Hello students, welcome to today's mathematics lesson. I am so happy to be here with you to learn about something really exciting – large numbers around us. Now, you might think you already know numbers, but wait until you see how big these numbers can get! We are going to explore lakhs, crores, and even bigger numbers that we use in our daily lives without even realizing it. Are you ready? Let's begin!
So students, let me start with a story. There is a farmer named Eshwarappa who lives in Chintamani, a town in Karnataka. He goes to the market regularly to buy seeds for his rice field. One day, he overheard a conversation between two people named Ramanna and Lakshmamma. Ramanna said, "Earlier our country had about a lakh varieties of rice. Farmers used to preserve different varieties of seeds and use them to grow rice. Now, we only have a handful of varieties. Also, farmers have to come to the market to buy seeds." Now, isn't that interesting? Just think about it – one lakh varieties of rice! That is an enormous number, isn't it?
Lakshmamma then said, "There is a seed bank near my house. So far, they have collected about a hundred indigenous varieties of rice seeds from different places. You can also buy seeds from there."
Now, you must have heard the word 'lakh' before. But do you really know how big one lakh is? Let us find out together. Eshwarappa shared this incident with his daughter Roxie and son Estu. Estu was surprised to know that there were about one lakh varieties of rice in this country. He wondered, "One lakh! So far I have only tasted 3 varieties. If we tried a new variety each day, would we even come close to tasting all the varieties in a lifetime of 100 years?"
Now, let us understand how much one lakh actually is. Students, look at the pattern we are going to fill in. The largest 3-digit number is 999. What happens when we add 1 to it? We get 1000, which is the smallest 4-digit number. Similarly, the largest 4-digit number is 9999, and adding 1 gives us 10,000, which is the smallest 5-digit number. The largest 5-digit number is 99,999, and adding 1 gives us 1,00,000, which is the smallest 6-digit number. And 1,00,000 is read as "One Lakh". So students, one lakh means 1 followed by five zeroes. That is 100,000. Now, isn't that a lot?
Let us think about Estu's question. If someone ate one variety of rice each day, how many varieties would they eat in 100 years? Well, in one year, there are 365 days (ignoring leap years). So in 100 years, they would eat 365 multiplied by 100, which is 36,500 varieties. That is nowhere close to one lakh! Roxie then asked, "What if we ate 2 varieties of rice every day? Would we then be able to eat 1 lakh varieties of rice in 100 years?" Let us calculate. If we eat 2 varieties per day for 100 years, that would be 2 times 365 times 100, which equals 73,000 varieties. Still less than one lakh! What about eating 3 varieties per day? That would be 3 times 365 times 100, which equals 1,09,500. Yes, that is more than one lakh! So students, if someone ate 3 varieties of rice every single day for 100 years, they would just barely cross one lakh varieties. But if they ate only 1 or 2 varieties per day, they would not be able to taste all the lakh varieties in their lifetime. Isn't that surprising?
Now, let us look at some real examples of large numbers around us. According to the 2011 Census, the population of the town of Chintamani was about 75,000. How much less than one lakh is 75,000? Well, 1,00,000 minus 75,000 equals 25,000. So 75,000 is 25,000 less than one lakh. The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000? It is 6,000 more than one lakh. And by how much did the population increase from 2011 to 2024? From 75,000 to 1,06,000, that is an increase of 31,000 people. These are all examples of large numbers that we encounter in real life.
And the Kunchikal waterfall in Karnataka is said to drop from a height of about 450 metres.
Imagine a boy named Somu who is 1 metre tall. Now, suppose there is a building where each floor is about four times his height, so about 4 metres per floor. If the building has 10 floors, its height would be 10 × 4 = 40 metres. Which is taller — the Statue of Unity or this building? The Statue of Unity is 180 metres tall, so it is 180 − 40 = 140 metres taller than the building. Now, the Kunchikal waterfall is about 450 metres tall. How many times taller is the waterfall than the building? 450 divided by 40 equals approximately 11.25 times. And how many floors would Somu's building need to be as high as the waterfall? 450 divided by 4 equals approximately 113 floors. So students, now you have a better sense of these large measurements! Now, let us think about whether one lakh is a very large number or not. Eshwarappa asked Roxie and Estu, "Is a lakh big or small?" Roxie feels that 1 lakh is a large number. She says, "We had one lakh varieties of rice—that is a lot." She also calculated that living 1 lakh days would mean living for 274 years, which is a really long time! And if 1 lakh people stood shoulder to shoulder in a line, they could stretch as far as 38 kilometres. That is like walking from one city to another almost!
