Large Numbers Around Us — Class 7 Mathematics Notes (NCERT)

Snapshot

This chapter takes you up to lakhs and crores (the Indian system) and to thousands, millions, billions (the international system).
A thousand thousands is a million; a hundred lakhs is a crore; ten lakhs equals one million.
The Indian place-value order is ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs, crores; commas go after $3$, then every $2$ digits.
You learn to read, write, compare and order large numbers, estimate by rounding, and use approximation for quick mental sums.
Smart play with numbers: making the greatest/least number from given digits, the Collatz-style number games, and patterns like palindromes.
Standard form / expanded form connects every digit to its place value ($5$ in $58,00,000$ stands for $50$ lakh).
Weightage: ~8 marks/year — reading-writing, place value, estimation and a word problem on large quantities.

Detailed Notes

1. Why large numbers?

India's population is more than $140$ crore. A cricket stadium holds about $1,00,000$ people. The distance to the Sun is roughly $15$ crore kilometres. To talk about populations, money, distances and data, we must be comfortable reading and writing very large numbers and making sense of how big they really are. This chapter builds that comfort step by step, starting from the numbers you already know and stretching up to crores and beyond.

The big skills are: (i) knowing the place value of every digit, (ii) reading a large numeral aloud correctly, (iii) comparing and ordering large numbers, and (iv) estimating so you can check whether an answer is sensible.

2. Place value and the Indian system

In our number system the value of a digit depends on its place. Moving one place to the left multiplies the place value by $10$. The Indian place names, from the right, are:

ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs, crores, ten-crores …

So the places have values $1,\ 10,\ 100,\ 1000,\ 10000,\ 1{,}00{,}000\ (\text{lakh}),\ 10{,}00{,}000\ (\text{ten lakh}),\ 1{,}00{,}00{,}000\ (\text{crore})$.

Crore	Ten lakh	Lakh	Th	H	T	O
3	2	5	408

The number $3{,}25{,}408$ has $3$ in the lakh-times-ten... let us read it carefully below. The key rule: a lakh is $1$ followed by $5$ zeros ($1{,}00{,}000$) and a crore is $1$ followed by $7$ zeros ($1{,}00{,}00{,}000$).

$1$ lakh $=100$ thousand $=10^{5}$ · $1$ crore $=100$ lakh $=10^{7}$

3. Commas and reading numbers (Indian system)

To read a large number, we group the digits using commas. In the Indian system the first comma comes after $3$ digits from the right (the thousands), and then after every 2 digits.

Indian grouping: xx , xx , xx , xxx (crore, lakh, thousand, hundreds)

Worked example — read $50732891$

Place commas Indian-style from the right: $5{,}07{,}32{,}891$. Now read by groups: five crore, seven lakh, thirty-two thousand, eight hundred ninety-one.

Worked example — write in figures

"Twenty-three lakh forty thousand six" $=23{,}40{,}006$. Place value check: $2\to$ ten-lakh, $3\to$ lakh, $4\to$ ten-thousand, $0\to$ thousand, $0\to$ hundred, $0\to$ ten, $6\to$ one.

4. The international system — millions and billions

Most of the world groups digits in threes: ones, thousands, millions, billions. Commas come after every $3$ digits.

$1$ million $=10$ lakh $=10^{6}$ · $1$ billion $=100$ crore $=10^{9}$

Indian	In figures	International
1 lakh	100000	100 thousand
10 lakh	1000000	1 million
1 crore	10000000	10 million
100 crore	1000000000	1 billion

Worked example — convert

$3$ crore $=30$ million (since $1$ crore $=10$ million). And $7$ million $=70$ lakh (since $1$ million $=10$ lakh).

5. Expanded form and standard place value

Expanded form writes a number as the sum of (each digit $\times$ its place value). This shows clearly what each digit is "worth".

Worked example — expand $58{,}06{,}209$

$58{,}06{,}209 = 5\times10000000$? No — recount the places.

$=50{,}00{,}000+8{,}00{,}000+0+6{,}000+200+0+9$

$=5\times10^{6}+8\times10^{5}+6\times10^{3}+2\times10^{2}+9.$ So the $5$ stands for fifty lakh and the $8$ for eight lakh.

Each step left $\Rightarrow \times10$: $10^{5}=$ lakh, $\ 10^{6}=$ ten lakh, $\ 10^{7}=$ crore.

6. Comparing and ordering large numbers

To compare two large numbers:

The number with more digits is greater (e.g. a $7$-digit number $>$ a $6$-digit number).
If the digit-count is the same, compare digits from the left (the highest place) until they differ.

