[{"data":1,"prerenderedAt":1275},["ShallowReactive",2],{"layer:number-system:extend":3},{"layer":4,"contentHash":1252,"dependencyHashes":1253,"approval":1269,"releaseId":1274},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1247,"reviewStatus":1248,"authoring":1249},1,"number-system","en","extend","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.",[13,14,15,16,17],"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.",55,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 55 minutes",{"label":29,"value":30},"Prior knowledge","Powers of ten (Go deeper)",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Place value, binary, bases, sprint, rounding",{"label":38,"value":39},"Stretch","Olympiad puzzles and open questions",[41,45,51,54,99,105,110,125,144,155,169,174,177,209,219,230,239,244,249,286,291,300,311,323,328,402,407,412,415,462,472,477,481,511,523,560,570,575,584,595,605,626,629,696,701,710,719,729,740,760,770,782,787,792,795,805,815,849,861,866,895,920,925,929,933,938,941,945,954,965,975,984,996,1005,1014,1034,1077,1200,1206,1211,1215,1219,1234],{"id":42,"type":43,"markdown":44},"intro-e","prose","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.\n\nPick the chapters that interest you. They do not have to be read in order.",{"id":46,"type":47,"title":48,"eyebrow":49,"navLabel":50},"ch01","chapter","Names beyond a billion","Chapter 01","1 Bigger names",{"id":52,"type":43,"markdown":53},"bigger","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**…",{"id":55,"type":56,"caption":57,"columns":58,"rows":63},"table-bigger","table","Big names in both systems",[59,60,61,62],"Power of ten","Digits","Indian","International",[64,69,74,79,84,89,94],[65,66,67,68],"10⁹","1,00,00,00,000","1 arab = 100 crore","1 billion",[70,71,72,73],"10¹⁰","10,00,00,00,000","10 arab","10 billion",[75,76,77,78],"10¹¹","1,00,00,00,00,000","1 kharab = 100 arab","100 billion",[80,81,82,83],"10¹²","10,00,00,00,00,000","10 kharab = 1 lakh crore","1 trillion",[85,86,87,88],"10¹³","1,00,00,00,00,00,000","1 neel (traditional) = 10 lakh crore","10 trillion",[90,91,92,93],"10¹⁵","1,00,00,00,00,00,00,000","1 padma (traditional) = 10 crore crore","1 quadrillion",[95,96,97,98],"10¹⁷","1,00,00,00,00,00,00,00,000","1 shankh (traditional)","100 quadrillion",{"id":100,"type":101,"variant":102,"title":103,"markdown":104},"nuance-long-scale","callout","nuance","A billion was not always a billion","Until 1974, official British usage said a **billion** was a million million (10¹²), the so-called long scale, and many European languages still use it: in French, *un milliard* is 10⁹ and *un billion* is 10¹². English today uses the short scale (billion = 10⁹) almost everywhere. When reading older books or translating, check which is meant. Indian names do not have this problem: a crore is always 10⁷.",{"id":106,"type":101,"variant":107,"title":108,"markdown":109},"ex-googol","example","Googol","A **googol** is 10¹⁰⁰: a 1 followed by a hundred zeros. The name was made up by nine-year-old Milton Sirotta, nephew of the American mathematician Edward Kasner, who made it famous in a 1940 book. A googol is far larger than the number of atoms in the observable universe, usually estimated at around 10⁸⁰. The company Google took its name from a misspelling of it.",{"id":111,"type":112,"prompt":113,"options":114,"explanation":124},"pred-kharab","prediction","Which is bigger: **1 kharab** or **50 billion**?",[115,118,121],{"id":116,"label":117},"a","1 kharab",{"id":119,"label":120},"b","50 billion",{"id":122,"label":123},"c","They are equal","**a: 1 kharab.