Mental Ability · Lesson 1

Numbers & Numeration

Big numbers are easy once you know where each digit lives. Learn place value, lakhs and millions, how to build the biggest and smallest numbers, rounding, and the secret code of Roman numerals.

~40 min read7 quick checksTest: 20 questions

1Place value and face value

We write every number with just ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The same digit can mean very different amounts. In 555, the first 5 means five hundred, the middle 5 means fifty and the last 5 means just five. What changes is the place where the digit sits.

Each place is 10 times the place on its right: ones → tens → hundreds → thousands → ten thousands → lakhs → ten lakhs → crores → ten crores.

Face value
The digit itself. It never changes, wherever the digit sits. Face value of 8 in 7,84,329 is 8.
Place value
Face value × value of its place. The 8 in 7,84,329 is in the ten thousands place, so its place value is 8 × 10,000 = 80,000.
L 3 TTh 0 Th 4 H 5 T 1 O 2 3,04,512
An abacus shows place value clearly: each rod is a place. This abacus shows 3,04,512. The empty ten-thousands rod stands for 0.
✏️ Worked example

In 7,84,329, find the difference between the place value and the face value of 8.

  1. 8 is in the ten thousands place. Place value = 8 × 10,000 = 80,000.
  2. Face value of 8 = 8.
  3. Difference = 80,000 − 8 = 79,992.
✏️ Worked example — "how many times?"

In 5,55,555, the place value of the 5 in the ten thousands place is how many times the place value of the 5 in the tens place?

  1. 5 in ten thousands place → 50,000. 5 in tens place → 50.
  2. 50,000 ÷ 50 = 1,000 times.
  3. Shortcut: count the places between them. Ten thousands is 3 places left of tens, so it is 10 × 10 × 10 = 1,000 times.
📌 Remember

The place value and the face value of 0 are always 0, in any place. And for a digit in the ones place, place value = face value.

What is the place value of 5 in 2,35,091?

  1. 5
  2. 500
  3. 5,000
  4. 50,000
Read from the right: 1 ones, 9 tens, 0 hundreds, 5 thousands. So its place value is 5 × 1,000 = 5,000.

2The Indian system: lakhs and crores

To read a long number, we split its digits into groups called periods, using commas. In the Indian system the periods are Ones, Thousands, Lakhs, Crores.

Indian system → 45,23,86,147 Crores Lakhs Thousands Ones TC 4 C 5 TL 2 L 3 TTh 8 Th 6 H 1 T 4 O 7 International system → 452,386,147 Millions Thousands Ones HM 4 TM 5 M 2 HTh 3 TTh 8 Th 6 H 1 T 4 O 7
The same nine digits in both systems. Thick lines show where the commas go. TC = ten crores, TL = ten lakhs, HM = hundred millions, TM = ten millions, HTh = hundred thousands.

Reading and writing number names

Read each period from the left and say its name. Do not say the name of the Ones period.

✏️ Worked example

Write 60540302 in words (Indian system).

  1. Put commas 3, 2, 2 from the right: 6,05,40,302.
  2. Crores period: 6 → "six crore". Lakhs period: 05 → "five lakh". Thousands period: 40 → "forty thousand". Ones period: 302 → "three hundred two".
  3. Answer: Six crore five lakh forty thousand three hundred two.
✏️ Worked example — from words to digits

Write "seven crore nine lakh six hundred eight" in numerals.

  1. Make a box for each place: Crores [ _ ] Lakhs [ _ _ ] Thousands [ _ _ ] Ones [ _ _ _ ].
  2. Fill in: crores = 7, lakhs = 09, thousands = 00 (none were said!), ones = 608.
  3. Answer: 7,09,00,608. Empty places must be filled with 0.
⚠️ Olympiad trap

Watch for missing periods. "Three crore two lakh nine thousand fifteen" is 3,02,09,015 — not 3,2,9,15 and not 3,20,90,015. Every period except the first must have its full number of digits (2 for lakhs and thousands, 3 for ones).

💡 Tip

We write "five lakh", "twenty crore", "forty thousand" — no "s" at the end, and no "and" in the middle. Two-digit parts get a hyphen: twenty-three, forty-five.

3The International system and conversions

Most countries use the International system. Its periods are Ones, Thousands, Millions (then Billions), and every period has 3 digits. So commas go after every 3 digits from the right: pattern 3, 3, 3.

