Number systemInvestigateabout 45 min
Testing big-number ideas
Predict first, then try it: shifting digits, rollovers, rounding traps and estimation errors
Make predictions about place value and then test them: what moving a digit does, how many numbers of each size exist, when a successor gains a digit, which numbers round to the same value, how far off an estimate can be, and why 6174 keeps appearing.
In this part you’ll
- Predict and test how moving a digit one place changes its value, and explain it with place value.
- Investigate patterns in counting numbers: how many n-digit numbers there are and how many digits it takes to write them.
- Find every number that rounds to a given value, and show why rounding twice can give a different answer.
- Compare estimates with exact answers and decide which rounding gives a good enough estimate.
- Test claims about forming numbers and Roman numerals, deciding whether they are always, sometimes or never true.
Mathematicians do not just follow rules: they poke at them. What happens if…? Is it always true? Can I find an example where it fails? This layer is built around that habit. Each chapter starts with a prediction: commit to an answer before you read on. Then test it with a lab, a table or a calculation, and finally explain why it happens using place value.
Wrong predictions are not failures. They are the most useful kind, because they show you exactly where your picture of numbers needs fixing.
Chapter 01
What happens when a digit moves?
Predict first
| Operation | Result | Place value of the 5 |
|---|---|---|
| Start | 2,508 | 500 |
| × 10 | 25,080 | 5,000 |
| × 100 | 2,50,800 | 50,000 |
| × 1,000 | 25,08,000 | 5,00,000 |
| × 10,000 | 2,50,80,000 | 50,00,000 |
Lab
Build 2,508 and then build it ten, hundred, thousand and ten thousand times bigger; watch every digit shift left.
Crores
Lakhs
Thousands
Ones
Tip: tap a digit, then use ↑ ↓ or type 0–9. Ten in one column swaps for one in the next.
Indian name: two thousand five hundred eight
International name: two thousand five hundred eight
Expanded form
2,000 + 500 + 8
Tap any digit to compare its face value and place value.
Build-the-number challenge
5 numbers to build. Each one is written a different way!
Text version of this activity
The lab starts at 2,508 in an eight-column Indian chart (up to crores), and shows the name in both systems. The challenges ask for 25,080, 2,50,800, 25,08,000 and 2,50,80,000. Each time, the same pattern of counters (2, 5, 0, 8) moves one column further left, and a new 0 fills the ones column.
In words: two thousand five hundred eight; twenty-five thousand eighty; two lakh fifty thousand eight hundred; twenty-five lakh eight thousand; two crore fifty lakh eighty thousand. Internationally, 25,080,000 is twenty-five million eighty thousand.
The last challenge, 8,052, uses the same digits in reverse order and is a completely different number: digits alone mean nothing without their places.
Predict first
Why “multiply by 10” means “put a 0 on the end”
- Step 01Every digit moves up one place
Ten ones make a ten, ten tens make a hundred: so ten copies of any place fill exactly the next place up.
- Step 02The ones place is left empty
After the shift nothing is in the ones place, so we write 0 there as a placeholder.
- Step 03It works for 100 and 1,000 too
Multiplying by 100 is multiplying by 10 twice: two shifts, two zeros. By 1,000: three shifts, three zeros.
- Step 04Dividing undoes it
43,750 ÷ 10 = 4,375: every digit moves one place right and the 0 disappears. (If the ones digit is not 0, the answer is not a whole number.)
Try it
Chapter 02
Is more digits always bigger?
Predict first
| Largest n-digit number | Successor | Digits in successor |
|---|---|---|
| 9 | 10 | 2 |
| 99 | 100 | 3 |
| 999 | 1,000 | 4 |
| 99,999 | 1,00,000 | 6 |
| 9,99,999 | 10,00,000 | 7 |
| 99,99,999 | 1,00,00,000 | 8 |
| 99,99,99,999 | 1,00,00,00,000 | 10 |
Lab
Explore rollovers: build the largest numbers of each size, look at their successors, and see a new place appear.
Crores
Lakhs
Thousands
Ones
Tip: tap a digit, then use ↑ ↓ or type 0–9. Ten in one column swaps for one in the next.