** 1 kharab = 10¹¹ = 100 billion, twice as much as 50 billion. Convert through powers of ten whenever the names come from different systems: 50 billion = 5 × 10¹⁰, and 10¹¹ = 10 × 10¹⁰.",{"id":126,"type":127,"component":128,"componentVersion":5,"config":129,"objective":142,"textAlternative":143},"lab-pv-extend","interactive","place-value",{"places":130,"system":131,"initial":132,"show":133,"challenges":135},9,"international",140000000,{"names":134,"expanded":134,"neighbours":134,"bothSystems":134},true,[136,137,138,139,140,141],384400,149600000,35786000,999999999,606060606,100000001,"Build space and population numbers in the International chart, then read them in both systems.","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.\n\nChallenges: 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).",{"id":145,"type":112,"prompt":146,"options":147,"explanation":154},"pred-googol","Which is bigger: a **googol** (10¹⁰⁰) or the number of **atoms in the observable universe** (roughly 10⁸⁰)?",[148,150,152],{"id":116,"label":149},"The atoms, by a lot",{"id":119,"label":151},"About the same",{"id":122,"label":153},"A googol, by a factor of about 10²⁰","**c.** 10¹⁰⁰ ÷ 10⁸⁰ = 10²⁰, a hundred billion billion. If every atom in the universe held a whole universe's worth of atoms, you would have about 10¹⁶⁰ atoms, far past a googol; but one universe falls short by a factor of 10²⁰. Numbers written with a few symbols can be far bigger than anything that physically exists.",{"id":156,"type":157,"itemId":158,"prompt":159,"check":160,"hints":164,"feedback":166},"pr-kharab","practice","number-system.extend-kharab","How many crore make one kharab?",{"kind":161,"answer":162,"tolerance":163},"number",10000,0,[165],"Write both as powers of ten.",{"correct":167,"incorrect":168},"Right: 1 kharab = 10¹¹ and 1 crore = 10⁷, so 10¹¹ ÷ 10⁷ = 10⁴ = 10,000 crore.","Kharab = 10¹¹, crore = 10⁷. Divide: 10¹¹⁻⁷ = 10⁴ = 10,000.",{"id":170,"type":47,"title":171,"eyebrow":172,"navLabel":173},"ch02","India in large numbers","Chapter 02","2 India in numbers",{"id":175,"type":43,"markdown":176},"india-nums","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.",{"id":178,"type":56,"caption":179,"columns":180,"rows":184},"table-india","Some large numbers from Indian life (approximate)",[181,182,183],"Quantity","Indian system","International system",[185,189,193,197,201,205],[186,187,188],"Population counted in the 2011 census","1,21,08,54,977 (about 121 crore)","1,210,854,977 (about 1.21 billion)",[190,191,192],"Population today (UN estimate, mid-2020s)","about 140–146 crore","about 1.4–1.46 billion",[194,195,196],"Registered voters, 2024 Lok Sabha election","about 97 crore","about 970 million",[198,199,200],"UPI payments in a month (late 2024)","more than 1,500 crore","more than 15 billion",[202,203,204],"Seats at the Narendra Modi Stadium","about 1,32,000","about 132,000",[206,207,208],"Length of Indian Railways track (all tracks)","about 1,35,000 km (2024)","about 135,000 km (2024)",{"id":210,"type":211,"title":212,"problem":213,"steps":214},"we-voters","worked_example","How 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.",[215,216,217,218],"66% of 97 crore ≈ 2\u002F3 of 97 crore (because 66% is close to two-thirds).","97 ÷ 3 ≈ 32.3, and 2 × 32.3 ≈ 64.7, so about **64 crore** people.","In millions: 1 crore = 10 million, so 64 crore ≈ **640 million**.","For comparison, that is about twice the whole population of the USA, voting in one election.",{"id":220,"type":157,"itemId":221,"prompt":222,"check":223,"hints":225,"feedback":227},"pr-census-crore","number-system.extend-census-crore","The 2011 census counted 1,21,08,54,977 people. Round this to the nearest crore and give the number of crores.",{"kind":161,"answer":224,"tolerance":163},121,[226],"Read the crores period: 1,21.",{"correct":228,"incorrect":229},"Right: the deciding digit (ten lakhs) is 0, so it rounds down to 1,21,00,00,000, which is 121 crore.","Rounding to crores, look at the ten lakhs digit, which is 0. Round down to 121 crore (1.21 