But Estu thinks it is not that big. He says, "Do you know that the cricket stadium in Ahmedabad has a seating capacity of more than 1 lakh? One lakh people in such a small area!" He also mentions that most humans have 80,000 to 1,20,000 hairs on their heads. Imagine, 1 lakh hairs fit in such a tiny space! And he heard that there are some species of fish where a female fish can lay almost one lakh eggs at once very comfortably. Some even lay tens of lakhs of eggs at a time. So students, whether one lakh is big or small depends on what we are comparing it with. In some contexts, it seems huge, and in others, it seems small. That is the interesting thing about numbers!
Now, let us learn about reading and writing large numbers. We have already been using commas for 5-digit numbers like 45,830 in the Indian place value system. As numbers grow bigger, using commas helps in reading the numbers easily. We use a comma in between the digits representing the "ten thousands" place and the "one lakh" place. For example, the number 12,78,830 is read as "twelve lakh seventy eight thousand eight hundred thirty". Similarly, the number 15,75,000 in words is "fifteen lakh seventy five thousand". Let us practice writing some numbers in words. The number 3,00,600 is "three lakh six hundred". The number 5,04,085 is "five lakh four thousand eighty five". The number 27,30,000 is "twenty seven lakh thirty thousand". And the number 70,53,138 is "seventy lakh fifty three thousand one hundred thirty eight".
Now, let us do the reverse – writing numbers from words. "One lakh twenty three thousand four hundred and fifty six" is written as 1,23,456. "Four lakh seven thousand seven hundred and four" is 4,07,704. "Fifty lakhs five thousand and fifty" is 50,05,050. And "Ten lakhs two hundred and thirty five" is 10,00,235.
Now students, let us move on to a very interesting section called "Land of Tens". In this land, there are special calculators with special buttons. Let us explore each one of them.
First, we have "The Thoughtful Thousands" which only has a +1000 button. How many times should it be pressed to show different numbers? To show three thousand, we need to press it 3 times. To show 10,000, we need to press it 10 times. To show fifty three thousand, we need to press it 53 times. To show 90,000, we need to press it 90 times. To show one lakh, we need to press it 100 times. And if we press it 153 times, we get 1,53,000. So students, how many thousands are required to make one lakh? The answer is 100, because 100 times 1000 equals 1,00,000.
Next, we have "The Tedious Tens" which only has a +10 button. To show five hundred, we need to press it 50 times. To show 780, we need to press it 78 times. To show 1000, we need to press it 100 times. To show 3700, we need to press it 370 times. To show 10,000, we need to press it 1000 times. To show one lakh, we need to press it 10,000 times. And to show 4350, we need to press it 435 times.
Then we have "The Handy Hundreds" which only has a +100 button. To show four hundred, we press it 4 times. To show 3,700, we press it 37 times. To show 10,000, we press it 100 times. To show fifty three thousand, we press it 530 times. To show 90,000, we press it 900 times. To show 97,600, we press it 976 times. To show 1,00,000, we press it 1000 times. And to show 58,200, we press it 582 times. Now, how many hundreds are required to make ten thousand? It is 100, because 100 times 100 equals 10,000. And how many hundreds are required to make one lakh? It is 1000, because 1000 times 100 equals 1,00,000.
Now, here is an interesting question. Handy Hundreds says, "There are some numbers which Tedious Tens and Thoughtful Thousands can't show but I can." Is this statement true? Let us think about it. Tedious Tens can show any number that is a multiple of 10. Thoughtful Thousands can show any number that is a multiple of 1000. Handy Hundreds can show any number that is a multiple of 100. Now, can Tedious Tens show the number 500? Yes, by pressing the +10 button 50 times. Can Thoughtful Thousands show 500? No, because 500 is not a multiple of 1000. But Handy Hundreds can show 500 by pressing the +100 button 5 times. Similarly, can Tedious Tens show 3700? Yes, by pressing 370 times. Can Thoughtful Thousands show 3700? No, because 3700 is not a multiple of 1000. But Handy Hundreds can show 3700 by pressing 37 times. So students, Handy Hundreds can show numbers like 500, 3700, 9700, etc., which are multiples of 100 but not multiples of 1000. These numbers cannot be shown by Thoughtful Thousands. However, can Tedious Tens show all the numbers that Handy Hundreds can show? Yes, because every multiple of 100 is also a multiple of 10. So the statement "There are some numbers which Tedious Tens can't show but I can" is false. But the statement "There are some numbers which Thoughtful Thousands can't show but I can" is true. So Handy Hundreds is partially correct.