Indian name: ninety-nine thousand nine hundred ninety-nine
International name: ninety-nine thousand nine hundred ninety-nine
Tap any digit to compare its face value and place value.
Build-the-number challenge
6 numbers to build. Each one is written a different way!
Text version of this activity
The lab starts at 99,999 with the predecessor (99,998) and successor (1,00,000) shown. Add one more counter to the ones column and every column rolls over: ten ones become a ten, ten tens a hundred, and so on until a single counter lands in the lakhs column.
The challenges pair each largest number with its successor: 9,99,999 and 10,00,000 (nine lakh ninety-nine thousand nine hundred ninety-nine, then ten lakh, which is one million); 99,99,999 and 1,00,00,000 (one crore, ten million); and finally 9,99,99,999 (nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine, or 99,999,999 internationally). Each time the successor has one more digit, and it is always 1 followed by zeros.
Try it
Chapter 03
How many numbers? How many digits?
Predict first
| Up to | Count of numbers | Digits used |
|---|---|---|
| 10 | 10 | 11 |
| 100 | 100 | 192 |
| 1,000 | 1,000 | 2,893 |
| 10,000 | 10,000 | 38,894 |
| 1,00,000 | 1,00,000 | 4,88,895 |
Worked example
0 / 4 steps shownHow many pages?
A printer used 1,002 digits to number the pages of a book, starting from page 1. How many pages does the book have?
Try it
Chapter 04
Where the two systems agree and differ
Predict first
| Digits | Indian example | Indian commas | International example | International commas |
|---|---|---|---|---|
| 3 | 999 | 0 | 999 | 0 |
| 4 | 9,999 | 1 | 9,999 | 1 |
| 5 | 99,999 | 1 | 99,999 | 1 |
| 6 | 9,99,999 | 2 | 999,999 | 1 |
| 7 | 99,99,999 | 2 | 9,999,999 | 2 |
| 8 | 9,99,99,999 | 3 | 99,999,999 | 2 |
| 9 | 99,99,99,999 | 3 | 999,999,999 | 2 |
Worked example
0 / 4 steps shownA rule for the number of commas
Find a rule for how many commas an n-digit number needs in each system (for n of 4 or more), and test it on a 12-digit number.
Try it
Predict first
Chapter 05
Playing with digits
Predict first
Predict first
Here is a famous investigation described in the 1950s by the Indian mathematician D. R. Kaprekar, a school teacher in Devlali, Maharashtra (it is often dated to a talk he gave in 1949 and was published in 1955). Take any 4-digit number whose digits are not all the same. Make the greatest and the smallest numbers from its digits (keep any zeros, so the smallest may start with 0), and subtract. Repeat with the answer.
| Step | Greatest | Smallest | Difference |
|---|---|---|---|
| 1 | 5,432 | 2345 | 3,087 |
| 2 | 8,730 | 0378 | 8,352 |
| 3 | 8,532 | 2358 | 6,174 |
Try it
Chapter 06
Which numbers round to the same value?
Predict first
| Rounded value | Rounded to nearest | Smallest | Largest |
|---|---|---|---|
| 70 | 10 | 65 | 74 |
| 500 | 100 | 450 | 549 |
| 5,000 | 1,000 | 4,500 | 5,499 |
| 40,000 | 10,000 | 35,000 | 44,999 |
| 3,00,000 | 1,00,000 | 2,50,000 | 3,49,999 |
Lab
Sort numbers by whether they round to 5,000, to discover the band of numbers from 4,500 to 5,499.
Rounded to the nearest thousand, does each number become 5,000?
10 cards, 2 bins, 60 seconds. Tap a card, then tap its bin. You can also drag, or press a bin’s number key.
Text version of this activity
Ten number cards, 60 seconds. Rounds to 5,000: 4,500, 5,499, 4,950, 5,050, 5,001 and 4,501. Does not: 4,499 (rounds to 4,000), 5,500 (rounds to 6,000), 4,099 (4,000) and 6,000 (already 6,000).
The deciding digit when rounding to thousands is the hundreds digit. The band that rounds to 5,000 runs from 4,500 (the lowest halfway point, which rounds up) to 5,499 (the highest number still below the next halfway point, 5,500).