billion).",{"id":231,"type":211,"title":232,"problem":233,"steps":234},"we-upi","UPI 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?",[235,236,237,238],"Seconds in 30 days: 30 × 86,400 = 25,92,000, about 26 lakh.","Payments: 1,500 crore = 15,00,00,00,000 = 15 billion.","15,00,00,00,000 ÷ 25,92,000 ≈ 5,787.","So roughly **5,800 payments every second**, day and night. Working with powers: 1.5 × 10¹⁰ ÷ 2.6 × 10⁶ ≈ 0.58 × 10⁴ ≈ 5,800. ✓",{"id":240,"type":101,"variant":241,"title":242,"markdown":243},"careful-data","careful","Big numbers need dates and sources","A population figure without a year is almost useless: India adds well over a crore people every few years. When you quote a large number, say **what** it counts, **when**, and **who** measured it. The figures in this chapter are rounded and flagged as approximate for exactly this reason.",{"id":245,"type":47,"title":246,"eyebrow":247,"navLabel":248},"ch03","Space numbers: ISRO and beyond","Chapter 03","3 Space numbers",{"id":250,"type":251,"title":252,"note":253,"scale":254,"rungs":255},"ladder-space","ladder","Distances from Earth (approximate)","Log scale: each step up is ten times farther.","log",[256,260,264,268,271,275,278,282],{"label":257,"value":258,"display":259},"Kármán line (edge of space)",100,"100 km",{"label":261,"value":262,"display":263},"International Space Station",400,"≈ 400 km",{"label":265,"value":266,"display":267},"Geostationary satellites",35786,"≈ 35,786 km",{"label":269,"value":136,"display":270},"The Moon (Chandrayaan-3)","≈ 3,84,400 km",{"label":272,"value":273,"display":274},"Aditya-L1 at the L1 point",1500000,"≈ 15 lakh km",{"label":276,"value":137,"display":277},"The Sun","≈ 15 crore km",{"label":279,"value":280,"display":281},"Mars at its closest (approx.)",55000000,"≈ 5.5 crore km",{"label":283,"value":284,"display":285},"One light-year",9460000000000,"≈ 9.46 lakh crore km",{"id":287,"type":101,"variant":288,"title":289,"markdown":290},"obs-ladder-order","observation","Notice the order","The ladder is sorted by value, so Mars at its closest (about 5.5 crore km) sits below the Sun (about 15 crore km). Mars is sometimes much farther away than the Sun: on the far side of its orbit it can be about 40 crore km from Earth.",{"id":292,"type":211,"title":293,"problem":294,"steps":295},"we-mangalyaan","Mangalyaan’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?",[296,297,298,299],"Days from 5 November 2013 to 24 September 2014: 323.","Estimate: 66 crore km ÷ 320 days. Round to 64 crore ÷ 320 to make it easy.","64,00,00,000 ÷ 320 = 20,00,000.","So roughly **20 lakh km per day**, or about 2 million km a day. Its path was a long curve around the Sun, not a straight line to Mars, which is why the journey is much longer than the distance between the planets. Because the distance flown is quoted differently by different sources, treat the answer as an estimate of the right size, not an exact figure.",{"id":301,"type":157,"itemId":302,"prompt":303,"check":304,"hints":306,"feedback":308},"pr-moon-trips","number-system.extend-moon-trips","The Sun is about 15,00,00,000 km from Earth and the Moon about 3,84,400 km. Roughly how many times farther is the Sun? Round the Moon distance to the nearest lakh first and give a whole number.",{"kind":161,"answer":305,"tolerance":163},375,[307],"Cancel the same number of zeros from both numbers before dividing.",{"correct":309,"incorrect":310},"Right: 3,84,400 ≈ 4,00,000, and 15,00,00,000 ÷ 4,00,000 = 1,500 ÷ 4 = 375. (Using the exact figures gives about 390.)","Round 3,84,400 to 4,00,000. Then 15,00,00,000 ÷ 4,00,000: cancel five zeros to get 1,500 ÷ 4 = 375.",{"id":312,"type":157,"itemId":313,"prompt":314,"check":315,"hints":318,"feedback":320},"pr-aditya","number-system.extend-aditya","Aditya-L1 studies the Sun from about 