Number systemExtendabout 55 min
Beyond a billion, and beyond base ten
Arab, kharab and trillion; ISRO distances; binary and other bases; puzzles and projects
Stretch the number system in every direction: bigger names in both systems, real Indian large numbers from elections to Mars, number systems of the Babylonians, Maya and Egyptians, binary as a place-value system, olympiad-style puzzles, Fermi estimates, projects and open questions.
In this part you’ll
- Use arab, kharab, trillion and powers of ten to read and convert very large numbers.
- Work with real large numbers from Indian life and space missions, converting and estimating sensibly.
- Explain how other cultures wrote numbers, and read and write numbers in binary and base five.
- Solve multi-step, olympiad-style puzzles about digits, place value and counting.
- Plan and carry out a large-number investigation or Fermi estimate of your own.
You now know the number system well enough to break out of it. This layer goes bigger (names beyond a billion), wider (real numbers from elections, railways and space), sideways (number systems with bases other than ten, including the binary inside every phone) and harder (olympiad-style puzzles). It ends with projects to do and questions nobody has fully answered.
Pick the chapters that interest you. They do not have to be read in order.
Chapter 01
Names beyond a billion
The Indian system continues after crore with a new name every two places: arab (100 crore) and kharab (100 arab). Traditional lists continue with neel, padma and shankh, though these are rarely used today, and people more often say "lakh crore". The International system continues every three places: trillion, quadrillion, quintillion…
| Power of ten | Digits | Indian | International |
|---|---|---|---|
| 10⁹ | 1,00,00,00,000 | 1 arab = 100 crore | 1 billion |
| 10¹⁰ | 10,00,00,00,000 | 10 arab | 10 billion |
| 10¹¹ | 1,00,00,00,00,000 | 1 kharab = 100 arab | 100 billion |
| 10¹² | 10,00,00,00,00,000 | 10 kharab = 1 lakh crore | 1 trillion |
| 10¹³ | 1,00,00,00,00,00,000 | 1 neel (traditional) = 10 lakh crore | 10 trillion |
| 10¹⁵ | 1,00,00,00,00,00,00,000 | 1 padma (traditional) = 10 crore crore | 1 quadrillion |
| 10¹⁷ | 1,00,00,00,00,00,00,00,000 | 1 shankh (traditional) | 100 quadrillion |
Predict first
Lab
Build space and population numbers in the International chart, then read them in both systems.
Millions
Thousands
Ones
Tip: tap a digit, then use ↑ ↓ or type 0–9. Ten in one column swaps for one in the next.
International name: one hundred forty million
Indian name: fourteen crore
Expanded form
100,000,000 + 40,000,000
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
A nine-column International chart (hundred millions to ones). It starts at 140,000,000: one hundred forty million, or fourteen crore. The largest number the chart can hold is 999,999,999; its successor, one billion (100 crore, one arab), would need a tenth column, a reminder that every chart has a limit but numbers do not.
Challenges: 384,400 (the Moon's average distance in km: three lakh eighty-four thousand four hundred); 149,600,000 (the Sun's distance in km: one hundred forty-nine million six hundred thousand, or fourteen crore ninety-six lakh); 35,786,000 (the geostationary height in metres: three crore fifty-seven lakh eighty-six thousand); 999,999,999 (the largest nine-digit number); 606,060,606 (six hundred six million sixty thousand six hundred six; sixty crore sixty lakh sixty thousand six hundred six); and 100,000,001 (ten crore one).
Predict first
Try it
Chapter 02
India in large numbers
Real data is where the number system earns its keep. The figures below are approximate and rounded; exact values change every year and depend on the source. Treat them as a starting point for your own checking.
| Quantity | Indian system | International system |
|---|---|---|
| Population counted in the 2011 census | 1,21,08,54,977 (about 121 crore) | 1,210,854,977 (about 1.21 billion) |
| Population today (UN estimate, mid-2020s) | about 140–146 crore | about 1.4–1.46 billion |
| Registered voters, 2024 Lok Sabha election | about 97 crore | about 970 million |
| UPI payments in a month (late 2024) | more than 1,500 crore | more than 15 billion |
| Seats at the Narendra Modi Stadium | about 1,32,000 | about 132,000 |
| Length of Indian Railways track (all tracks) | about 1,35,000 km (2024) | about 135,000 km (2024) |
Worked example
0 / 4 steps shownHow many voted?
