Large numbers in real life

Big numbers are not just textbook curiosities. They appear every time we talk about a country, a planet, a budget, or a year.

Idea

A few "anchor" large numbers help you feel sizes:

• The population of India is about $1{,}40{,}00{,}00{,}000 = 1.4 \times 10^9$ (about $140$ crore or $1.4$ billion).
• The population of a city like Mumbai is about $2$ crore $= 2 \times 10^7$.
• The distance from the Earth to the Sun is about $15$ crore km $= 1.5 \times 10^8$ km.
• The number of seconds in a day is $24 \times 60 \times 60 = 86{,}400 \approx 10^5$.
• The number of seconds in a year is about $3.15 \times 10^7$ , surprisingly close to $1$ crore.

You can answer many "Is that possible?" questions just by comparing such anchors.

A second key idea is the unit. The same big number is read very differently when you change unit. For instance, $1{,}00{,}00{,}000$ rupees is $1$ crore, but expressed in lakhs it is "$100$ lakhs", and in thousands it is "$1$ lakh thousands". A clever student always asks: What is the unit?

A third idea: when a large number is divided by another large number, the answer may be small. India's population is huge, but the population per square kilometre is only a few hundred. Always look at per , per person, per second, per square metre , to get a feeling for the number.

Worked examples

Example 1. If India's population is $140$ crore and the country has $7000$ railway stations, how many people are there on average per station?

$\frac{140 \times 10^7}{7000} = \frac{14 \times 10^8}{7 \times 10^3} = 2 \times 10^5 = 2 \text{ lakh per station}.$

Example 2. Light travels about $3{,}00{,}000$ km per second. How far does it travel in one minute?

$3{,}00{,}000 \times 60 = 1{,}80{,}00{,}000$ km, i.e., $1.8$ crore km, or $1.8 \times 10^7$ km.

Example 3. A book has $250$ pages and a library has $40{,}000$ such books. How many pages in total?

$250 \times 40{,}000 = 1{,}00{,}00{,}000 = 1$ crore pages.

Example 4. A country's annual budget is $\rupee 30$ lakh crore. Express this in rupees and in billion rupees.

$30 \text{ lakh crore} = 30 \times 10^5 \times 10^7 = 30 \times 10^{12} = 3 \times 10^{13}$ rupees.

In billions: $1$ billion $= 10^9$, so the budget is $\frac{3 \times 10^{13}}{10^9} = 3 \times 10^4 = 30{,}000$ billion rupees.

Try it yourself

1. How many seconds in $10$ minutes? Express in scientific form.
2. A city of $50$ lakh people uses $200$ litres of water per person per day. How many litres does the city use daily?
3. If a stadium seats $80{,}000$ people, about how many such stadiums fill the population of India?
4. The Moon is about $3{,}84{,}000$ km away. How many times farther is the Sun ($15$ crore km)?
5. A printing press prints $1200$ newspapers per minute. How many in an $8$-hour shift?
6. India's land area is about $33$ lakh square km. India's population density is about $420$ per square km. Estimate the total population.
7. Express $1$ trillion ($10^{12}$) in crores and in lakh crores.
8. A YouTuber gets $5$ million views per video; in a year she posts $50$ videos. How many total views?

Activity

Make a "size ladder" poster: write on a long strip of paper the powers of $10$ from $10^0$ to $10^{12}$. Beside each, paste a real-world example (e.g., $10^4$ , students in a large school, $10^9$ , people in India). Hang it in your study area.

Pick a country other than India. Find its population and area online; compute people per square km. Compare with the value for India.