15 lakh km away from Earth. How many million km is that?",{"kind":161,"answer":316,"tolerance":317},1.5,0.01,[319],"10 lakh = 1 million.",{"correct":321,"incorrect":322},"Right: 10 lakh = 1 million, so 15 lakh = 1.5 million km.","15 lakh = 15,00,000 = 1,500,000 = 1.5 million. Divide lakhs by 10 to get millions.",{"id":324,"type":47,"title":325,"eyebrow":326,"navLabel":327},"ch04","How other cultures wrote numbers","Chapter 04","4 Other systems",{"id":329,"type":330,"title":331,"prompt":332,"options":333},"explorer-systems","explorer","Number systems around the world","Pick a system to see how it worked and what it lacked.",[334,347,359,371,381,392],{"id":335,"label":336,"chain":337,"badge":343,"note":346},"egypt","Egyptian",[338,339,340,341,342],"Stroke = 1","Heel bone = 10","Coil of rope = 100","Lotus = 1,000","Add the symbols",{"text":344,"tone":345},"No place value","no","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.",{"id":348,"label":349,"chain":350,"badge":355,"note":358},"babylon","Babylonian",[351,352,353,354],"Wedge marks","Base 60","Places: 1, 60, 3,600","Gap, later a mark, for empty",{"text":356,"tone":357},"Place value, base 60","yes","About 4,000 years ago in Mesopotamia (modern Iraq), scribes pressed wedges into clay using **base 60**: each place is worth 60 times the place to its right. We still use it: 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle. For a long time an empty place was just a gap, which caused confusion; a placeholder sign came later, but not a true zero used as a number.",{"id":360,"label":361,"chain":362,"badge":368,"note":370},"maya","Maya",[363,364,365,366,367],"Dot = 1","Bar = 5","Shell = 0","Mostly base 20","Written top to bottom",{"text":369,"tone":357},"Place value, base 20, with zero","The Maya of Central America independently invented a place-value system with a **shell symbol for zero**, more than 1,500 years ago. They counted mostly in twenties (perhaps fingers and toes). Places were stacked vertically. For calendars, the third place was 18 × 20 = 360 instead of 400, close to the length of a year.",{"id":372,"label":373,"chain":374,"badge":379,"note":380},"roman","Roman",[375,376,377,378],"I V X L C D M","Add and subtract","No zero","Abacus for sums",{"text":344,"tone":345},"Letters with fixed values, added and sometimes subtracted. Fine for labels and short numbers, hard for calculation, so Romans did arithmetic on a counting board or abacus and wrote down only the result.",{"id":382,"label":383,"chain":384,"badge":389,"note":391},"chinese","Chinese rods",[385,386,387,388],"Bamboo rods","Base 10","Alternate vertical\u002Fhorizontal","Blank for empty",{"text":390,"tone":357},"Place value, base 10","Chinese counting rods, used for over 2,000 years, laid out digits on a counting board in base ten, alternating vertical and horizontal forms in neighbouring places so the places could be told apart. An empty place was left blank; a written round zero appears in Chinese texts much later, in the 1200s.",{"id":393,"label":394,"chain":395,"badge":399,"note":401},"hindu","Hindu–Arabic",[396,386,397,398],"Ten digits 0–9","Zero as digit and number","Every place reuses the same digits",{"text":400,"tone":357},"Place value, base 10, with zero","Developed in India and spread by Arabic-writing scholars: ten symbols, one of them zero, reused in every place. Zero is both a placeholder and a number with rules (Brahmagupta, 628 CE). This combination is what made written calculation fast enough to become universal.",{"id":403,"type":101,"variant":404,"title":405,"markdown":406},"aha-base","aha","Ten is not special","Nothing in mathematics forces **ten**. Our base almost certainly comes from our ten fingers. Place value works with any whole number base of 2 or more: base 60 for the Babylonians, base 20 for the Maya, base 2 for computers. Change the base and every rule you learned (expanded form, comparing, successors, rollovers) still works, just with a different number in place of 10.",{"id":408,"type":47,"title":409,"eyebrow":410,"navLabel":411},"ch05","Binary: place value with two digits","Chapter 05","5 Binary",{"id":413,"type":43,"markdown":414},"binary","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\".