be counted by making a one-to-one mapping between them and the standard sequence, following the sequence order.
Since there is no end to numbers, the challenge is to come up with an unending standard sequence or number system that is easy to count with. Using sticks gives an unending standard sequence or number system. However, it is not convenient to count larger collections, as we will need as many sticks as the number of objects being counted. Using the sounds of the letters of a language, as in Method 2, is convenient for the counting process but is not an unending standard sequence or number system because we run out of letters. The standard sequence or number system given in Method 3 was actually the system used in Europe before it got replaced by the Hindu number system. It is called the Roman number system. It was widely used in Europe for centuries and was convenient for many purposes, but had the similar drawback that one cannot write arbitrarily large numbers without introducing more and more symbols. We will learn more about this system of writing numbers later on.
As illustrated by the three methods, history gives us examples of number systems formed using physical objects (such as sticks, pebbles, body parts, etc.), names, and written symbols. Some groups of people had numbers represented both by physical objects as well as by names, while others like the Chinese had all three forms of representation. The symbols occurring in a written number system are called numerals. For example, 0, 1, 5, 36, 193, and so on, are some of the numerals occurring in the Hindu number system. Numerals representing "smaller" numbers always had names, and so a number system composed of written symbols always went hand in hand with a number system composed of names, as is the case with the modern-day Hindu system.
Now, students, let us explore some of the early number systems that different groups of people used across the world. This will help us understand how our modern number system evolved.
First, let us look at the use of body parts for counting. Many groups of people across the world have used their hands and body parts for counting. Here is how a group of people in Papua New Guinea used and still use their body parts as the standard sequence or number system. Students, isn't it fascinating that people used their own bodies as counting tools? Think about it — when you count on your fingers, you are using a method that dates back thousands of years!
Next, students, let us look at one of the oldest methods of number representation — making notches or marks cut on a surface such as a bone or a wall of a cave. These marks are also called tally marks. In this method, a mark is made for each object that is being counted. So the final collection of marks represents the total number of objects. This method is very similar to the method of using sticks to count, except for the fact that a mark is made instead of adding a stick. Archaeologists have discovered bones dating back more than 20,000 years that seem to have tally marks. The oldest known such bones with markings that are thought to represent numbers are the Ishango bone and the Lebombo bone. The Ishango bone, dating back 20,000 to 35,000 years, was discovered in the Democratic Republic of Congo. It features notches arranged in columns, possibly indicating calendrical systems. The Lebombo bone, discovered in South Africa, is an even older tally stick with 29 notches, estimated to be around 44,000 years old. It is considered one of the oldest mathematical artefacts and may have served as a tally stick or lunar calendar. Students, just imagine — these bones are older than any civilization we know of, and people were already counting!
Now, students, let us look at a very interesting number system used by a group of indigenous people in Australia called the Gumulgal. They had the following words for their numbers. The number 1 is urapon. The number 2 is ukasar. The number 3 is ukasar-urapon. The number 4 is ukasar-ukasar. The number 5 is ukasar-ukasar-urapon. The number 6 is ukasar-ukasar-ukasar. Can you see how their number names are formed? The number name for 3 is composed of number names of 2 and 1. The number name for 4 is composed of two occurrences of the number name for 2. The numbers are counted in 2s, using which the number names are formed: 3 equals 2 plus 1, 4 equals 2 plus 2, 5 equals 2 plus 2 plus 1, 6 equals 2 plus 2 plus 2. Gumulgal called any number greater than 6 by the word ras.