Now, let us meet "Creative Chitti", a different kind of calculator. It has the following buttons: +1, +10, +100, +1000, +10000, +100000 and +1000000. It always has multiple ways of doing things. To get the number 321, it presses +10 thirty two times and +1 once. That gives us 32 times 10 plus 1, which is 320 plus 1, equals 321. Alternatively, it can press +100 two times and +10 twelve times and +1 once. That gives us 2 times 100 plus 12 times 10 plus 1, which is 200 plus 120 plus 1, equals 321. So there are multiple ways to get the same number!
Now, let us look at two different ways to get 5072. The first way is: press +1000 five times, +100 zero times, +10 seven times, and +1 two times. That gives us 5 times 1000 plus 0 times 100 plus 7 times 10 plus 2 times 1, which is 5000 plus 0 plus 70 plus 2, equals 5072. This can be expressed as (5 × 1000) + (7 × 10) + (2 × 1) = 5072. The second way is: press +1000 three times, +100 twenty times, +10 zero times, and +1 seventy-two times. That gives us 3 times 1000 plus 20 times 100 plus 0 times 10 plus 72 times 1, which is 3000 plus 2000 plus 0 plus 72, equals 5072. This can be expressed as (3 × 1000) + (20 × 100) + (72 × 1) = 5072. So students, can you find a different way to get 5072? Let us try another way. We could do (4 × 1000) + (10 × 100) + (7 × 10) + (2 × 1). That is 4000 + 1000 + 70 + 2, which equals 5072. Or we could do (2 × 1000) + (40 × 100) + (3 × 10) + (2 × 1). That is 2000 + 4000 + 30 + 2, which also equals 5072. There are so many ways!
For 367813, one way is (3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1) = 300,000 + 60,000 + 7,000 + 800 + 10 + 3 = 367,813. Another way could be (30 × 10000) + (67 × 1000) + (8 × 100) + (1 × 10) + (3 × 1) = 300,000 + 67,000 + 800 + 10 + 3 = 367,813. Yes, that works!
Now, here is an interesting question. If you have to make exactly 30 button presses, what is the largest 3-digit number you can make? And what is the smallest 3-digit number you can make?
To make the largest 3-digit number with exactly 30 presses, we want to maximize the value while using exactly 30 presses. The largest 3-digit number is 999, which would use 9 + 9 + 9 = 27 presses. We need 3 more presses. If we add +1 three more times, we get 1002, which is 4 digits. So we need to reduce one of the other digits. If we use (9 × 100) + (8 × 10) + (13 × 1) = 900 + 80 + 13 = 993, that uses 9 + 8 + 13 = 30 presses exactly. So 993 is the largest 3-digit number with exactly 30 presses.
To make the smallest 3-digit number with exactly 30 presses, we want to minimize the value while using exactly 30 presses. The smallest 3-digit number is 100, which uses only 1 press. We need 29 more presses. We can use (1 × 100) + (29 × 1) = 100 + 29 = 129. That uses 1 + 29 = 30 presses. Can we get smaller? If we try to use +10, we would need to use fewer +1 presses, but that increases the value. For example, (1 × 100) + (1 × 10) + (28 × 1) = 138, which is larger. So 129 is the smallest 3-digit number with exactly 30 presses.
Now, the second question: 997 can be made using 25 clicks. Can you make 997 with a different number of clicks? Yes, we can. For example, (8 × 100) + (19 × 10) + (7 × 1) = 800 + 190 + 7 = 997. The number of clicks is 8 + 19 + 7 = 34. So we can make 997 with 34 clicks instead of 25.