\n\nSo 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.",{"id":416,"type":56,"caption":417,"columns":418,"rows":422},"table-binary","Counting in binary",[419,420,421],"Number","Binary","Places used",[423,425,427,430,433,436,440,444,447,451,454,458],[424,424,424],"0",[426,426,426],"1",[428,429,428],"2","10",[431,33,432],"3","2 + 1",[434,435,434],"4","100",[437,438,439],"5","101","4 + 1",[441,442,443],"7","111","4 + 2 + 1",[445,446,445],"8","1000",[448,449,450],"15","1111","8 + 4 + 2 + 1",[452,453,452],"16","10000",[455,456,457],"31","11111","16 + 8 + 4 + 2 + 1",[459,460,461],"45","101101","32 + 8 + 4 + 1",{"id":463,"type":211,"title":464,"problem":465,"steps":466},"we-to-binary","Writing 2026 in binary","Write 2,026 in binary.",[467,468,469,470,471],"List the powers of 2 up to 2,026: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. (2,048 is too big.)","2,026 − 1,024 = 1,002; − 512 = 490; − 256 = 234; − 128 = 106; − 64 = 42; − 32 = 10.","10 = 8 + 2. So the places used are 1,024, 512, 256, 128, 64, 32, 8 and 2.","Write 1 for used and 0 for unused places, from 1,024 down to 1: **11111101010**.","Check: 1,024 + 512 + 256 + 128 + 64 + 32 + 8 + 2 = 2026. ✓",{"id":473,"type":101,"variant":474,"title":475,"markdown":476},"mis-binary-ten","misconception","“10 in binary means ten”","The digits **1 0** mean \"one of the next place, none of this place\". In base ten the next place is ten, but in binary it is **two**, so binary 10 is two, binary 100 is four and binary 1000 is eight. When a number could be in another base, label it: 10₂ = 2, 10₁₀ = 10. The same digits mean different numbers in different bases.",{"id":478,"type":101,"variant":288,"title":479,"markdown":480},"obs-binary-rules","Old rules, new base","Everything from this topic carries over:\n\n- **Rollover:** the successor of 111 (7) is 1000 (8), just as 999 + 1 = 1,000.\n- **Largest n-digit binary number** is n ones = 2ⁿ − 1 (1111 = 15 = 16 − 1).\n- **Shifting left** (adding a 0 at the end) multiplies by 2 instead of 10: 101 is 5, 1010 is 10.\n- **Comparing:** more digits wins; otherwise the first difference from the left decides.",{"id":482,"type":127,"component":483,"componentVersion":5,"config":484,"objective":509,"textAlternative":510},"lab-match-binary","match-pairs",{"prompt":485,"mode":486,"pairs":487},"Match each binary number with its value in base ten.","memory",[488,491,494,497,500,503,506],{"a":489,"b":490},"110 (binary)","6",{"a":492,"b":493},"1001 (binary)","9",{"a":495,"b":496},"1100 (binary)","12",{"a":498,"b":499},"10011 (binary)","19",{"a":501,"b":502},"11001 (binary)","25",{"a":504,"b":505},"100000 (binary)","32",{"a":507,"b":508},"110010 (binary)","50","Play a memory game matching binary numbers with their base-ten values by adding place values 1, 2, 4, 8, 16, 32.","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.",{"id":512,"type":112,"prompt":513,"options":514,"explanation":522},"pred-byte","A **byte** is 8 binary digits, like 10110010. How many different values can one byte hold?",[515,516,517,519],{"id":116,"label":445},{"id":119,"label":452},{"id":122,"label":518},"255",{"id":520,"label":521},"d","256","**d: 256.