Now, students, here is a very interesting and puzzling historical phenomenon associated with this number system. Look at the following number systems of a group of indigenous people in South America called the Bakairi, and the Bushmen of South Africa. In Bakairi, 1 is xa, 2 is t'oa, 3 is quo, 4 is t'oa-t'oa, 5 is t'oa-t'oa-t'a, 6 is t'oa-t'oa-t'oa. In Bushmen, 1 is urapon, 2 is ukasar, 3 is ukasar-urapon, 4 is ukasar-ukasar, 5 is ukasar-ukasar-urapon, 6 is ukasar-ukasar-ukasar. And in Gumulgal, it is exactly the same! Despite being so far apart geographically, and with no trace of contact between them, these three groups have developed equivalent number systems! Historians have wondered how this happened. One theory is that these three groups of people may have had common ancestors who used this number system. In course of time, their descendants migrated to these places. Isn't that remarkable, students?
Even though the number system of Gumulgal had number names for numbers only till 6, we can see the emergence of an important idea here. Counting in 2s is more efficient for representing numbers than, for example, a tally system. A general form which this idea has taken in different number systems is as follows: count in groups of a certain number (like 2 in the case of Gumulgal's system), and use the word or symbol associated with this group size to represent bigger numbers. Some of the commonly used group sizes in different number systems have been 2, 5, 10, and 20. You can find the idea of counting by 5s in the Roman system, which we will discuss shortly.
This idea of counting in a certain group size and using it to represent numbers is an important idea in the history of the evolution of number systems. One of the phenomena that could have led people to this idea might be the human limit for immediately knowing the size of a collection at a glance. Let us try out the following activity. Quickly count the number of objects in each of the following boxes. Up to what group size could you immediately see the number of objects without counting? Most humans find it difficult to count groups having 5 or more objects in a single glance. This limit of perception could have prompted people using tally marks to replace every group of, say, 5 marks, with a new symbol, as seen in the Roman system.
Now, students, let us discuss one of the most famous number systems from history — the Roman Numerals. We have already seen the Roman number system till 20. We have seen that it uses I for 1, V for 5, X for 10. To get the Roman numeral for any number till 39, it is first grouped into as many 10s as possible, the remaining is grouped into as many 5s as possible, and finally the remaining is grouped into 1s. Let us take the number 27 as an example. 27 equals 10 plus 10 plus 5 plus 1 plus 1. So, 27 in Roman numerals is XXVII. Instead of representing 50 as XXXXX, a new symbol is given to it: L. Following the way the number 4 is represented as 1 less than 5 — that is, as IV — 40 is represented as 10 less than 50 — that is, as XL. However, people using this system were not always consistent with this practice. Sometimes, 40 was also represented as XXXX.
The Roman number system introduces newer symbols to represent certain bigger numbers. Let us call all these numbers that have a new basic symbol as landmark numbers. Here are some of the landmark numbers of the Roman system and their associated numerals. I equals 1, V equals 5, X equals 10, L equals 50, C equals 100, D equals 500, M equals 1000. These symbols are used to denote other numbers as well. For example, consider the number 2367. Writing it as a sum of landmark numbers starting from 1000 such that we take as many 1000s as possible, 500s as possible, and so on, we get 2367 equals 1000 plus 1000 plus 100 plus 100 plus 100 plus 50 plus 10 plus 5 plus 1 plus 1. So in Roman numerals, this number is MMCCCLXVII.
Now, students, we see how vastly efficient this system is compared to some of the previous number systems that we have seen. This system seems to have evolved out of the ancient Greek number system in around the 8th century BCE in Rome, and evolved over time. It spread throughout Europe with the expansion of the Roman empire. The efficiency of this system is due to the grouping of a given number by not just one group size, but a sequence of group sizes that we call landmark numbers, and then using these landmark numbers to represent the given number. This idea is the next important breakthrough in the history of the evolution of number systems.
Despite the relative efficiency of the Roman system, it doesn't lend itself to an easy performance of arithmetic operations, particularly multiplication and division. Let us try adding the following numbers without converting them to Hindu numerals: CCXXXII plus CCCXIII. Let us find the total number of Is, Xs, and Cs, and group them starting from the largest landmark number. Apparently, it looks like the largest landmark number is C, but note that 5 Cs (100s) make a D (500). So the sum is DCXLV. Now, students, try doing this yourself: LXXXVII plus LXXVIII. How will you multiply two numbers given in Roman numerals, without converting them to Hindu numerals? Try to find the product of the following pairs of landmark numbers: V times L, L times D, V times D, VII times IX. People using the Roman system made use of a calculating tool called the abacus to perform their arithmetic operations. However, only specially trained people used this tool for calculation.