Worked example — arrange in ascending order

$4{,}50{,}312;\ \ 45{,}03{,}120;\ \ 4{,}05{,}213.$ Counting digits: the second has $7$ digits, so it is largest. Among the two $6$-digit numbers, compare from the left: $4{,}05{,}213$ vs $4{,}50{,}312$ → the second digit $0<5$, so $4{,}05{,}213$ is smaller. Ascending: $4{,}05{,}213 < 4{,}50{,}312 < 45{,}03{,}120.$

7. Greatest and least numbers from given digits

A favourite puzzle: using each given digit once, build the greatest and the least number.

Greatest: arrange the digits in descending order.
Least: arrange in ascending order — but if $0$ is present, it cannot be the first digit, so put the next smallest digit first.

Worked example — digits 7, 0, 4, 2, 9

Greatest: $97{,}420$. Least: putting $0$ first is not allowed, so $20{,}479$ (start with $2$, then $0,4,7,9$).

8. Rounding and estimation

Estimation gives a quick, sensible answer without exact calculation — very useful for checking. To round a number to a given place, look at the digit just to the right:

If the next digit is $5$ or more → round up; if it is less than $5$ → round down (keep the same). Replace the rest with zeros.

Worked example — round to nearest thousand

$47{,}812$: the hundreds digit is $8\ (\ge5)$, so round up the thousands: $48{,}000$. $52{,}349$: hundreds digit $3\ (<5)$, so $52{,}000$.

Worked example — estimate a sum

Estimate $3{,}87{,}219 + 5{,}96{,}480$ to the nearest lakh. Round each: $4{,}00{,}000 + 6{,}00{,}000 = 10{,}00{,}000$. The exact sum is $9{,}83{,}699$, so $10$ lakh is a good estimate.

9. Number play and patterns

The chapter is full of "number play" that builds number sense.

Palindromes: a number that reads the same forwards and backwards, like $12321$ or $4554$.
Reach-100 / Collatz-style games: apply a rule repeatedly (e.g. even → halve, odd → $3n+1$) and watch the chain.
Estimating crowds & quantities: if a hall has $25$ rows of $40$ seats, that is about $1000$ seats; multiply up for larger venues.

Worked example — how many seconds in a day?

$60\times60\times24 = 86{,}400$ seconds — roughly $86$ thousand, useful to remember as "under a lakh".

10. Word problems with large quantities

Worked example — total cost

A factory makes $1{,}25{,}000$ bottles a day. How many in $30$ days? $1{,}25{,}000\times30 = 37{,}50{,}000$ → thirty-seven lakh fifty thousand bottles.

Worked example — difference

City A has $48{,}62{,}300$ people; City B has $39{,}74{,}800$. How many more in A? $48{,}62{,}300 - 39{,}74{,}800 = 8{,}87{,}500$ → eight lakh eighty-seven thousand five hundred more.

11. Common mistakes to avoid

Placing commas every $3$ digits in the Indian system — after the first $3$, group by 2.
Mixing up $1$ million ($=10$ lakh) with $1$ crore ($=100$ lakh).
Letting $0$ be the leading digit when forming the least number.
Rounding using the wrong reference digit — always check the digit to the right of the rounding place.
Mis-counting zeros: lakh has $5$ zeros, crore has $7$ zeros.

12. Quick revision checklist

Indian places: ones → thousands → lakhs → crores; commas after $3$ then every $2$ digits.
$1$ lakh $=10^{5}$, $1$ crore $=10^{7}$, $1$ million $=10^{6}$, $1$ billion $=10^{9}$.
$10$ lakh $=1$ million; $1$ crore $=10$ million; $100$ crore $=1$ billion.
Compare by digit-count first, then left-to-right.
Round: look right of the place; $\ge5$ up, else down.

Practice MCQs

1. The number of zeros in one crore is:

Answer: (C) $1$ crore $=1{,}00{,}00{,}000=10^{7}$, which has $7$ zeros.

2. $10$ lakh is equal to:

$1$ million
$1$ crore
$1$ billion
$100$ thousand

Answer: (A) $10$ lakh $=10{,}00{,}000=10^{6}=1$ million.

3. In the Indian system, the correct comma placement for $40500670$ is:

$40{,}500{,}670$
$4{,}05{,}00{,}670$
$4{,}0500{,}670$
$40{,}5006{,}70$

Answer: (B) Commas after $3$ digits then every $2$: $4{,}05{,}00{,}670$.

4. The place value of $7$ in $7{,}38{,}29{,}016$ is:

$7$ lakh
$7$ crore
$70$ lakh
$7$ ten-crore

Answer: (B) $7$ sits in the crore place, so its value is $7$ crore.