Predict first
Try it
Try it
Lab
Race the clock: round numbers up to 99,999 to the nearest 10, 100, 1,000 or 10,000 in 15 seconds each.
A number appears on a number line. Race to pick what it rounds to! Rounding to the nearest 10, 100, 1,000, 10,000, 12 rounds, 15 seconds each.
Halfway? It rounds up.
Text version of this activity
Twelve rounds with 15 seconds each. A number between 100 and 99,999 appears on a number line with the two nearest multiples of 10, 100, 1,000 or 10,000 marked. Tap the one it rounds to. Correct answers build a streak; the clock rewards quick, confident rounding.
Strategy: find the deciding digit (one place to the right of the rounding place) and ignore everything after it. For 67,452 to the nearest 10,000 the deciding digit is 7, so the answer is 70,000; to the nearest 1,000 the deciding digit is 4, so 67,000; to the nearest 100 it is 5, so 67,500; to the nearest 10 it is 2, so 67,450.
Chapter 07
How good is an estimate?
Predict first
| Product | Estimate | Exact | Error (as % of exact) |
|---|---|---|---|
| 438 × 267 | 400 × 300 = 1,20,000 | 1,16,946 | 3% |
| 4,812 × 3,276 | 5,000 × 3,000 = 1,50,00,000 | 1,57,64,112 | 5% |
| 149 × 149 | 100 × 100 = 10,000 | 22,201 | 55% |
| 68 × 72 | 70 × 70 = 4,900 | 4,896 | 0.1% |
| 912 × 48 | 900 × 50 = 45,000 | 43,776 | 3% |
Look at the error column. The estimate is excellent when the rounding errors cancel (one number rounded up, the other down, like 68 × 72 → 70 × 70) and poor when both are pushed the same way by a lot (149 × 149 → 100 × 100). A good estimator watches which way each number was rounded, and adjusts: both rounded down, so the real answer is bigger than my estimate.
| Round each number to the nearest | Estimated sum | Error |
|---|---|---|
| 10 | 17,980 | 3 |
| 100 | 18,000 | 23 |
| 1,000 | 17,000 | 977 |
Lab
Estimate, then calculate, with bigger numbers and a three-minute clock; see how close your estimates get.
12 questions on addition, subtraction, multiplication with some word problems mixed in, against a 180-second clock. Estimate first, then work it out exactly.
Get three in a row and the numbers level up!
Text version of this activity
Twelve questions in three minutes. Each shows a sum, difference or product with a first number up to 99,999 and a second up to 9,999. You give an estimate first and then the exact answer; the game shows the gap.
Some rounds are word problems that ask only for the exact answer; estimate them in your head first: 48,750 + 52,380 tickets (estimate 49,000 + 52,000 = 1,01,000; exact 1,01,130); 1,25,000 − 87,600 litres (estimate 1,25,000 − 88,000 = 37,000; exact 37,400); 365 × ₹48 notebooks (estimate 400 × 50 = ₹20,000, which is too high because both numbers were rounded up; exact ₹17,520).
Try to notice when both numbers were rounded in the same direction and say whether your estimate is too high or too low.
Try it
Chapter 08
How big is a crore, really?
Predict first
Log scale. Each step is ten times the one below.
- 1 thousand seconds≈ 17 minutes
- 1 lakh seconds≈ 28 hours
- 10 lakh = 1 million seconds≈ 11.6 days
- 1 crore seconds≈ 116 days
- 10 crore seconds≈ 3.2 years
- 100 crore = 1 billion seconds≈ 31.7 years
Chapter 09
Investigating Roman numerals
Predict first
Predict first
| Digit | Ones | Tens | Hundreds | Symbols |
|---|---|---|---|---|
| 1 | I | X | C | 1 |
| 2 | II | XX | CC | 2 |
| 3 | III | XXX | CCC | 3 |
| 4 | IV | XL | CD | 2 |
| 5 | V | L | D | 1 |
| 6 | VI | LX | DC | 2 |
| 7 | VII | LXX | DCC | 3 |
| 8 | VIII | LXXX | DCCC | 4 |
| 9 | IX | XC | CM | 2 |
The table reveals a secret: Roman numerals quietly use place value after all. Every digit from 1 to 9 has a fixed pattern (I, II, III, IV, V, VI, VII, VIII, IX), and the tens and hundreds just repeat that pattern with different letters (X-L-C and C-D-M instead of I-V-X). What Romans lacked was a single set of symbols reused in every place, and a zero to hold an empty place.