Suppose about 97 crore people are registered to vote and about 66% of them vote. Roughly how many people vote? Give the answer in crore and in millions.
Try it
Worked example
0 / 4 steps shownUPI payments per second
Suppose there are about 1,500 crore UPI payments in a 30-day month. About how many payments is that per second?
Chapter 03
Space numbers: ISRO and beyond
Log scale: each step up is ten times farther.
- Kármán line (edge of space)100 km
- International Space Station≈ 400 km
- Geostationary satellites≈ 35,786 km
- The Moon (Chandrayaan-3)≈ 3,84,400 km
- Aditya-L1 at the L1 point≈ 15 lakh km
- The Sun≈ 15 crore km
- Mars at its closest (approx.)≈ 5.5 crore km
- One light-year≈ 9.46 lakh crore km
Worked example
0 / 4 steps shownMangalyaan’s journey
India's Mars Orbiter Mission (Mangalyaan) was launched on 5 November 2013 and entered orbit around Mars on 24 September 2014, after a long curved journey of roughly 66 crore km (published figures for the distance flown differ, from about 65 crore to about 78 crore km). About how many kilometres did it travel per day?
Try it
Try it
Chapter 04
How other cultures wrote numbers
Explore
Number systems around the world
Pick a system to see how it worked and what it lacked.
- Stroke = 1
- Heel bone = 10
- Coil of rope = 100
- Lotus = 1,000
- Add the symbols
No place value
From about 3000 BCE, Egyptian hieroglyphs used a separate picture for each power of ten up to a million (a kneeling god with raised arms). To write 2,345 you drew 2 lotus flowers, 3 coils, 4 heel bones and 5 strokes, in any order. Simple to read, but long and slow to calculate with, and there was no zero.
Chapter 05
Binary: place value with two digits
In binary (base 2) there are only two digits, 0 and 1, and each place is worth twice the place to its right: 1, 2, 4, 8, 16, 32, 64, 128… A 1 means "this place is used" and a 0 means "not used".
So binary 101101 means 32 + 8 + 4 + 1 = 45. Computers use binary because a circuit can easily be in one of two states, on or off. Every photo, song and message on a phone is stored as long strings of 0s and 1s.
| Number | Binary | Places used |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 10 | 2 |
| 3 | 11 | 2 + 1 |
| 4 | 100 | 4 |
| 5 | 101 | 4 + 1 |
| 7 | 111 | 4 + 2 + 1 |
| 8 | 1000 | 8 |
| 15 | 1111 | 8 + 4 + 2 + 1 |
| 16 | 10000 | 16 |
| 31 | 11111 | 16 + 8 + 4 + 2 + 1 |
| 45 | 101101 | 32 + 8 + 4 + 1 |
Worked example
0 / 5 steps shownWriting 2026 in binary
Write 2,026 in binary.
Lab
Play a memory game matching binary numbers with their base-ten values by adding place values 1, 2, 4, 8, 16, 32.
Match each binary number with its value in base ten.
14 face-down cards hide 7 pairs. Flip two at a time and remember where things are!
Text version of this activity
Seven pairs are hidden under fourteen cards. The pairs are: 110 (binary) = 6; 1001 (binary) = 9; 1100 (binary) = 12; 10011 (binary) = 19; 11001 (binary) = 25; 100000 (binary) = 32; 110010 (binary) = 50. To read a binary number, write the place values 32, 16, 8, 4, 2, 1 under its digits (right-aligned) and add the places with a 1. For example 10011 = 16 + 2 + 1 = 19, and 110010 = 32 + 16 + 2 = 50.