Now, students, let us move on to a very important concept — the idea of a base. We are now going to see a written number system that the Egyptians developed around 3000 BCE. In this system, we see the use of landmark numbers to group and represent a given number. However, what makes this system special is its sequence of landmark numbers.
Imagine making collections of pebbles. The first landmark number is 1. Group together 10 collections of the previous landmark number (1). Its size is the second landmark number which is 10. Group together 10 collections of the previous landmark number (10). Its size is the third landmark number which is 10 times 10 equals 100, and so on. Each landmark number is 10 times the previous one. Since 1 is the first landmark number, they are all powers of 10. The following are the symbols given to these numbers: 1 is represented by a special symbol, 10 is represented by another symbol, 10² (which is 100) by yet another symbol, 10³ (which is 1000) by another, and so on, up to 10⁷.
As in the case of Roman numbers, a given number is counted in groups of the landmark numbers, starting from the largest landmark number less than the given number. This is then used to assign the numeral. For example, 324, which equals 100 plus 100 plus 100 plus 10 plus 10 plus 4, is written using the Egyptian symbols.
Now, students, here is an interesting question. Instead of grouping together 10 collections of size equal to the previous landmark number (as in the case of the Egyptian system), can we get a number system by grouping together 5 collections of size equal to the previous landmark number? Can this 5 be replaced by any positive integer? Let us examine this possibility. Let 1 be the first landmark number. Group together 5 collections of size equal to the previous landmark number (1). Its size is the second landmark number which is 5. Group together 5 collections of size equal to the previous landmark number (5). Its size is the third landmark number which is 5 times 5 equals 25. Group together 5 collections of size equal to the previous landmark number (5). Its size is the fourth landmark number which is 5 times 25 equals 125. Thus, we have a new number system where each landmark number is 5 times the previous one. Since 1 is the first landmark number, they are all powers of 5. 5⁰ equals 1, 5¹ equals 5, 5² equals 25, 5³ equals 125, 5⁴ equals 625, 5⁵ equals 3125.
Now, let us express the number 143 in this new system. Let us start grouping, starting with the size 5³ equals 125, as this is the largest landmark number smaller than 143. We get 143 equals 125 plus 5 plus 5 plus 5 plus 1 plus 1 plus 1. So the number 143 in the new system is represented using symbols for 1, 5, 25, and 125.
Now, students, here is a very important definition. Number systems having landmark numbers in which the first landmark number is 1, and every next landmark number is obtained by multiplying the current landmark number by some fixed number n, is said to be a base-n number system. The Egyptian number system is a base-10 system, and the number system that we created is a base-5 system. A base-10 number system is also called a decimal number system.
Now, what is the advantage of having landmark numbers that are all the powers of a number? To understand this, let us perform some arithmetic operations using them. Let us add the following Egyptian numerals. Let us find the total number of ones and tens and group them starting from the largest possible landmark number. It has a total of 15 ones and 15 tens. Since 10 ones gives the next landmark number, the sum can be regrouped. Since 10 ones gives a ten, we have the final sum. Students, contrast the addition done in a base-n number system with that done in the Roman system. In the Roman system, the grouping and rearranging has to be done carefully as it is not always by the same size that each landmark number has to be grouped to get the next one.