5. The greatest $5$-digit number formed from $3,0,6,1,8$ (each once) is:

$86{,}310$
$83{,}610$
$86{,}301$
$80{,}631$

Answer: (A) Descending order of digits gives $86{,}310$.

6. The least $5$-digit number formed from $3,0,6,1,8$ (each once) is:

$01{,}368$
$10{,}368$
$13{,}068$
$10{,}386$

Answer: (B) $0$ cannot lead, so smallest non-zero ($1$) first: $10{,}368$.

7. $1$ billion equals how many crores?

$10$
$100$
$1000$
$1$

Answer: (B) $1$ billion $=10^{9}=100\times10^{7}=100$ crore.

8. Rounded to the nearest thousand, $46{,}728$ is:

$46{,}000$
$47{,}000$
$46{,}700$
$50{,}000$

Answer: (B) Hundreds digit is $7\ (\ge5)$, so round up to $47{,}000$.

9. Which number is the largest?

$9{,}87{,}654$
$10{,}23{,}456$
$9{,}99{,}999$
$8{,}88{,}888$

Answer: (B) It is the only $7$-digit number, hence the largest.

10. The expanded form of $3{,}05{,}060$ is:

$3\times10^{5}+5\times10^{3}+6\times10^{1}$
$3\times10^{5}+5\times10^{4}+6\times10^{2}$
$3\times10^{4}+5\times10^{3}+6$
$3\times10^{6}+5\times10^{3}+6$

Answer: (A) $300000+5000+60=3\times10^{5}+5\times10^{3}+6\times10^{1}$.

11. $1$ million is the same as:

$1$ lakh
$10$ lakh
$1$ crore
$100$ crore

Answer: (B) $1$ million $=10^{6}=10$ lakh.

12. Which of these is a palindrome?

$12345$
$45654$
$67890$
$23456$

Answer: (B) $45654$ reads the same forwards and backwards.

13. The successor of the greatest $6$-digit number is:

$9{,}99{,}998$
$10{,}00{,}000$
$1{,}00{,}000$
$99{,}99{,}999$

Answer: (B) Greatest $6$-digit number is $9{,}99{,}999$; its successor is $10{,}00{,}000$.

14. Estimate $5{,}94{,}300 + 4{,}08{,}900$ to the nearest lakh:

$9$ lakh
$10$ lakh
$11$ lakh
$8$ lakh

Answer: (B) $6$ lakh $+4$ lakh $=10$ lakh.

15. How many lakhs make one crore?

$10$
$100$
$1000$
$50$

Answer: (B) $1$ crore $=100$ lakh.

Important Questions

Q1. Write $5{,}07{,}32{,}891$ in words (Indian system) and in the international system. (2 marks)

Answer: Indian: five crore, seven lakh, thirty-two thousand, eight hundred ninety-one. International: $50{,}732{,}891$ → fifty million, seven hundred thirty-two thousand, eight hundred ninety-one.

Q2. Using digits $5,2,0,8,1$ once each, form the greatest and least $5$-digit numbers and find their difference. (3 marks)

Answer: Greatest $=85{,}210$; least $=10{,}258$ ($0$ cannot lead). Difference $=85{,}210-10{,}258=74{,}952$.

Q3. A water tank holds $2{,}45{,}000$ litres. A town uses $18$ such tanks daily. Estimate the daily use to the nearest lakh and give the exact figure. (3 marks)

Answer: Exact: $2{,}45{,}000\times18=44{,}10{,}000$ litres. Estimate: round to $2$ lakh $\times18\approx36$ lakh (rough); or round product to nearest lakh $=44$ lakh litres.

Q4. Arrange in descending order: $36{,}07{,}812;\ 3{,}67{,}812;\ 36{,}70{,}812;\ 3{,}60{,}871.$ (2 marks)

Answer: $36{,}70{,}812 > 36{,}07{,}812 > 3{,}67{,}812 > 3{,}60{,}871.$ Compare digit-count first (two $7$-digit), then left to right.

Q5. Explain why $1$ crore $=10$ million using place value. (2 marks)

Answer: $1$ crore $=1{,}00{,}00{,}000=10^{7}$. $1$ million $=10^{6}$. So $10^{7}=10\times10^{6}$, i.e. $1$ crore $=10$ million.

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More Class 7 Mathematics chapters

Arithmetic Expressions · A Peek Beyond the Point · Expressions using Letter-Numbers · Parallel and Intersecting Lines · Number Play · A Tale of Three Intersecting Lines · Working with Fractions · Geometric Twins