Predict first
| Name | Size | As a power | Roughly |
|---|---|---|---|
| bit | 1 binary digit | 2⁰ digit | 0 or 1 |
| byte | 8 bits | 2⁸ = 256 patterns | one letter of text |
| kilobyte (KB) | 1,000 bytes (older usage: 1,024 = 2¹⁰) | 10³ or 2¹⁰ | a short paragraph |
| megabyte (MB) | 10,00,000 bytes | 10⁶ | a photo or a minute of music |
| gigabyte (GB) | 100 crore bytes | 10⁹ | a film in ordinary quality |
| terabyte (TB) | 1 lakh crore bytes | 10¹² | a large hard disk |
Try it
Chapter 06
Other bases: five, twelve and sixty
Worked example
0 / 4 steps shownCounting on one hand: base five
In base five the places are 1, 5, 25, 125, 625. Write 2,026 in base five.
Try it
Worked example
0 / 5 steps shownFrom base five back to base ten, and a check
A child counting on one hand writes 4,301 in base five. What number is that in base ten?
Try it
Base twelve survives in dozens (12) and grosses (144 = 12 × 12): eggs, bananas and pencils are often sold by the dozen. Twelve is handy because it divides evenly by 2, 3, 4 and 6, while ten divides only by 2 and 5.
Base sixty survives in time and angles. 2 hours 15 minutes 30 seconds is a three-place base-sixty number: 2 × 3,600 + 15 × 60 + 30 = 8,130 seconds. Converting between hours, minutes and seconds is exactly place-value conversion in base 60.
Lab
Sort everyday examples by the base (2, 10, 12, 20 or 60) that they come from.
Which base is hiding in each everyday thing?
12 cards, 5 bins. Tap a card, then tap its bin. You can also drag, or press a bin’s number key.
Text version of this activity
Twelve cards and five bins. Base 2: computer memory switches, barcode stripes, 1,024 bytes in a kilobyte (older usage). Base 10: phone keypad digits, 100 paise in a rupee, metric units. Base 12: dozens and grosses, months in a year. Base 20: Maya calendar, French quatre-vingts for 80. Base 60: seconds and minutes, 360 degrees in a circle. Traces of old bases survive in the way we measure time, angles, and goods.
Chapter 07
Olympiad-style puzzles
Worked example
0 / 4 steps shownHow many 1s?
How many times is the digit 1 written when you write every number from 1 to 1,000?
Worked example
0 / 4 steps shownSmallest number with digit sum 30
What is the smallest whole number whose digits add up to 30?
Worked example
0 / 5 steps shownCounting palindromes
A palindrome reads the same forwards and backwards, like 4,774. How many palindromes are there from 1 to 99,999?
Try it
Try it
Worked example
0 / 5 steps shownThe 1,000th digit
Write the counting numbers in a row: 123456789101112131415… What is the 1,000th digit?
Try it
Chapter 08
Fermi estimates: huge numbers from small facts
The physicist Enrico Fermi liked to ask questions like "How many piano tuners are there in Chicago?" and answer them with nothing but sensible guesses and multiplication. Such Fermi estimates are about getting the right number of digits, not the exact answer. Round every guess to one or two digits, multiply, and ask whether the result is sensible.
Worked example
0 / 5 steps shownHow many rotis does India eat in a day?
Make a Fermi estimate of the number of rotis (chapatis) eaten in India in one day.
Worked example
0 / 5 steps shownHow many cricket balls fill a classroom?
Make a Fermi estimate of how many cricket balls would fill a classroom 8 m long, 6 m wide and 3 m high.
Lab
Estimate, then calculate products and quotients with large numbers, including real-life word problems.
10 questions on multiplication, division with some word problems mixed in. Estimate first, then work it out exactly.
Get three in a row and the numbers level up!
Text version of this activity
Ten rounds of multiplication and division with a first number from 1,000 to 99,999 and a second from 2 to 99; divisions always come out exactly. For generated questions you estimate first, then give the exact answer. Word problems ask only for the exact answer, so estimate in your head before calculating.
Word problems: 1,248 passengers × 365 trips (about 1,000 × 400 = 4,00,000; exact 4,55,520); 1,32,000 seats ÷ 24 blocks (about 1,20,000 ÷ 24 = 5,000; exact 5,500); ₹2,52,000 ÷ 840 students (about 2,40,000 ÷ 800 = 300; exact ₹300); 7,500 bicycles × 12 months (exact 90,000).
