Number names for powers

Two naming systems, one shared start. Both systems agree that thousand $=10^3$. After that they diverge because they group digits differently when writing commas.

Indian system introduces a new name every $2$ digits past thousand:

where: each name labels the power of ten reached when you multiply the previous unit by $100$.

International system introduces a new name every $3$ digits:

where: each name is reached by multiplying the previous unit by $1000$.

Commas follow the grouping. Indian style places a comma after the first $3$ digits from the right, then after every $2$ digits: $12{,}34{,}56{,}789$. International style places a comma after every $3$ digits: $123{,}456{,}789$ — same number, different-looking layout, because the two systems chop the digit string at different points.

Because every name is tied to an exact exponent, converting a named quantity into exponential (standard) form, or comparing two quantities named in different systems, reduces to comparing exponents rather than counting digits.

Why this matters beyond the exam. News reports, government census data, and Indian textbooks routinely quote figures in lakhs and crores — a state's budget, a district's population, the cost of a bridge. International scientific sources, most spreadsheets, and calculators default to thousands, millions, and billions. A student who can only read one system will misjudge the size of a number the moment it is quoted in the other. Being fluent in both, and knowing exactly how they line up power-by-power, is therefore not just a syllabus point but a practical reading skill for anything from a newspaper headline to a scientific abstract about astronomical distances.

A quick sanity check that avoids counting digits. Rather than counting zeros one by one, it helps to remember the ladder: thousand, lakh, crore in the Indian system sit at $10^3, 10^5, 10^7$, each two powers of ten apart after the first step; thousand, million, billion in the International system sit at $10^3, 10^6, 10^9$, each three powers of ten apart. Any number that is "one power of ten larger" than lakh (that is, $10^6$) does not have its own separate Indian name — it is simply called "ten lakh" — while that very same value, $10^6$, is exactly one million in the International system. Spotting overlaps like this is often the fastest way to check a conversion.

A city's population is reported as "42 lakh." Write it in ordinary form with correct commas, in exponential form, and convert it to millions.

Step 1. Recall $1\text{ lakh}=10^5=100{,}000$, so $42\text{ lakh}=42\times10^5$.

Step 2. Multiply out: $42\times10^5 = 42\times100{,}000 = 42{,}00{,}000$ using Indian comma grouping (comma after the first $3$ digits, then every $2$), which is the same number as $4{,}200{,}000$ in International comma grouping.

Step 3. Convert to standard exponential form: $42\times10^5 = 4.2\times10\times10^5 = 4.2\times10^6$.

Step 4. Convert to millions using $1\text{ million}=10^6$: since $4.2\times10^6$ already matches $10^6$ exactly, the population is $4.2$ million. The city's population is $42{,}00{,}000 = 4{,}200{,}000 = 4.2\times10^6 = 4.2$ million.

Competitive stretch

Convert $7$ crore into millions. $7$ crore $=7\times10^7$. Since $10^7=10\times10^6$, this is $7\times10\times10^6=70\times10^6=70$ million $=7.0\times10^7$ — matching the original exponent confirms the conversion is correct. Notice this also tells us directly that $1$ crore $=10$ million, a useful fact to keep handy since crore and million are the two units most commonly mismatched in news reports about budgets and populations. The same four-step method works in reverse. Suppose a report states a country's exports as "$250$ million" and a student needs this in crores. Step 1 gives $250\times10^6$. Step 2 already has the mantissa outside the $1$-to-$10$ range in the usual sense but the exponent form $2.5\times10^8$ is fine as standard form. Step 3 rewrites the exponent to match crore's power, $10^7$: $2.5\times10^8 = 25\times10^7$, so the answer is $25$ crore. Step 4 checks the arithmetic makes sense: $25$ crore is $25\times10^7=2.5\times10^8=250\times10^6=250$ million, which matches the original figure, confirming the conversion.

• Treating "lakh" and "million" as the same size: a lakh is $10^5$ while a million is $10^6$, a factor of $10$ apart, so $1$ million $=10$ lakh, not $1$ lakh. This is the single most common error in this topic and is worth double-checking every time.
• Mixing comma-grouping rules, e.g. writing an Indian-system number with International-style comma spacing (or vice versa) — always match the comma pattern to the naming system being used, since a misplaced comma changes how a reader parses the number's size at a glance.
• Forgetting that thousand ($10^3$) is the one name shared by both systems, which causes confusion right at the start of a conversion — students sometimes assume every unit has a different name in each system, when in fact only the units above thousand diverge.
• Guessing the conversion factor between two named units instead of going back to their exact powers of ten (e.g. crore $=10^7$ and lakh $=10^5$ differ by exactly $10^2=100$, so $1$ crore $=100$ lakh, never $10$ lakh).
• Losing track of the exponent when converting the coefficient into standard form $k\times10^n$ — always re-check that $1\le k<10$ once the shifting is done, otherwise the final power of ten will be off by one.

Ask the tutor: Confused converting between lakh, crore, million, and billion? Ask the AI tutor to walk through a quick conversion drill with your own numbers.