Lab
Connect Roman numerals with the numbers they stand for, splitting each numeral into place-value chunks.
Match each Roman numeral with its value.
7 pairs are hiding in two mixed-up columns. Pick one from each side to join them.
Text version of this activity
Seven Roman numerals and seven numbers to connect. The pairs are: XIV = 14; XLIX = 49; XCIV = 94; CDXLIV = 444; MDCLXVI = 1,666; MMXXVI = 2,026; MMMCMXCIX = 3,999. Strategy: split each numeral into thousands, hundreds, tens and ones chunks. For example CDXLIV splits as CD | XL | IV = 400 + 40 + 4 = 444, and MMMCMXCIX splits as MMM | CM | XC | IX = 3,000 + 900 + 90 + 9 = 3,999, the largest number you can write with the standard symbols.
Worked example
0 / 4 steps shownOrdering Roman numerals
Arrange XCIX, CI, LXXXIX, XC and CX from smallest to largest.
Try it
Try it
Chapter 10
Investigating real big numbers
Real large numbers are messy: they come rounded, in different systems, and sometimes with mistakes. A good investigator checks them. Here is one way to test a claim you see in the news or in a quiz: convert it to plain digits, check the size with a rough estimate, and ask whether the answer is sensible.
Worked example
0 / 4 steps shownChecking a claim
A social media post says: "India has 140 crore people, that is 14 billion!" Is it right?
Worked example
0 / 4 steps shownA crore of steps?
A child walks about 8,000 steps a day. Roughly how many days would it take to walk one crore steps?
Chapter 11
Estimation detectives: real Indian contexts
In real life nobody hands you a neat sum. You meet a situation, decide which numbers matter, round them sensibly, and check whether the answer is believable. This chapter works through four such situations. In each, predict first, then estimate, then compare with the exact answer.
Predict first
Worked example
0 / 4 steps shownHow much rain falls on a roof?
Mumbai receives very roughly 2,200 mm of rain in a year, most of it in the monsoon. How many litres fall on a flat roof of 100 square metres?
Worked example
0 / 4 steps shownSeats on a long-distance train
A train has 22 coaches. Most are sleeper coaches with 72 berths; suppose all 22 are. It runs every day of the year. Estimate the number of berths it offers in a year, then calculate exactly.
Try it
Try it
Reflect
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Words to know
All maths vocabulary →Investigation words
- Prediction
- A definite guess made before testing, so the test can show whether you were right.
- Always / sometimes / never true
- Ways to classify a claim. One counterexample is enough to show a claim is not always true.
- Counterexample
- An example that shows a claim is false.
- Example: 999 shows that "the successor has the same number of digits" is not always true.
- Kaprekar’s constant
- 6,174: the number every 4-digit number (digits not all equal) reaches by repeatedly subtracting its smallest arrangement from its largest.
- Example: 7,641 − 1,467 = 6,174
- Double rounding
- Rounding a number in two stages (for example to tens, then to hundreds), which can give a different answer from rounding once.
- Example: 2,449 → 2,450 → 2,500, but 2,449 → 2,400 directly
- Rounding band
- The set of all numbers that round to the same value.
- Example: 450 to 549 round to 500
- Estimation error
- The difference between an estimate and the exact value, sometimes given as a percentage of the exact value.
- Arrangement (permutation)
- One way of ordering a set of digits or objects.
- Example: 258 and 852 are two arrangements of 2, 5, 8
Quick check
Investigate check
10 questions · answer what you can, then check. Getting one wrong is useful.
Related to
Number and shape patternsCounting digits, Kaprekar’s routine and the multiples of 9 from reversed numbers are number patterns you can predict and explain.
Related to
Prime and composite numbersDifferences like 9 × (a − b) are always multiples of 9, a first taste of reasoning about factors.