✏️ Worked example

Write 60540302 with commas and in words (International system).

  1. Groups of 3 from the right: 60,540,302.
  2. Millions: 60 → "sixty million". Thousands: 540 → "five hundred forty thousand". Ones: 302 → "three hundred two".
  3. Answer: Sixty million five hundred forty thousand three hundred two.

The conversion table

NumberIndianInternational
1,0001 thousand1 thousand
1,00,000 = 100,0001 lakh100 thousand
10,00,000 = 1,000,00010 lakh1 million
1,00,00,000 = 10,000,0001 crore10 million
10,00,00,000 = 100,000,00010 crore100 million
Remember: "1 lakh is 100 thousand, 10 lakh is 1 million, 1 crore is 10 million." So to change millions into lakhs, multiply by 10. To change crores into millions, multiply by 10.
✏️ Worked example — quick conversions
  1. 5 million = 5 × 10 lakh = 50 lakh.
  2. 3 crore = 3 × 10 million = 30 million.
  3. 1 crore = 100 lakh = 10,000 thousand.
  4. 25 lakh = 25,00,000 = 2,500,000 (two million five hundred thousand).
✏️ Worked example — read an Indian number the International way

How is 70,70,707 read in the International system?

  1. Remove the Indian commas: 7070707.
  2. Put commas every 3 digits from the right: 7,070,707.
  3. Read: seven million seventy thousand seven hundred seven.

Which shows 4,52,03,160 correctly in the International system?

  1. 45,203,160
  2. 4,520,3160
  3. 452,031,60
  4. 4,52,03,160
Remove the commas (45203160) and regroup in threes from the right: 45,203,160. Option D is the Indian form, not the International one.

3 crore is equal to how many millions?

  1. 3 million
  2. 30 million
  3. 300 million
  4. 3,000 million
1 crore = 10 million, so 3 crore = 3 × 10 = 30 million (3,00,00,000 = 30,000,000).

4Expanded form and "tens, hundreds" puzzles

The expanded form writes a number as the sum of the place values of its digits.

✏️ Worked example
  1. 3,46,215 = 3,00,000 + 40,000 + 6,000 + 200 + 10 + 5.
  2. 4,07,05,060 = 4,00,00,000 + 7,00,000 + 5,000 + 60. (Zeros are skipped, but they still hold their places.)
  3. Going back: 80,000 + 700 + 3 = 80,703. Build it place by place: 8 ten thousands, 0 thousands, 7 hundreds, 0 tens, 3 ones.

Olympiads love hiding numbers inside words like "900 tens" or "45 hundreds". Always multiply by the place first.

⚠️ Olympiad trap: 1 million − 900 tens
  1. 1 million = 1,000,000.
  2. 900 tens = 900 × 10 = 9,000 (not 900!).
  3. 1,000,000 − 9,000 = 991,000.

The wrong answer 999,100 comes from forgetting to multiply 900 by 10.

✏️ Worked example

What number is 5 lakh + 45 hundreds + 3 tens?

  1. 5 lakh = 5,00,000.
  2. 45 hundreds = 45 × 100 = 4,500.
  3. 3 tens = 30.
  4. Total = 5,00,000 + 4,500 + 30 = 5,04,530.

5Comparing and ordering numbers

  1. Count the digits. The number with more digits is greater. 1,00,000 (6 digits) > 99,999 (5 digits).
  2. Same number of digits? Compare from the left, place by place. The first place where the digits differ decides.
✏️ Worked example

Arrange in ascending order (smallest first): 4,05,321 · 4,50,123 · 45,321 · 4,05,312

  1. 45,321 has only 5 digits, so it is the smallest.
  2. The other three have 6 digits. Lakhs digit: all 4. Ten-thousands digit: 0, 5, 0 → 4,50,123 is the biggest.
  3. 4,05,321 vs 4,05,312: same until the tens place, where 2 > 1, so 4,05,321 is bigger.
  4. Ascending: 45,321 < 4,05,312 < 4,05,321 < 4,50,123.
💡 Tip

Ascending = going up, smallest to greatest (like climbing stairs). Descending = coming down, greatest to smallest. Symbols: < means "is less than", > means "is greater than". The open mouth always faces the bigger number.