Now, let us meet "Systematic Sippy", another calculator. It has the following buttons: +1, +10, +100, +1000, +10000, +100000. It wants to be used as minimally as possible. So we want to get numbers using as few button clicks as possible.
How can we get 5072 using as few button clicks as possible? One way is to use the Indian place value notation. 5072 = (5 × 1000) + (7 × 10) + (2 × 1). That uses 5 + 7 + 2 = 14 button clicks. Is there a better way? What about (4 × 1000) + (10 × 100) + (7 × 10) + (2 × 1)? That is 4000 + 1000 + 70 + 2 = 5072. That uses 4 + 10 + 7 + 2 = 23 button clicks. That is more than 14. What about (3 × 1000) + (20 × 100) + (7 × 10) + (2 × 1)? That is 3000 + 2000 + 70 + 2 = 5072. That uses 3 + 20 + 7 + 2 = 32 button clicks. Even more! So the best way is the first one with 14 clicks. And notice that 14 is actually the sum of the digits of 5072. That is interesting!
Similarly, for 8300, the best way is (8 × 1000) + (3 × 100) = 8300, which uses 8 + 3 = 11 button clicks. And 11 is the sum of the digits of 8300.
Now, students, do you see the connection? The smallest number of button clicks needed to get a number is equal to the sum of its digits! This is because the most efficient way is to use the Indian place value system, where we use each digit once with its corresponding place value. For example, for 40629, the best way is (4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1) = 40,000 + 600 + 20 + 9 = 40,629. That uses 4 + 6 + 2 + 9 = 21 button clicks. And 21 is the sum of the digits 4 + 0 + 6 + 2 + 9 = 21. For 56354, the best way is (5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1) = 50,000 + 6000 + 300 + 50 + 4 = 56,354. That uses 5 + 6 + 3 + 5 + 4 = 23 button clicks. And 5 + 6 + 3 + 5 + 4 = 23. For 66666, the best way is (6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1) = 60,000 + 6000 + 600 + 60 + 6 = 66,666. That uses 6 + 6 + 6 + 6 + 6 = 30 button clicks. And 6 + 6 + 6 + 6 + 6 = 30. For 367813, the best way is (3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1) = 300,000 + 60,000 + 7000 + 800 + 10 + 3 = 367,813. That uses 3 + 6 + 7 + 8 + 1 + 3 = 28 button clicks. And 3 + 6 + 7 + 8 + 1 + 3 = 28. So students, the connection is clear: the minimum number of button clicks is equal to the sum of the digits of the number. And this makes sense because the most efficient way is to use each digit according to its place value, which is exactly what the Indian place value notation does!
Now, what happens if we press the +10,00,000 button ten times? What number will come up? That would be 10 times 10,00,000, which is 1,00,00,000. How many zeroes does it have? It has eight zeroes. What should we call it? The number 1,00,00,000 is called one crore. And 10 crores would be 10,00,00,000, which is 10 followed by seven zeroes. And 100 crores would be 1,00,00,00,000, which is one arab or one hundred crores. An arab is 1 followed by nine zeroes, which is also 1 billion in the American system.
Now, let us learn about the Indian and American systems of naming numbers. In the Indian system, we use the words lakh and crore. One lakh is 1 followed by 5 zeroes, which is 100,000. One crore is 1 followed by 7 zeroes, which is 10,000,000. One arab is 1 followed by 9 zeroes, which is 1,000,000,000 or 1 billion.
In the American system, we use the words thousand, million, billion, etc. One thousand is 1 followed by 3 zeroes. One million is 1 followed by 6 zeroes. One billion is 1 followed by 9 zeroes.
Notice the placement of commas is different in the two systems. In the Indian system, commas are placed to group the digits in a 3-2-2-2 pattern from right to left. For example, 1,00,000 has a comma after the first 3 digits, then after every 2 digits. In the American system, commas are placed in a 3-3-3 pattern from right to left. For example, 100,000 has a comma after every 3 digits.
Let us read some numbers in both systems. The number 9,87,65,01,234 in the Indian system is read as "9 arab 87 crore 65 lakh 1 thousand and 234" or "987 crore 65 lakh 1 thousand 234". In the American system, it is 9,876,501,234, which is read as "9 billion 876 million 501 thousand 234".
