Ganita Prakash I · Chapter 1 — Large Numbers Around Us

Imagine a farmer who hears at the market that India once grew about a lakh different kinds of rice. If you tasted one new kind every day, would you finish them in your lifetime? Before we answer, we need a feel for how big one lakh really is.

Where a lakh comes from. Count up from the largest number of each size. The largest 3-digit number is 999, and one more gives 1000, the smallest 4-digit number. The largest 4-digit number is 9999, and one more gives 10,000. The largest 5-digit number is 99,999, and one more gives 1,00,000, the smallest 6-digit number. We read it as one lakh. It is 1 followed by five zeros.

The rice question. A year has 365 days (leaving out leap years), so in y years there are 365 × y days. In 100 years there are 365 × 100 = 36,500 days. Tasting one kind a day, you would reach only 36,500, far short of one lakh. Even two kinds a day gives 73,000. With three kinds a day you would reach 1,09,500, which just crosses one lakh.

Comparing with things we know. A big number is easier to imagine when we compare it with something familiar.

• One lakh days is about 274 years, because 1,00,000 ÷ 365 is close to 274.
• A large cricket stadium can seat more than one lakh people.
• A person has roughly 80,000 to 1,20,000 hairs on the head, so about a lakh hairs fit on one small head.
• A town of 75,000 people is 25,000 short of one lakh; a town of 1,06,000 is 6,000 more than one lakh.

So is a lakh big or small? It depends on what we are counting. A lakh days is several lifetimes, but a lakh hairs fit on one head. Mathematicians and scientists always ask, ‘big compared with what?’

Using a measuring stick. The same trick works for heights. Suppose a boy is 1 m tall and each floor of a building is about 4 times his height, so 4 m. A building of 10 floors is then about 40 m tall. A famous statue about 180 m tall is close to 4 or 5 such buildings stacked up, and a waterfall about 450 m high would need more than 100 such floors. Comparing in steps like this turns a hard-to-picture number into a picture.

Large numbers are hard to read when the digits run together. Try reading 1278830 aloud. It is much easier as 12,78,830. The commas split the number into groups that match the names we say.

The Indian place value chart. From the right, the places are ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores, ten crores, and so on. In the Indian system we put the first comma after three digits from the right (to mark off the hundreds), and then a comma after every two digits (thousands, lakhs, crores).

So 12,78,830 is read twelve lakh seventy-eight thousand eight hundred thirty. Each comma group is read as one number followed by its name: ‘12’ lakh, ‘78’ thousand, then ‘830’.

More examples.

• 15,75,000 → fifteen lakh seventy-five thousand.
• 3,00,600 → three lakh six hundred (the thousands group is 00, so we say nothing for it).
• 5,04,085 → five lakh four thousand eighty-five.
• 70,53,138 → seventy lakh fifty-three thousand one hundred thirty-eight.

From words to digits. Work group by group. ‘Four lakh seven thousand seven hundred four’ has 4 in the lakhs group, 07 in the thousands group and 704 in the last group: 4,07,704. ‘Ten lakh two hundred thirty-five’ has 10 lakhs, 00 thousands and 235: 10,00,235. A group that is missing in the words must still be filled with zeros, or every digit after it slides into the wrong place.

A quick check. Count the digits. One lakh has 6 digits, ten lakh has 7 digits and one crore has 8 digits. If ‘fifty lakh five thousand fifty’ comes out with fewer than 7 digits, a zero is missing. The correct form is 50,05,050.

Why commas help. In everyday life we see big numbers on cheques, in newspaper reports and in census tables. A misplaced comma or a missing zero can change a number a hundred times, so reading and writing with care is an important skill.

Picture a land where every calculator has only a few special buttons. One has just a +1000 button, one has just +10, one has just +100. Playing with them shows how our place value system is built.

The +1000 calculator. To show 53,000 we press +1000 fifty-three times. To show one lakh we must press it 100 times, because 100 thousands make one lakh. Pressing it 153 times shows 1,53,000.

The +10 calculator. To show 780 we press 78 times. To show 10,000 we press 1000 times. To reach one lakh we would press 10,000 times, a very tiring job.

The +100 calculator. To show 3,700 we press 37 times; to show one lakh, 1000 times. Note that 100 hundreds make ten thousand, and 1000 hundreds make one lakh. This calculator can show 400 or 3,700, which the +1000 calculator cannot, but it cannot show 780. Every machine can only reach multiples of its own button.

A calculator with many buttons. Now imagine one with +1, +10, +100, +1000, +10,000, +1,00,000 and +10,00,000. There are many ways to make the same number. For 5072:

• 50 presses of +100, 7 of +10 and 2 of +1: (50 × 100) + (7 × 10) + (2 × 1) = 5072.
• 3 presses of +1000, 20 of +100 and 72 of +1: (3 × 1000) + (20 × 100) + (72 × 1) = 5072.
• 5 presses of +1000, 6 of +10 and 12 of +1: 5000 + 60 + 12 = 5072, using 23 presses.

