Comparing and ordering large numbers

Is $1$ crore bigger than $9$ million? Is $20{,}48{,}317$ greater than $20{,}84{,}137$? These look intimidating until you know the simple rule.

Idea

When comparing two whole numbers, follow this short routine:

1. Count the digits. The one with more digits is larger. (E.g., $1$ crore has $8$ digits, $9$ million has $7$ digits, so $1$ crore wins.)
2. If they have the same number of digits, compare from the left. Look at the leftmost digit; if equal, look at the next column, and so on until they differ.
3. The first column where they differ decides the winner. The larger digit means the larger number.

To order several numbers (smallest to largest or vice versa), compare them pairwise using the same rule. A neat way is to first sort by digit count, then within each group sort by leftmost difference.

When numbers are written with commas (Indian or international), comparing is even easier because the commas line up the columns. Always rewrite numbers in the same system before comparing , never compare $1{,}25{,}00{,}000$ directly to $12{,}500{,}000$ until you notice they are the same number.

A common student mistake: confusing $10$ lakh with $10$ million. They are not equal , $10$ lakh is $10^6$ (one million), while $10$ million is $10^7$ (one crore). Always convert mentally to powers of $10$.

Worked examples

Example 1. Which is greater, $87{,}45{,}600$ or $8{,}74{,}56{,}000$?

The first has $7$ digits; the second has $8$ digits. So $8{,}74{,}56{,}000 > 87{,}45{,}600$.

Example 2. Arrange in increasing order: $5$ lakh, $50000$, $5$ million, $5$ crore.

Convert each to a numeral: $5$ lakh $= 5{,}00{,}000$; $50000 = 50{,}000$; $5$ million $= 50{,}00{,}000$; $5$ crore $= 5{,}00{,}00{,}000$.

By digit count: $50{,}000$ ($5$ digits), $5{,}00{,}000$ ($6$ digits), $50{,}00{,}000$ ($7$ digits), $5{,}00{,}00{,}000$ ($8$ digits). So the increasing order is $50{,}000 < 5{,}00{,}000 < 50{,}00{,}000 < 5{,}00{,}00{,}000.$

Example 3. Compare $4{,}28{,}617$ and $4{,}28{,}706$.

Both have $7$ digits. Compare left to right: $4 = 4$, $2 = 2$, $8 = 8$, $6 = 7$ , wait, the next column gives $6$ vs $7$ in the hundreds place. So $4{,}28{,}706 > 4{,}28{,}617$.

Example 4. A state's budget is $2.5$ lakh crore. Another's is $30$ billion. Which is larger?

$2.5$ lakh crore $= 2.5 \times 10^5 \times 10^7 = 2.5 \times 10^{12}$. And $30$ billion $= 3 \times 10^{10}$. So $2.5 \times 10^{12} > 3 \times 10^{10}$ , the first is larger (by a factor of about $83$).

Try it yourself

1. Which is greater, $9{,}99{,}999$ or $10{,}00{,}001$?
2. Arrange in decreasing order: $1$ crore, $99$ lakh, $10$ million, $1$ billion.
3. Put $<$ or $>$: $4{,}57{,}832 \,\square\, 4{,}57{,}823$.
4. Order from smallest to largest: $20$ lakh, $200000$, $0.2$ million, $20000$.
5. How many lakhs are equal to $0.5$ crore?
6. A library has $2{,}50{,}000$ books; a bigger one has $1.2$ million. Which has more?
7. Is $9 \times 10^7$ greater or less than $1 \times 10^8$?
8. Write the smallest and largest $7$-digit number using the digits $0, 1, 2, 3, 4, 5, 6$ exactly once.

Activity

Take any newspaper. Find five large numbers in it and write each on a slip of paper. Without looking at the original story, shuffle the slips and arrange them in increasing order. Then check by re-reading the stories.