Now, let us practice reading some numbers. The number 40,50,678 in the Indian system is "forty lakh fifty thousand six hundred seventy-eight". In the American system, it is 4,050,678, which is "four million fifty thousand six hundred seventy-eight". The number 4,81,21,620 in the Indian system is "four crore eighty-one lakh twenty-one thousand six hundred twenty". In the American system, it is 48,121,620, which is "forty eight million one hundred twenty one thousand six hundred twenty". The number 2,00,22,002 in the Indian system is "two crore twenty-two thousand two". In the American system, it is 20,022,002, which is "twenty million twenty-two thousand and two". The number 24,68,13,579 in the Indian system is "twenty four crore sixty eight lakh thirteen thousand five hundred seventy nine". In the American system, it is 246,813,579, which is "two hundred forty six million eight hundred thirteen thousand five hundred seventy nine". The number 34,50,00,543 in the Indian system is "thirty four crore fifty lakh five hundred forty three". In the American system, it is 345,000,543, which is "three hundred forty five million five hundred forty three". The number 1,02,03,04,050 in the Indian system is "one arab two crore three lakh four thousand fifty". In the American system, it is 1,020,304,050, which is "one billion twenty millions three hundred four thousand fifty".
Now, let us write some numbers in Indian place value notation. "One crore one lakh one thousand ten" is written as 1,01,01,010. "One billion one million one thousand one" is written as 1,00,10,01,001. "Ten crore twenty lakh thirty thousand forty" is written as 10,20,30,040. "Nine billion eighty million seven hundred thousand six hundred" is written as 9,08,07,00,600.
Now, let us compare some numbers. Is 30 thousand less than, equal to, or greater than 3 lakhs? 30 thousand is 30,000, and 3 lakhs is 3,00,000. So 30 thousand is less than 3 lakhs. Is 500 lakhs less than, equal to, or greater than 5 million? 500 lakhs is 5,00,00,000, which is 500 million or half a billion. 5 million is 50,00,000. So 500 lakhs is greater than 5 million. Is 800 thousand less than, equal to, or greater than 8 million? 800 thousand is 8,00,000. 8 million is 80,00,000. So 800 thousand is less than 8 million. Is 640 crore less than, equal to, or greater than 60 billion? 640 crore is 6,40,00,00,000, which is 6.4 billion. 60 billion is 60,00,00,00,000. So 640 crore is less than 60 billion.
Now, students, we often do not need exact numbers and can use approximations. For example, according to the 2011 census, the population of Chintamani town is 76,068. Instead, saying that the population is about 75,000 is enough to give an idea of how big the quantity is.
There are situations where it makes sense to round up a number. For example, if a school has 732 people including students, teachers, and staff, the principal might order 750 sweets instead of 700 sweets, to make sure there are enough for everyone.
There are situations where it is better to round down. For example, if the cost of an item is ₹470, the shopkeeper may say that the cost is around ₹450 instead of saying it is around ₹500, because ₹450 is closer to the actual price in this case.
Can you think of situations where it is appropriate to round up, round down, either one is okay, and when exact numbers are needed? For rounding up: buying food or fruits for a group, where you want to have extra. For rounding down: estimating time remaining to leave for school to catch the bus. For either: estimating distance between places, especially far-off places. For exact numbers: handling money in banks, measuring ingredients for a recipe.
Now, let us learn about nearest neighbours of large numbers. With large numbers, it is useful to know the nearest thousand, lakh, or crore. For example, the nearest neighbours of the number 6,72,85,183 are shown in the table. The nearest thousand is 6,72,85,000. The nearest ten thousand is 6,72,90,000. The nearest lakh is 6,73,00,000. The nearest ten lakh is 6,70,00,000. The nearest crore is 7,00,00,000.
Now, let us find the nearest neighbours for 3,87,69,957. The nearest thousand is 3,87,70,000. The nearest ten thousand is 3,87,70,000. The nearest lakh is 3,88,00,000. The nearest ten lakh is 3,90,00,000. The nearest crore is 4,00,00,000.
For 29,05,32,481, the nearest thousand is 29,05,32,000. The nearest ten thousand is 29,05,30,000. The nearest lakh is 29,05,00,000. The nearest ten lakh is 29,10,00,000. The nearest crore is 29,00,00,000.