The fewest presses. Which way uses the fewest presses? For 5072 the best is 5 × 1000 + 7 × 10 + 2 × 1: only 5 + 7 + 2 = 14 presses. Look closely: the numbers of presses are exactly the digits 5, 0, 7, 2. For 40,629 the fewest presses are 4 of +10,000, 6 of +100, 2 of +10 and 9 of +1, which is 21 presses, the sum of the digits.

Why the digits give the fewest presses. If you ever press a button 10 or more times, you could replace ten of those presses by a single press of the next bigger button and save 9 presses. So in the best plan every button is pressed at most 9 times. That is exactly what place value does: each place holds one digit from 0 to 9, and the number is the sum of (digit × place value).

Writing it as an expression. 3,67,813 = 3 × 1,00,000 + 6 × 10,000 + 7 × 1000 + 8 × 100 + 1 × 10 + 3 × 1. This is called the expanded form. The button game is just expanded form in disguise.

Press the +10,00,000 (ten lakh) button ten times and you get 1,00,00,000. That is 100 lakhs, and we call it one crore. It is 1 followed by seven zeros.

Two naming systems. India and some neighbouring countries name and group numbers one way; many other countries use the International (also called American) system. Compare them:

Groups of digits. The Indian system groups digits 3-2-2-2 from the right: thousands, lakhs, crores, arabs. The International system groups them 3-3-3: thousands, millions, billions. The words lakh and crore come from the Sanskrit words lakṣha and koṭi.

Useful links between the systems.

• 1 million = 10 lakh.
• 1 crore = 10 million.
• 1 billion = 100 crore = 1 arab.
• 1 lakh = 100 thousand; 1 crore = 100 lakh; 1 arab = 100 crore.
• A billion is 10,000 lakhs.

Reading a ten-digit number both ways. Take 9876501234. Indian commas: 9,87,65,01,234, read as 987 crore 65 lakh 1 thousand 234 (or 9 arab 87 crore 65 lakh 1 thousand 234). International commas: 9,876,501,234, read as 9 billion 876 million 501 thousand 234. It is the same number; only the grouping and the names change.

Comparing across systems. Is 500 lakh more or less than 5 million? 5 million = 50 lakh, so 500 lakh is greater. Is 640 crore more or less than 60 billion? 60 billion = 6000 crore, so 640 crore is less. Change both into the same unit first, then compare.

Counting zeros. A lakh has 5 zeros, a million 6, a crore 7 and a billion 9. A thousand lakh is 1,000 × 1,00,000 = 10,00,00,000, which has 8 zeros, so it is ten crore.

A headline says, ‘One lakh people visited the book fair.’ Did exactly 1,00,000 people come? Probably not. Very often we only need an approximate value, a round number that tells us how big something is.

Exact or approximate? The 2011 census counted 76,068 people in a town. Saying ‘about 75,000’ gives a clear idea. But when you dial a phone number, pay an exact bill or count medicine doses, only the exact number will do.

Rounding up and rounding down. When the approximate value is more than the actual number, we have rounded up. When it is less, we have rounded down. Which is better depends on the situation:

• A school with 732 people orders 750 sweets, not 700, so no one is left out (round up).
• A shopkeeper says an item costing ₹470 is ‘around ₹450’ to make it sound cheaper (round down).
• Buying cloth or rope, round up so you do not fall short.

Nearest neighbours. For a large number we often want the nearest thousand, ten thousand, lakh, ten lakh or crore. For 6,72,85,183:

How to find a nearest neighbour. Look at the place you are rounding to, and then at the digit just to its right. If that digit is 5 or more, round up; if it is 4 or less, round down. For the nearest lakh of 6,72,85,183, the lakhs digit is 2 and the next digit (ten thousands) is 8, so we round up to 6,73,00,000. For the nearest ten lakh, the ten-lakhs digit is 7 and the next digit is 2, so we round down to 6,70,00,000.

Another example. For 3,87,69,957 the nearest thousand and the nearest ten thousand are both 3,87,70,000, the nearest lakh is 3,88,00,000, the nearest ten lakh is 3,90,00,000 and the nearest crore is 4,00,00,000. A number very close to a round number can share several neighbours.

A good habit. Before saying a rounded value, ask what it will be used for. Round up when running short would be a problem, round down when a lower figure is safer, and keep the exact value when every unit matters.

Rounding helps us check a calculation before we do it. If the estimate and the answer are far apart, we know something has gone wrong.

Estimating a sum. Take 4,63,128 + 4,19,682. One friend says the sum is near 8,00,000 and more than it; another says near 9,00,000 and less than it. Both are right! 4 lakh + 4 lakh = 8 lakh, and both numbers have more beyond their lakhs, so the sum is more than 8 lakh. Their extra parts (63 thousand and 19 thousand) add to about 82 thousand, so the sum is less than 9 lakh. In fact it is closer to 9 lakh. Is it more than 8,50,000? Yes, because 63 + 19 thousand is more than 50 thousand. The exact sum is 8,82,810.

Estimating a difference. For 14,63,128 − 4,90,020, think 14.6 lakh minus about 4.9 lakh, which is about 9.7 lakh. So the difference is near 10,00,000 and below it, and also near 9,00,000 and above it. It is more than 9,50,000. The exact value is 9,73,108.