Lab
Round crore-sized numbers to the nearest lakh, the way newspapers report budgets and populations.
A number appears on a number line. Race to pick what it rounds to! Rounding to the nearest 1,00,000, 8 rounds, 20 seconds each.
Halfway? It rounds up.
Text version of this activity
Eight rounds, 20 seconds each. A number between 10,00,000 and 99,99,99,999 appears; round it to the nearest lakh by looking at the ten-thousands digit. For example 23,47,86,512 lies between 23,47,00,000 and 23,48,00,000; the ten-thousands digit is 8, so it rounds up to 23,48,00,000, which a newspaper might print as "23.48 crore". A number such as 5,99,53,000 rounds up to 6,00,00,000 because the carry ripples through the 9s.
Chapter 09
Projects and people who use big numbers
Projects to try
- Step 01Newspaper hunt1 week
Collect 20 large numbers from newspapers or news sites. Rewrite each in both systems and in words, and sort them by size.
- Step 02Build a lakhclass project
Collect bottle caps or draw dots on grid paper: 100 per sheet, so 1,000 sheets make a lakh. How far does a class get in a month?
- Step 03Family timeline in seconds1 evening
Work out how many seconds old each family member is. Who is over a billion seconds (about 31.7 years)?
- Step 04Binary birthday30 minutes
Write your birth date and age in binary and base five. Make a binary bracelet with two colours of beads.
- Step 05Fermi challengegroup
Estimate: how many cricket balls would fill your classroom? How many litres of tea does your town drink a day? Compare answers between groups.
- Step 06Roman year walkouting
Find Roman numerals on buildings, clocks and foundation stones. Convert each and note what it labels.
| Career | Big numbers they use |
|---|---|
| Census and survey statistician | Counts of crores of people, households and farms; rounding and estimating from samples. |
| Space scientist or engineer (ISRO) | Distances in lakhs and crores of km, speeds, fuel masses, orbits. |
| Banker, accountant or economist | Budgets in lakh crore, interest on loans, national income. |
| Data scientist or software engineer | Billions of records, binary storage, gigabytes and terabytes. |
| Railway or airline planner | Lakhs of passengers a day, timetables, tickets and seats. |
| Election official | Tens of crores of voters, lakhs of polling stations, counting and cross-checking totals. |
Chapter 10
Open questions
Reflect
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Chapter 11
Challenge practice set
A final set for the strongest learners: every item needs more than one idea from the topic. Work on paper, use the hints only when stuck, and check each answer with a second method (an estimate, a count done another way, or a small case you can list completely).
Try it
Try it
Worked example
0 / 5 steps shownReverse and add until a palindrome
Start with 87. Add it to its reverse, and repeat with the answer until you reach a palindrome.
Worked example
0 / 4 steps shownA two-digit puzzle
A 2-digit number AB added to its reverse BA gives 121. How many such numbers are there?
Try it
Worked example
0 / 4 steps shownDriving to the Moon
If a car could drive to the Moon at a steady 60 km per hour without stopping, how long would the 3,84,400 km trip take?
Worked example
0 / 4 steps shownEstimating population growth
The 2011 census counted about 121 crore people, and UN estimates put India at roughly 146 crore in 2025. Roughly how many people were added per year, and per day?
Try it
Words to know
All maths vocabulary →Extend vocabulary
- Arab
- Indian name for 100 crore, 10⁹, equal to one billion.
- Kharab
- Indian name for 100 arab, 10¹¹, equal to 100 billion.
- Trillion
- In the short scale used today, a million million, 10¹². Equal to 1 lakh crore.
- Short scale / long scale
- Two meanings of billion: 10⁹ (short scale, modern English) or 10¹² (long scale, older British and many European languages).
- Googol
- The number 10¹⁰⁰, a 1 followed by 100 zeros.
- Base
- The number of digits a place-value system uses, and how many times bigger each place is than the next.
- Example: Base 2, base 10, base 60
- Binary
- Base 2: place values 1, 2, 4, 8, 16…, digits 0 and 1.