Now, here is an interesting puzzle: "I have a number for which all five nearest neighbours are 5,00,00,000. What could the number be? How many such numbers are there?" If all five nearest neighbours (nearest thousand, ten thousand, lakh, ten lakh, and crore) are all 5,00,00,000, then the number must be very close to 5,00,00,000. Let us think. If the nearest crore is 5,00,00,000, then the number must be between 4,50,00,001 and 5,49,99,999. If the nearest ten lakh is 5,00,00,000, then the number must be between 4,95,00,001 and 5,04,99,999. If the nearest lakh is 5,00,00,000, then the number must be between 4,99,50,001 and 5,00,49,999. If the nearest ten thousand is 5,00,00,000, then the number must be between 4,99,95,001 and 5,00,04,999. If the nearest thousand is 5,00,00,000, then the number must be between 4,99,99,500 and 5,00,00,499. So the number must be between 4,99,99,500 and 5,00,00,499. That is a range of 1000 numbers. So there are 1000 such numbers!
Now, let us estimate the values of some expressions. First, let us estimate 4,63,128 + 4,19,682. Roxie says, "The sum is near 8,00,000 and is more than 8,00,000." Estu says, "The sum is near 9,00,000 and is less than 9,00,000." Whose estimate is closer? Let us add the numbers: 4,63,128 + 4,19,682 = 8,82,810. So the exact sum is 8,82,810. Roxie's estimate was 8,00,000, and the difference is 82,810. Estu's estimate was 9,00,000, and the difference is 17,190. So Estu's estimate is closer! Will the sum be greater than 8,50,000 or less than 8,50,000? Since 8,82,810 is greater than 8,50,000, it is greater. Will the sum be greater than 8,83,128 or less than 8,83,128? 8,82,810 is less than 8,83,128, so it is less.
Now, let us estimate 14,63,128 – 4,90,020. Roxie says, "The difference is near 10,00,000 and is less than 10,00,000." Estu says, "The difference is near 9,00,000 and is more than 9,00,000." Let us subtract: 14,63,128 – 4,90,020 = 9,73,108. So the exact difference is 9,73,108. Roxie's estimate was 10,00,000, and the difference is 26,892. Estu's estimate was 9,00,000, and the difference is 73,108. So Roxie's estimate is closer! Will the difference be greater than 9,50,000 or less than 9,50,000? Since 9,73,108 is greater than 9,50,000, it is greater. Will the difference be greater than 9,63,128 or less than 9,63,128? 9,73,108 is greater than 9,63,128, so it is greater.
Now, students, we looked at populations of some Indian cities. We saw that Mumbai has the highest population, followed by New Delhi, Bengaluru, Hyderabad, and so on. Most cities have seen a significant rise in population from 2001 to 2011. Bengaluru showed the maximum increase in population. Some cities like Bengaluru, Hyderabad, Surat, and Vadodara have almost doubled their population.
Now, let us learn about some multiplication shortcuts. Roxie evaluated 116 × 5 as follows: 116 × 5 = 116 × 10/2 = (116 ÷ 2) × 10 = 58 × 10 = 580. Why does this work? Because multiplying by 5 is the same as multiplying by 10 and then dividing by 2. And since 10 ÷ 2 = 5, this shortcut works!
Estu evaluated 824 × 25 as follows: 824 × 25 = 824 × 100/4 = (824 ÷ 4) × 100 = 206 × 100 = 20,600. This works because 25 = 100/4.
Now, let us find quick ways to calculate some products. For 2 × 1768 × 50, we can do 2 × 50 × 1768 = 100 × 1768 = 1,76,800. For 72 × 125, we know that 125 = 1000/8, so 72 × 125 = 72 × 1000/8 = (72 ÷ 8) × 1000 = 9 × 1000 = 9000. For 125 × 40 × 8 × 25, we can do 125 × 8 × 40 × 25 = 1000 × 1000 = 10,00,000.
Now, let us calculate some products quickly. 25 × 12 = 25 × (12) = (100/4) × 12 = 100 × (12/4) = 100 × 3 = 300. 25 × 240 = (100/4) × 240 = 100 × (240/4) = 100 × 60 = 6000. 250 × 120 = (1000/4) × 120 = 1000 × (120/4) = 1000 × 30 = 30,000. 2500 × 12 = (10000/4) × 12 = 10000 × (12/4) = 10000 × 3 = 30,000. And for something like 120,000,000, we could do 1200 × 1,00,000 = 120,000,000.