Reading a table of populations. Census tables are full of large numbers. Here are a few Indian cities:

Questions we can answer by rounding.

• Pune grew from about 25 lakh to about 31 lakh, an increase of about 6 lakh. (Exactly 5,76,958.)
• Bengaluru grew by about 41 lakh, from about 43 lakh to 84 lakh. Its population almost doubled.
• Vadodara went from about 17 lakh to about 36 lakh, a little more than double.
• Mumbai has about 1 crore 24 lakh people and Patna about 17 lakh. Mumbai is roughly 7 times Patna, since 7 × 17 lakh = 119 lakh.

Why estimate first? Estimating is quick, can be done in the head, and catches mistakes such as a missing zero. A good estimate also tells you whether the exact value matters at all: to decide which city grew faster, the rounded numbers are enough.

Steps. Round each number to a convenient place, do the easy calculation, then use the leftover parts to decide whether the true answer is a bit more or a bit less.

Multiplying by 10, 100 or 1000 is easy: just add zeros. We can use this to make other products easy too.

Multiplying by 5. Since 5 = 10 ÷ 2, multiplying by 5 is the same as halving and then multiplying by 10. So 116 × 5 = 58 × 10 = 580. Why does this work? Five groups of 116 is half of ten groups of 116.

Multiplying by 25 and 125. Since 25 = 100 ÷ 4, we get 824 × 25 = 206 × 100 = 20,600. Since 125 = 1000 ÷ 8, we get 72 × 125 = 9 × 1000 = 9000.

Regrouping factors. In a long product we may multiply in any order, so we pair numbers that make 10, 100 or 1000.

• 2 × 1768 × 50 = 1768 × (2 × 50) = 1768 × 100 = 1,76,800.
• 125 × 40 × 8 × 25 = (125 × 8) × (40 × 25) = 1000 × 1000 = 10,00,000.
• 25 × 12 = 300, so 25 × 240 = 6000, 250 × 120 = 30,000 and 2500 × 12 = 30,000.

How many digits will the product have? Can two 2-digit numbers have a 5-digit product? Think of the extremes. The smallest product is 10 × 10 = 100 (3 digits). Every 2-digit number is less than 100, so every product is less than 100 × 100 = 10,000; the largest is 99 × 99 = 9801 (4 digits). So the product of two 2-digit numbers has 3 or 4 digits.

The general pattern.

If one number has m digits and the other has n digits, the product has m + n − 1 or m + n digits. This is a quick way to check a big multiplication: 3-digit × 3-digit can never give a 4-digit answer.

Surprising facts. Easy products can uncover interesting large numbers. 1250 × 380 = 4,75,000, the number of songs a famous 15th-century composer is said to have composed. 2100 × 70,000 = 14,70,00,000, roughly the distance in kilometres from the Earth to the Sun (147 million km). 6400 × 62,500 = 40,00,00,000, about the number of litres of water the Amazon pours into the sea every second.

Some of the best questions sound a little silly. Could the whole of Mumbai fit into one lakh buses? Could you walk to the Moon? Such thought experiments make us estimate, multiply in stages and make sensible assumptions.

Mumbai in buses. Assume a bus holds 50 people. Then 1 lakh buses hold 1 lakh × 50 = 50 lakh people. Mumbai has more than 1 crore 24 lakh people, so one lakh buses are not enough. What about ships that carry about 2500 passengers each? 5000 ships hold 2500 × 5000 = 1,25,00,000 people, just a little more than Mumbai's 1,24,42,373. They would only just fit!

Travelling to the Moon. The Moon is about 3,84,400 km away. If you travel 100 km every day, in a year you cover 100 × 365 = 36,500 km, and in 10 years 3,65,000 km. That is still about 19,400 km short. Working in stages (one day, one year, ten years) keeps the numbers easy.

More quick experiments.

• If one sheet of paper weighs 5 g, one lakh sheets weigh 5,00,000 g = 500 kg. No one can lift that.
• If 250 babies are born every minute, then in a day 250 × 60 × 24 = 3,60,000 are born, much less than a million (10 lakh).
• Counting one coin per second, a day has only 24 × 60 × 60 = 86,400 seconds. A million coins would take about 11.6 days of non-stop counting.
• A stack of coins each 1 mm thick, as tall as a 180 m statue, needs 180 × 1000 = 1,80,000 coins.

Making assumptions. Every thought experiment needs some guesses: how many seats in a bus, how thick a coin is. Choose values that are reasonable and say them clearly. A different but sensible assumption may change the answer a little, but usually not the conclusion.

Division facts. Dividing large numbers also uncovers facts. 13,95,000 ÷ 150 = 9300, about the length in km of the world's longest single train journey across Russia. 10,50,00,000 ÷ 700 = 1,50,000, about the mass in kg that an adult blue whale can reach.

Why this matters. Thought experiments train you to judge whether a claim in the news is believable. If someone says a crowd of 10 lakh fitted into a small ground, a quick estimate tells you whether to be doubtful.