The advantage of a number system with a base becomes more evident when we consider multiplication. How to multiply two numbers in Egyptian numerals? Let us first consider the product of two landmark numbers. What is any landmark number multiplied by 10? Find the following products: 10 times 10, 100 times 10, 1000 times 10, 10000 times 10. Each landmark number is a power of 10, and so multiplying it with 10 increases the power by 1, which is the next landmark number. What is any landmark number multiplied by 100? Find the following products: 10 times 100, 100 times 100, 1000 times 100, 10000 times 100. Similarly, find the following products: 10 times 10000, 100 times 1000, 1000 times 1000, 10000 times 10000. Thus, the product of any two landmark numbers is another landmark number! Does this property hold true in the base-5 system that we created? Does this hold for any number system with a base? What can we conclude about the product of a number and 10 in the Egyptian system?
Let us take an example. What is 200 times 10? 200 is the same as 200. So, 200 times 10 equals 200 times 10. As these are numbers, the distributive law holds. So, 200 times 10 equals (200) times 10, which equals the next landmark number. Similarly, what is 241 times 10? 241 is the same as 200 plus 40 plus 1. Thus, 241 times 10 equals (200 plus 40 plus 1) times 10. Will the distributive property hold here? For the same reason that it holds for (a plus b) times n, it also holds when one of the numbers has more than 2 terms. For example, (a plus b plus c) times n equals a times n plus b times n plus c times n. So, (200 plus 40 plus 1) times 10 equals (200 times 10) plus (40 times 10) plus (1 times 10) which equals 2000 plus 400 plus 10, which equals 2410. Thus, students, what would be a simple rule to multiply a number with 10? Simply put, you just add a zero at the end! This is exactly what we do in our Hindu number system, isn't it?
As has been seen, a process of multiplying two numbers involves the multiplication of landmark numbers. When the landmark numbers are powers of a number, then their product is another landmark number. This fact simplifies the process of multiplication. However, this is not the case with the Roman numerals, which is why multiplication using them is difficult. Thus, a number system whose landmark numbers are powers of a number, that is, a number system with a base, is efficient not only in number representation but also in its utility in carrying out arithmetic operations. The idea of a number system with a base was a turning point in the history of the evolution of number systems. Our modern Hindu number system is built on this structure.
Now, students, let us talk about a calculating device that makes use of the decimal system. In around the 11th century, even the people still using the Roman numerals started using a calculating device — the abacus — constructed using a decimal system. It was a board with lines. Starting from the line that stood for 1, each successive line stood for a successive power of 10. Numbers were represented in it as follows: the given number was first grouped into the landmark numbers (powers of 10), in exactly the same way we have been grouping them so far. For each power of 10, as many counters were placed on its line as the number of times it occurred in the grouping. The presence of a counter above a line contributed a value of 5. For example, let us take the number 3426. It can be grouped as 3426 equals 1000 plus 1000 plus 1000 plus 100 plus 100 plus 100 plus 100 plus 10 plus 10 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1. This number was represented on the abacus. Notice how the 6 ones are represented. To get an idea of how the abacus was used for calculations, let us consider a simple addition problem: 2907 plus 43. The two numbers were taken on either side of the vertical partition. How would you use this to find the sum? The counters along each line were brought together. What is to be done if the total in a line exceeded 10? In this problem, the 7 ones and the 3 ones together make 10 ones, which contributes a counter to the line representing 10s. Students, the abacus is a wonderful example of how people used the idea of base-10 even before they fully adopted the Hindu number system!
Now, students, let us discuss the shortcomings of the Egyptian system. Despite being a number system that enabled relatively efficient number representations for numbers till a crore (10⁷), and relatively easy computations, the Egyptian system had a drawback. If larger and larger numbers needed to be represented, then there was a need for inventing an unending sequence of symbols for higher and higher powers of 10. Here we see the original challenge of number representation resurfacing in a different form! The next and the final idea in the history of the evolution of number systems not only solves this problem but also remarkably simplifies number representation and computations!
Now, students, we come to one of the most important concepts in the entire chapter — the concept of Place Value Representation. This is the idea that completely changed how we write numbers.