Now, let us observe some interesting patterns in multiplication. Look at these: 11 × 11 = 121, 111 × 111 = 12321, 1111 × 1111 = 1234321. Do you see the pattern? When we multiply ones by ones, we get a palindromic number! Similarly, 66 × 61 = 4026, 666 × 661 = 440,226, 6666 × 6661 = 44,422,626. There is a pattern here too.
Now, look at these: 3 × 5 = 15, 33 × 35 = 1155, 333 × 335 = 111,555. And 101 × 101 = 10201, 102 × 102 = 10404, 103 × 103 = 10609.
Now, let us think about the number of digits in the product of two numbers. Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct? Let us check. The smallest 2-digit number is 10. The smallest product of two 2-digit numbers is 10 × 10 = 100, which is a 3-digit number. The largest 2-digit number is 99. The largest product is 99 × 99 = 9801, which is a 4-digit number. So yes, Roxie is correct! We did not need to try all possible multiplications; we just needed to check the smallest and largest possible products.
Can multiplying a 3-digit number with another 3-digit number give a 4-digit number? The smallest 3-digit number is 100. The smallest product is 100 × 100 = 10,000, which is a 5-digit number. So no, it cannot give a 4-digit number. The minimum is 5 digits, and the maximum would be 6 digits (999 × 999 = 998001).
Can multiplying a 4-digit number with a 2-digit number give a 5-digit number? The smallest 4-digit number is 1000, and the smallest 2-digit number is 10. The smallest product is 1000 × 10 = 10,000, which is a 5-digit number. So yes, it can give a 5-digit number.
Now, students, let us look at the general pattern. A 1-digit number multiplied by a 1-digit number gives a 1-digit or 2-digit number. A 2-digit number multiplied by a 1-digit number gives a 2-digit or 3-digit number. A 2-digit number multiplied by a 2-digit number gives a 3-digit or 4-digit number. A 3-digit number multiplied by a 3-digit number gives a 5-digit or 6-digit number. A 5-digit number multiplied by a 5-digit number gives a 9-digit or 10-digit number. An 8-digit number multiplied by a 3-digit number gives a 10-digit or 11-digit number. A 12-digit number multiplied by a 13-digit number gives a 24-digit or 25-digit number.
The maximum digits in the product will be the sum of the digits of the two numbers. The minimum digits in the product will be one less than their sum.
Now, let us look at some fascinating facts about large numbers. 1250 × 380 = 4,75,000, which is the number of kirtanas composed by Purandaradāsa according to legends. Purandaradāsa was a composer and singer in the 15th century. His kirtanas spanned social reform, bhakti, and spirituality.
2100 × 70,000 = 14,70,00,000, which is approximately 147 million kilometers. This is an approximation of the average distance between the Earth and the Sun. The actual distance keeps varying throughout the year because Earth's orbit is elliptical, not circular. The farthest distance (aphelion) is about 152 million kilometers, and the closest distance (perihelion) is about 147 million kilometers.
6400 × 62,500 = 40,00,00,000, which is the average number of litres of water the Amazon river discharges into the Atlantic Ocean every second. The river's flow into the Atlantic is so much that drinkable freshwater is found even 160 kilometers into the open sea.
Now, let us look at some division facts. 13,95,000 ÷ 150 = 9300, which is the distance in kilometers of the longest single-train journey in the world. The train runs in Russia between Moscow and Vladivostok. The duration of this journey is about 7 days.
10,50,00,000 ÷ 700 = 1,50,000, which is the weight in kilograms of an adult blue whale. A newborn blue whale weighs around 2,700 kg, which is similar to the weight of an adult hippopotamus.
52,00,00,00,000 ÷ 130 = 40,00,00,000, which was the weight in tonnes of global plastic waste generated in the year 2021.
In a single gram of healthy soil, there can be 100 million to 1 billion bacteria and 1 lakh to 1 million fungi, which can support plants' growth and health.
Now, let us do some thought experiments. Estu asked, "Could the entire population of Mumbai fit into 1 lakh buses?" Let us assume a bus can accommodate 50 people. Then 1 lakh buses can accommodate 1 lakh × 50 = 50 lakh people. The population of Mumbai is more than 1 crore 24 lakhs, which is 124 lakh. So the entire population of Mumbai cannot fit in 1 lakh buses.