6Forming numbers, successor and predecessor

Greatest and smallest numbers from given digits

⚠️ Olympiad trap: zero at the front

02,479 is not a 5-digit number — it is really 2,479, a 4-digit number. When 0 is one of the digits, the smallest number starts with the smallest non-zero digit, followed by 0.

✏️ Worked example — even and odd twists

Using 1, 4, 7, 8, 5 once each, form the greatest 5-digit even number.

  1. Greatest arrangement is 87,541 — but it ends in 1, so it is odd.
  2. An even number must end in an even digit: 4 or 8.
  3. Keep the big digits at the front. Put the smaller even digit (4) at the end: 8, 7, 5, 1, then 4.
  4. Answer: 87,514. (Ending in 8 gives only 75,418.)
✏️ Worked example

Using 0, 3, 6, 8, 5 once each, form the smallest 5-digit odd number.

  1. It must end in an odd digit: 3 or 5.
  2. We want the smallest possible first digit, 3, so save 3 for the front and put 5 at the end.
  3. Remaining digits 0, 6, 8 go in ascending order after the 3: 3, 0, 6, 8, then 5.
  4. Answer: 30,685. (Ending in 3 would give 50,683, which is bigger.)

Successor and predecessor

The successor comes just after a number (add 1). The predecessor comes just before it (subtract 1).

✏️ Worked example — counting numbers

How many 4-digit numbers are there?

  1. Smallest 4-digit number = 1,000. Greatest = 9,999.
  2. Count = greatest − smallest + 1 = 9,999 − 1,000 + 1 = 9,000.
  3. Pattern: 9,000 four-digit numbers, 90,000 five-digit numbers, 9,00,000 six-digit numbers.

What is the smallest 5-digit number that can be formed using 6, 0, 2, 9, 4 (each once)?

  1. 02,469
  2. 20,469
  3. 24,069
  4. 20,496
Smallest non-zero digit (2) first, then 0, then the rest in ascending order: 2, 0, 4, 6, 9 → 20,469. "02,469" is really a 4-digit number.

7Rounding and estimation

Rounding gives a "nice", nearby number that is easy to work with. Use this rule:

  1. Find the place you are rounding to.
  2. Look at the digit just to its right.
  3. If it is 0, 1, 2, 3 or 4 → round down: keep the digit, change everything to its right to 0.
  4. If it is 5, 6, 7, 8 or 9 → round up: add 1 to the digit, change everything to its right to 0.
3,62,700 3,62,750 3,62,800 halfway 3,62,748 closer to 3,62,700
Rounding to the nearest hundred: 3,62,748 is before the halfway mark (3,62,750), so it rounds down to 3,62,700.
✏️ Worked example — one number, four roundings

Round 3,62,748 to the nearest 10, 100, 1,000 and 10,000.

  1. Nearest 10: look at ones digit 8 (≥ 5) → round up → 3,62,750.
  2. Nearest 100: look at tens digit 4 (< 5) → round down → 3,62,700.
  3. Nearest 1,000: look at hundreds digit 7 (≥ 5) → round up → 3,63,000.
  4. Nearest 10,000: look at thousands digit 2 (< 5) → round down → 3,60,000.
⚠️ Olympiad trap: the carry chain

Round 9,960 to the nearest hundred. The tens digit is 6, so round up: 9 hundreds + 1 = 10 hundreds. The answer is 10,000, not 9,900. Also: look only at the one digit next door — 449 rounds to the nearest hundred as 400, even though 49 "looks" close to 50.

Estimation

To estimate a sum or difference, round each number first, then calculate. It helps you check if an answer is sensible.

✏️ Worked example

Estimate 6,784 + 2,351 by rounding to the nearest hundred.

  1. 6,784 → 6,800 (tens digit 8). 2,351 → 2,400 (tens digit 5).
  2. Estimated sum = 6,800 + 2,400 = 9,200.
  3. Check: the exact sum is 9,135 — very close.

Round 84,562 to the nearest thousand.

  1. 84,000
  2. 84,500
  3. 85,000
  4. 80,000
The thousands digit is 4. The digit to its right (hundreds) is 5, so round up: 85,000.

8Roman numerals

The ancient Romans wrote numbers with letters. There are only seven symbols — and no symbol for zero.

I 1 V 5 X 10 L 50 C 100 D 500 M 1000
Green symbols (I, X, C, M) may repeat up to 3 times. Pink symbols (V, L, D) are never repeated and never subtracted.