Used in
Data handlingChecking whether real data makes sense, and rounding it sensibly, is the first step of any data investigation.
Keep this
What we found
- Multiplying by 10 shifts every digit one place left, so each place value becomes ten times as big.
- Reversing a 2-digit number changes it by 9 × (difference of the digits).
- More digits means bigger for whole numbers written normally, because 10ⁿ is one more than the largest n-digit number.
- The successor gains a digit only when the number is all 9s.
- Digits to write 1 to 100: 192. There are 9 × 10ⁿ⁻¹ numbers with n digits.
- The systems first differ at one lakh: from 1,000 to 99,999 the commas are identical.
- Kaprekar’s constant: 4-digit numbers (digits not all equal) reach 6,174 in at most 7 steps.
- Rounding bands: 450–549 round to 500 (nearest 100). Round once, from the original number: double rounding changes 5% of results.
- Estimates of products can be far off when both numbers are rounded the same way; note the direction of rounding.
- A crore seconds ≈ 116 days; a billion seconds ≈ 32 years.
- Real estimates: round gently when the answer is close to a limit, coarsely when only the size matters; judge an estimate by whether it answers your question.
- Roman numerals reuse a digit pattern in each place but have no zero; 88 = LXXXVIII is the longest up to 100.
Where this comes from
Sources
Ganita Prakash, Mathematics textbook for Grade 6 — Chapter 3: Number Play (opens another website) — NCERTawaiting owner check
Supports the Class 6 treatment of numbers: comparing and ordering numbers (supercells, smallest and largest), forming numbers from given digits, digit sums, digit patterns and palindromes, and estimating with numbers of a few digits.
Ganita Prakash, Mathematics textbook for Grade 7 — Chapter 1: Large Numbers Around Us (opens another website) — NCERTawaiting owner check
Supports lakh, crore and arab; the Sanskrit origin of the words lakh and crore; reading large numbers in the Indian (3-2-2) and American/International systems; rounding to the nearest thousand, lakh or crore; and estimating with real-world large numbers.
Place value (Arithmetic) (opens another website) — Khan Academyawaiting check
Supports place value, expanded form, comparing numbers by place value and rounding whole numbers.
Roman Numerals (opens another website) — Math is Funawaiting owner check
Supports the seven Roman symbols, the rule not to repeat a symbol more than three times, the subtraction pairs (IV, IX, XL, XC, CD, CM) and the bar over a symbol to multiply it by 1,000.
Indian numbering system (opens another website) — Wikipediaawaiting owner check
Supports lakh (10⁵), crore (10⁷), arab (10⁹) and kharab (10¹¹), the 3-2-2 comma grouping, lakh crore = 10¹², and the equivalences with the short-scale thousand, million and billion.
Place Value (opens another website) — Math is Funawaiting owner check
Supports place value, the rule that each column on the left is ten times the column on its right, expanded form (143 = 1 × 100 + 4 × 10 + 3 × 1), and the place names from ones up to millions.
End of Investigate
What you just read
- Predict and test how moving a digit one place changes its value, and explain it with place value.
- Investigate patterns in counting numbers: how many n-digit numbers there are and how many digits it takes to write them.
- Find every number that rounds to a given value, and show why rounding twice can give a different answer.
- Compare estimates with exact answers and decide which rounding gives a good enough estimate.
- Test claims about forming numbers and Roman numerals, deciding whether they are always, sometimes or never true.
- Next depthGo deeper: Go deeperMechanisms, reasoning, calculations and nuance.
- Practise83 questionsHints and a worked solution for every question — or play a 10-question round.
- Step backUnderstandGo back over the ground before this one — you can move up and down as often as you like.
- TopicAll of number systemThe whole ladder, the connections and the words to know, on one page.
The web
Explore a connection
Helps you understand
Four operationsPlace value is what makes column addition, carrying and long division work.
Helps you understand
Data handlingReading, comparing and rounding numbers comes first when you sort data and round a mean.
Related to
Number and shape patternsPlace-value charts are full of patterns: each place is ten times the one to its right.
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Revision 1 · release preview-7e1cbbcc4f · accepted 20/09/2026