What about fitting the population of Mumbai into 5000 such ships? Each ship carries about 2500 passengers. 5000 ships would carry 5000 × 2500 = 1,25,00,000, which is 1.25 crore. That is more than Mumbai's population of 1.24 crore. So yes, the entire population of Mumbai could fit into 5000 such ships!
Now, Roxie wondered, "If I could travel 100 kilometers every day, could I reach the Moon in 10 years?" The distance between the Earth and the Moon is 3,84,400 km. How far would she travel in a year? 100 km × 365 = 36,500 km (ignoring leap years). How far would she travel in 10 years? 36,500 × 10 = 3,65,000 km. That is less than 3,84,400 km. So she would not reach the Moon in 10 years. She would need about 384400/36500 = 10.53 years, so almost 11 years.
Now, can you reach the Sun in a lifetime if you travel 1000 kilometers every day? The distance between the Earth and the Sun is about 15 crore kilometers (150,000,000 km). In a lifetime of about 70 years, you would travel 1000 × 365 × 70 = 2,55,50,000 km, which is about 2.55 crore km. That is nowhere near 15 crore km. You would need about 150,000,000/365,000 = 411 years! So you cannot reach the Sun in a lifetime.
Now, let us think about some more questions. If a single sheet of paper weighs 5 grams, could you lift one lakh sheets of paper together at the same time? One lakh sheets would weigh 1,00,000 × 5 = 5,00,000 grams, which is 500 kg. Since a person cannot lift 500 kg, we cannot lift one lakh sheets of paper at the same time.
If 250 babies are born every minute across the world, will a million babies be born in a day? In one day, there are 60 × 24 = 1440 minutes. So babies born in one day = 250 × 1440 = 3,60,000. That is 3.6 lakh, which is less than one million (10 lakh). So a million babies cannot be born in a day.
Can you count 1 million coins in a day? Assume you can count 1 coin every second. In one day, there are 60 × 60 × 24 = 86,400 seconds. So you can count 86,400 coins in a day. That is less than 1 million. So you cannot count 1 million coins in a day.
Now, students, we have covered a lot of concepts in this chapter. Let me summarize everything we have learned.
We started by understanding what a lakh is. One lakh is 1 followed by 5 zeroes, which is 100,000. We learned that one lakh varieties of rice would take more than a lifetime to taste if we ate only 1 or 2 varieties per day, but we could taste all of them if we ate 3 varieties per day for 100 years.
We learned about reading and writing large numbers in the Indian place value system. We learned that commas are placed in a 3-2-2 pattern from right to left in the Indian system, and in a 3-3-3 pattern in the American system.
We learned about lakh, crore, and arab. One lakh is 1 followed by 5 zeroes. One crore is 1 followed by 7 zeroes. One arab is 1 followed by 9 zeroes, which is also 1 billion in the American system.
We learned about the Land of Tens, where we explored different calculators with different buttons. We learned that the most efficient way to get a number is to use the Indian place value system, and the number of button clicks needed is equal to the sum of the digits of the number.
We learned about rounding numbers. Sometimes we round up, sometimes we round down, and sometimes exact numbers are needed.
We learned about nearest neighbours of large numbers – the nearest thousand, ten thousand, lakh, ten lakh, and crore.
We learned about multiplication shortcuts, like multiplying by 5 is the same as multiplying by 10 and dividing by 2, and multiplying by 25 is the same as multiplying by 100 and dividing by 4.
We learned about patterns in products, and how to determine the number of digits in the product of two numbers.
We learned fascinating facts about large numbers, like the distance from the Earth to the Sun, the discharge of the Amazon river, the weight of blue whales, and the number of bacteria in soil.
We did thought experiments about whether the population of Mumbai can fit in buses or ships, whether we can reach the Moon or the Sun in a certain number of years, and whether we can lift one lakh sheets of paper or count one million coins.
So students, large numbers are all around us. We see them in populations, distances, measurements, and many other places. It is important to understand them and be able to read, write, and work with them. I hope you enjoyed this lesson as much as I enjoyed teaching it. Thank you for your attention, and keep exploring the wonderful world of numbers!