The four rules

  1. Repeat rule: I, X, C, M can be repeated up to 3 times in a row (III = 3, XXX = 30, CCC = 300, MMM = 3,000). V, L, D are never repeated (VV is wrong — 10 is X).
  2. Add rule: a smaller symbol written after a bigger one is added. VI = 5 + 1 = 6; LX = 60; MC = 1,100.
  3. Subtract rule: a smaller symbol written before a bigger one is subtracted — but only these six pairs: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900. I only before V and X; X only before L and C; C only before D and M.
  4. One at a time: only one smaller symbol may be subtracted. 8 is VIII, never IIX. V, L and D are never subtracted: 95 is XCV, never VC.
Remember the order: "I Value Xylophones Like Cows Dig Milk" → I, V, X, L, C, D, M = 1, 5, 10, 50, 100, 500, 1000.

Converting to Roman numerals

Break the number into thousands, hundreds, tens and ones. Convert each part separately, then join them.

✏️ Worked examples
  1. 489 = 400 + 80 + 9 = CD + LXXX + IX = CDLXXXIX.
  2. 1947 = 1000 + 900 + 40 + 7 = M + CM + XL + VII = MCMXLVII.
  3. 2026 = 2000 + 20 + 6 = MM + XX + VI = MMXXVI.
  4. 49 = 40 + 9 = XL + IX = XLIX (not IL — I can only go before V or X).
  5. 99 = 90 + 9 = XCIX (not IC).
✏️ Worked example — Roman to our numbers

Find the value of MCMLXXXIV.

  1. Read left to right, grouping subtract-pairs: M | CM | L | XXX | IV.
  2. 1000 + 900 + 50 + 30 + 4 = 1984.
✏️ Worked example — Roman arithmetic

Find CLX − LXVI. Give the answer in Roman numerals.

  1. CLX = 100 + 50 + 10 = 160.
  2. LXVI = 50 + 10 + 5 + 1 = 66.
  3. 160 − 66 = 94.
  4. 94 = 90 + 4 = XC + IV = XCIV.
⚠️ Olympiad trap: "which is correct?"

Wrong forms examiners use: IIII (4 = IV), VV (10 = X), LL (100 = C), VX (5 = V), IL (49 = XLIX), IC (99 = XCIX), VC (95 = XCV), IIX (8 = VIII), XM (990 = CMXC). Always convert both ways to check.

🌟 Did you know?

With the normal rules the biggest Roman numeral is MMMCMXCIX = 3,999. For larger numbers, a bar drawn over a symbol multiplies it by 1,000 — V with a bar means 5,000. You still see Roman numerals on clocks, in book chapters and in names like "Class IV".

How is 49 written in Roman numerals?

  1. IL
  2. XLIX
  3. XLVIIII
  4. XXXXIX
49 = 40 + 9 = XL + IX = XLIX. IL breaks the rule that I can only be subtracted from V and X; XLVIIII and XXXXIX repeat a symbol 4 times.

Which Roman numeral has the greatest value?

  1. LXXXVIII
  2. XCIV
  3. XCII
  4. XCVI
LXXXVIII = 88, XCIV = 94, XCII = 92, XCVI = 96. The greatest is XCVI.

9Watch and explore

10Quick recap

Flip cards

QuestionPlace value of 7 in 3,74,590?
Answer70,000 (ten thousands place). Its face value is 7.
QuestionHow many lakhs make 1 million?
Answer10 lakh = 1,000,000 = 1 million.
QuestionWrite 8,05,00,017 in words.
AnswerEight crore five lakh seventeen.
Question1 million − 700 hundreds = ?
Answer700 hundreds = 70,000, so 1,000,000 − 70,000 = 930,000.
QuestionSmallest 6-digit number using 6, 0, 4, 9, 2, 7?
Answer2,04,679 — smallest non-zero digit first, then 0.
QuestionSuccessor of 99,999?
Answer1,00,000 — one lakh, the smallest 6-digit number.
QuestionRound 4,648 to the nearest hundred.
Answer4,600 (tens digit 4 is less than 5).
QuestionValue of CDXLIV?
AnswerCD + XL + IV = 400 + 40 + 4 = 444.
Question1990 in Roman numerals?
AnswerM + CM + XC = MCMXC.