Let us look at the Mesopotamian Number System. In the beginning, the number system used in ancient Mesopotamia had different symbols for different landmark numbers. In later times, it became a base-60 system, also called the sexagesimal system, with a very efficient number representation. It has puzzled many why they chose base 60. Different theories exist to explain this, ranging from the connection between 60 and the periods of some important events (like the length of their lunar month which had 30 days, or the time taken for the Sun to complete one revolution around the Earth when Earth is taken to be stationary), the ease of representing fractions, their earlier sequence of landmark numbers — 1, 10, 60, 600, 3600, 36000, and so on — getting reduced to only the powers of 60, and so on. The influence of the Mesopotamian sexagesimal system, also known as the Babylonian number system, can be seen even now in our units of time measurements — 1 hour equals 60 minutes and 1 minute equals 60 seconds. This system used one symbol for 1 and another symbol for 10.
Let us now briefly pause on the study of their number system, and think about how one can build an efficient number system using the Mesopotamian features seen so far. Let us give our own symbols to their landmark numbers: 1, 60, 60² equals 3600, 60³ equals 216000, and so on. Note that we have actually used Indian numerals in creating these symbols. We could have invented our own symbols, but for the sake of easy recall and use, we have chosen to take help of the familiar numerals 1, 2, 3, and so on.
Using the symbols for 1 and 10, numbers from 1 to 59 can be represented. For example, 1 is represented by one symbol for 1, 2 by two symbols for 1, 3 by three symbols for 1, and so on, up to 9. Then 10 is represented by one symbol for 10, 11 by one symbol for 10 and one symbol for 1, 12 by one symbol for 10 and two symbols for 1, 20 by two symbols for 10, 30 by three symbols for 10, 40 by four symbols for 10, 50 by five symbols for 10, and 59 by five symbols for 10 and nine symbols for 1.
Now, let us represent the number 640 in this system. Grouping it into landmark numbers, we see that 640 equals 10 times 60 plus 40. If we use the Egyptian idea, this number would be represented using 10 ones for the 60s, and 40 would be represented using 4 tens. Can we represent this more compactly? We can simply represent this number as 10 (for the 60s) and 40 (for the ones), which can be read as ten 60s and one 40, just as we have written in the equation.
Let us try another number — 7530. 7530 equals 2 times 3600 plus 5 times 60 plus 30. So, its representation would be 2 (for the 3600s), 5 (for the 60s), and 30 (for the ones).
Note that when a number is grouped into powers of 60 for its representation, no power of 60 can occur 60 or more times. If this happens, then 60 of them can be grouped to form the next power of 60. For example, consider the expression: 1 times 3600 plus 70 times 60 plus 2 equals 1 times 60² plus (60 plus 10) times 60 plus 2 equals 1 times 60² plus 60² plus 10 times 60 plus 2 equals 2 times 60² plus 10 times 60 plus 2. Therefore, any number can be represented using the numerals from 1 to 59, along with the numerals for landmark numbers.
Now, what if we make the representation even more compact by dropping the symbols for the different powers of 60 altogether? For 640, our earlier representation was something like 10 (for 60s) and 40 (for ones). Our compact representation would simply be 10, 40. For 7530, it would be 2, 5, 30. This is exactly what the Mesopotamians did! In their numeral, the rightmost set of symbols showed the number of 1s, the set of symbols to its left showed the number of 60s, the next showed the number of 3600s, and so on. Whenever there was no occurrence of a power of 60, a blank space was given in that position.
It does not seem that the Mesopotamians arrived at this idea in the same way we did. Some scholars suggest that the similarity of symbols given to the landmark numbers 1 and 60 in their earlier number system, and an accidental usage of them, might have made them stumble upon this idea.
Thus, we can see how the Mesopotamian system removes the need for generating an unending sequence of symbols for the landmark numbers by making use of the positions where the symbols are written. Such a number system (having a base) that makes use of the position of each symbol in determining the landmark number that it is associated with is called a positional number system or a place value system. This idea of place value marks the highest point in the history of evolution of number systems, and gives a very elegant solution to the problem of representing the unending sequence of numbers using only a finite number of different symbols!