- Example: 101 = 5
- Bit
- A single binary digit, 0 or 1.
- Palindrome
- A number that reads the same forwards and backwards.
- Example: 12,321
- Digit sum
- The sum of a number’s digits.
- Example: The digit sum of 3,999 is 30.
- Fermi estimate
- A rough estimate of a large quantity built from simple, sensible guesses multiplied together.
- Light-year
- The distance light travels in a year, about 9.46 lakh crore km (9.46 trillion km).
Quick check
Extend check
10 questions · answer what you can, then check. Getting one wrong is useful.
Related to
Number and shape patternsBinary counting, palindromes and Kaprekar-style routines are rich sources of number patterns.
Used in
Data handlingCensus data, election results and budgets are large-number data sets that data handling organises and summarises.
Related to
Order of operationsFermi estimates chain several multiplications and divisions, where the order of operations matters.
Keep this
Cheat sheet
- Indian: crore 10⁷, arab 10⁹, kharab 10¹¹; International: billion 10⁹, trillion 10¹²; 1 lakh crore = 1 trillion.
- Long scale (older British) billion = 10¹²; today billion = 10⁹.
- Googol = 10¹⁰⁰.
- Real figures: always state what, when and who; round honestly.
- Space: Moon ≈ 3,84,400 km; Sun ≈ 15 crore km (150 million km); light-year ≈ 9.46 lakh crore km.
- Other systems: Egyptian (no place value), Babylonian base 60, Maya base 20 with a shell zero, Chinese rods base 10, Roman (no place value).
- Binary: digits 0 and 1; places 1, 2, 4, 8, 16…; 101101 = 45; 2,026 = 11111101010.
- Any base works: rollovers, expanded form and comparing follow the same rules with 10 replaced by the base.
- Puzzle tools: count by place; fewest digits for smallest; palindromes are decided by their first half.
- Fermi estimates get the right number of digits from sensible guesses.
Where this comes from
Sources
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.
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.
Binary number (opens another website) — Wikipediaawaiting owner check
Supports binary as a positional notation with radix 2, each bit standing for an increasing power of 2 (1, 2, 4, 8, 16…), used by almost all modern computers because it is simple to build with logic gates.
Hindu-Arabic numerals (opens another website) — Encyclopaedia Britannicaawaiting owner check
Supports the set of ten symbols of the decimal system, their origin in India in the 6th or 7th century, and their introduction to Europe about the 12th century through Middle Eastern mathematicians, especially al-Khwarizmi and al-Kindi.
SI prefixes (opens another website) — BIPM (International Bureau of Weights and Measures)awaiting owner check
Supports metric prefixes as powers of ten: deca 10¹, hecto 10², kilo 10³, mega 10⁶, giga 10⁹, and deci 10⁻¹, centi 10⁻² and milli 10⁻³.
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.
Mars Orbiter Mission (opens another website) — Wikipediaawaiting owner check
Supports the launch of Mangalyaan on 5 November 2013, Mars orbit insertion on 24 September 2014, and a figure of about 780 million km for the distance travelled on the transfer to Mars.
Narendra Modi Stadium (opens another website) — Wikipediaawaiting owner check
Supports the seating capacity of 132,000 at the Narendra Modi Stadium in Ahmedabad and its description as the largest cricket stadium in the world.
Indian Railways (opens another website) — Wikipediaawaiting owner check
Supports the size of Indian Railways as at 31 March 2024: total track length 135,207 km (route length 69,181 km) and 6.905 billion passengers carried in 2023–24, which is close to two crore a day.
End of Extend
What you just read
- Use arab, kharab, trillion and powers of ten to read and convert very large numbers.
- Work with real large numbers from Indian life and space missions, converting and estimating sensibly.
- Explain how other cultures wrote numbers, and read and write numbers in binary and base five.
- Solve multi-step, olympiad-style puzzles about digits, place value and counting.
- Plan and carry out a large-number investigation or Fermi estimate of your own.
- Practise83 questionsHints and a worked solution for every question — or play a 10-question round.
- Step backGo deeperGo 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