The Mesopotamian system, however, cannot be considered a fully developed place value system. It has certain defects that lead to confusion while reading a number. Look at the representation of 60. What will be the representation for 3600? While writing the numerals, the spacing between symbols was not given the way we are giving it here. It was also difficult to maintain a consistent spacing for blanks across different manuscripts written by different people. These created ambiguities. For example, consider the representations of the following numbers: 1, 60, 3600, 12, 602, 36002. Because of the ambiguity in finding which symbols correspond to which powers of 60, the same numeral can be read in different ways. Even in our representation which uses uniform spacing between symbols for different powers of 60, it is difficult to know the number of blanks between two sets of symbols, as in the representation of 36002. To address the issue arising out of blank spaces, the later Mesopotamians used a brilliant idea of assigning a placeholder symbol to denote a blank space. This is like the 0 (zero) we use in our system. Thus, zero — the symbol that shows nothingness — is indispensable as a placeholder in a place value system in which numbers are written in an unambiguous manner. Even with the problem arising out of blank spaces solved, other ambiguities still remained in the system. For example, the placeholder symbol was primarily used in the middle of numbers and not at the end; so they would not use it to represent a number like 3600.
Now, students, let us look at another fascinating civilization — the Mayan civilization in Central America. They made great intellectual and cultural progress between the 3rd and 10th centuries CE. Among their intellectual achievements stands their place value system designed independently of those in Asia. They also made use of a placeholder symbol for the modern-day 0 that looked like a seashell.
The Mayan number system is almost a base-20 system. Their landmark numbers are 1, 20, 20 times 18 equals 360, 20² times 18 equals 7200, 20³ times 18 equals 144000. Wait a minute, students, do you notice something strange here? Their third landmark number is 360 rather than 400! Some scholars feel that this might have something to do with their calendars. They used a dot for 1 and a bar for 5. These were used to denote numbers from 1 to 19. The symbols associated with different landmark numbers were written one below the other with the lowermost set of symbols corresponding to the number of 1s, the set above corresponding to the number of 20s, the set above to the number of 360s, and so on. Because the Mayan system is not an actual base-20 system, it lacks the advantages that a system with a base has for computations. Nevertheless, their place value notation and their use of a placeholder symbol for zero is considered an important advance in the history of number systems. A curious fact is that we can still find the use of base-20 in the number names of some European languages.
Now, students, let us look at the Chinese Number System. The Chinese used two number systems — a written system for recording quantities, and a system making use of rods for performing computations. The numerals in the rod-based number system are called rod numerals. Here we discuss the rod numerals, which are more efficient in writing and computing with numbers than the written system of the Chinese. The rod numerals developed in China by at least the 3rd century AD and were used till the 17th century. It was a decimal system (base-10). The symbols for 1 to 9 were as follows. Note that there are two sets of symbols — the zongs represent units, hundreds, tens of thousands, etc., and the hengs represent tens, thousands, hundreds of thousands, etc. Like the Mesopotamians, the rod numerals used a blank space to indicate the skipping of a place value. However, because of the slightly more uniform sizes of the symbols for one through nine, the blank spaces were easier to locate than in the Mesopotamian system. Notice how similar the rod numerals are to the Hindu system. The Chinese system, with a symbol for zero, would be a fully developed place value system.
Now, students, we come to the most important part of our journey — the Hindu Number System. Where does the Hindu or Indian number system figure in the evolution of ideas of number representation? What are its landmark numbers? And does it use a place value system?
The Hindu number system is a base-10 or decimal system. It uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Let us take a numeral, for example, 375. How is this to be read? The leftmost digit 3 is in the 10² place, which means 3 times 10² equals 300. The next digit 7 is in the 10 place, which means 7 times 10 equals 70. The last digit 5 is in the 1 place, which means 5 times 1 equals 5. So the number is 300 plus 70 plus 5 equals 375. As can be seen, the Hindu number system is a place value system.
The Hindu number system has had a symbol for 0 at least as early as 200 BCE. Because of the use of 0 as a digit, and the use of a single digit in each position, this system does not lead to any kind of ambiguity when reading or writing numerals. It is for this reason that the Hindu number system is now used throughout the world.
The use of 0 as a digit, and indeed as a number, was a breakthrough that truly changed the world of mathematics and science. In Indian mathematics, indeed, zero was not just used as a placeholder in the place value system, but was also given the status of a number in its own right, on par with other numbers. The arithmetic properties of the number 0 (for example, that 0 plus any number is the same number, and that 0 times any number is zero) were explicitly used by Aryabhata in his Aryabhatiya in 499 CE to compute with and do elaborate scientific computations using Hindu numerals. The use of 0 as a number like any other number, on which one can perform the basic arithmetic operations, was codified by Brahmagupta in his work Brahmasphutasiddhanta in 628 CE. By introducing 0 as a number, along with the negative numbers, Brahmagupta created what in modern terms is called a ring — that is, a set of numbers that is closed under addition, subtraction, and multiplication (that is, any two numbers in the set can be added, subtracted, or multiplied to get another number in that set). These new ideas laid the foundations for modern mathematics, and particularly for the areas of algebra and analysis.
Hopefully, this gives you a sense of all the ideas that went into writing and computing with numbers in the way that we do today. The discovery of 0 and the resulting Indian number system is truly one of the greatest, most creative, and most influential inventions of all time — appearing constantly in our daily lives and forming the basis of much of modern science, technology, computing, accounting, surveying, and more. The next time you are writing numbers, think about the incredible history behind them and all the deep ideas that went into their discovery!
Now, students, let us summarise the evolution of ideas in number representation. First, we had the idea of counting in groups of a single number, like the Gumulgal system where they counted in 2s. Second, we had the idea of grouping using landmark numbers, like the Roman system with I, V, X, L, C, D, M. Third, we had the idea of choosing powers of a number as landmark numbers — the idea of a base — like the Egyptian system with 1, 10, 100, 1000, and so on. Fourth, we had the idea of using positions to denote the landmark numbers — the idea of place value system — like in the Mesopotamian, Mayan, Chinese, and Hindu systems. And finally, we had the idea of 0 as a positional digit and as a number, which truly completed the system.
Students, let us now review what we have learned in this chapter. To represent numbers, we need a standard sequence of objects, names, or written symbols that have a fixed order. This standard sequence is called a number system. The symbols representing numbers in a written number system are called numerals. In a number system, landmark numbers are numbers that are easily recognisable and used as reference points for understanding and working with other numbers. They serve as anchors within the number system, helping people to orient themselves and make sense of quantities, particularly larger ones.
A number system whose landmark numbers are the powers of a number n is referred to as a base-n number system. Number systems having a base that make use of the position of a symbol in determining the landmark number that it is associated with are called positional number systems or place value systems. Place value representations were used in the Mesopotamian (Babylonian), Mayan, Chinese, and Indian civilisations.
The system of numerals that we use throughout the world today is the Hindu number system (also sometimes called the Indian number system, or the Hindu-Arabic number system). It is a place value system with (usually) 10 digits, including the digit 0, which is treated on par with other digits. Due to its use of 0 as a number, the system enables the writing of all numbers unambiguously using just finitely many symbols, and also enables efficient computation. The system originated in India around 2000 years ago, and then spread across the world, and is considered one of human history's greatest inventions.
Students, I hope you now have a deep appreciation for the numbers you use every day. Remember that when you write 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, you are using a system that took thousands of years to develop, that was invented in India, and that represents one of the most important ideas in the history of human civilization. Thank you for listening so patiently, and I will see you in the next lesson!