** Each of the 8 places can be 0 or 1, so there are 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸ = 256 patterns, from 00000000 (0) to 11111111 (255). Same idea as 3-digit base-ten strings: 10 × 10 × 10 = 1,000 patterns, from 000 to 999.",{"id":524,"type":56,"caption":525,"columns":526,"rows":531},"table-bytes","Binary sizes in computing",[527,528,529,530],"Name","Size","As a power","Roughly",[532,537,542,547,552,556],[533,534,535,536],"bit","1 binary digit","2⁰ digit","0 or 1",[538,539,540,541],"byte","8 bits","2⁸ = 256 patterns","one letter of text",[543,544,545,546],"kilobyte (KB)","1,000 bytes (older usage: 1,024 = 2¹⁰)","10³ or 2¹⁰","a short paragraph",[548,549,550,551],"megabyte (MB)","10,00,000 bytes","10⁶","a photo or a minute of music",[553,554,65,555],"gigabyte (GB)","100 crore bytes","a film in ordinary quality",[557,558,80,559],"terabyte (TB)","1 lakh crore bytes","a large hard disk",{"id":561,"type":157,"itemId":562,"prompt":563,"check":564,"hints":565,"feedback":567},"pr-binary","number-system.extend-binary","What is the binary number **1100100** in base ten?",{"kind":161,"answer":258,"tolerance":163},[566],"Write 64, 32, 16, 8, 4, 2, 1 under the digits.",{"correct":568,"incorrect":569},"Right: 64 + 32 + 4 = 100.","Place values from the right: 1, 2, 4, 8, 16, 32, 64. The 1s are in the 64, 32 and 4 places: 64 + 32 + 4 = 100.",{"id":571,"type":47,"title":572,"eyebrow":573,"navLabel":574},"ch06","Other bases: five, twelve and sixty","Chapter 06","6 Other bases",{"id":576,"type":211,"title":577,"problem":578,"steps":579},"we-base5","Counting on one hand: base five","In base five the places are 1, 5, 25, 125, 625. Write 2,026 in base five.",[580,581,582,583],"2,026 ÷ 625 = 3, remainder 2,026 − 1,875 = 151. So the 625s digit is 3.","151 ÷ 125 = 1, remainder 26. The 125s digit is 1.","26 ÷ 25 = 1, remainder 1. The 25s digit is 1. The 5s digit is 0, and the ones digit is 1.","**2,026 = 31101 in base five.** Check: 3 × 625 + 1 × 125 + 1 × 25 + 0 × 5 + 1 = 2,026. ✓",{"id":585,"type":157,"itemId":586,"prompt":587,"check":588,"hints":590,"feedback":592},"pr-base5","number-system.extend-base5","What is the base-five number **1234** in base ten?",{"kind":161,"answer":589,"tolerance":163},194,[591],"The places are 125, 25, 5 and 1.",{"correct":593,"incorrect":594},"Right: 1 × 125 + 2 × 25 + 3 × 5 + 4 × 1 = 125 + 50 + 15 + 4 = 194.","Base-five places from the right are 1, 5, 25, 125. So 1 × 125 + 2 × 25 + 3 × 5 + 4 = 194.",{"id":596,"type":211,"title":597,"problem":598,"steps":599},"we-base5-back","From 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?",[600,601,602,603,604],"Base-five places from the right: 1, 5, 25, 125.","4 × 125 = 500; 3 × 25 = 75; 0 × 5 = 0; 1 × 1 = 1.","Total: 500 + 75 + 0 + 1 = **576**.","Check by converting back: 576 ÷ 125 = 4 remainder 76; 76 ÷ 25 = 3 remainder 1; 1 ÷ 5 = 0 remainder 1 → 4, 3, 0, 1. ✓","Note the placeholder zero works exactly as in base ten: the 0 says \"no fives\".",{"id":606,"type":157,"itemId":607,"prompt":608,"check":609,"hints":621,"feedback":623},"pr-base-digit","number-system.extend-base-digits","Which of these could **not** be a number written in base five?",{"kind":610,"options":611,"correct":620},"choice",[612,614,616,618],{"id":116,"label":613},"4,304",{"id":119,"label":615},"1,000",{"id":122,"label":617},"2,051",{"id":520,"label":619},"3,333",[122],[622],"Which digits does base five use?",{"correct":624,"incorrect":625},"Right: base five uses only the digits 0 to 4, so a 5 cannot appear. Five ones are regrouped into one five, just as ten ones become one ten in base ten.","In base five the digits are 0, 1, 2, 3 and 4. 2,051 contains a 5, so it cannot be a base-five numeral.",{"id":627,"type":43,"markdown":628},"base12-60","**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.\n\n**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.",{"id":630,"type":127,"component":631,"componentVersion":5,"config":632,"objective":694,"textAlternative":695},"lab-sort-bases","sort-game",{"prompt":633,"bins":634,"items":648,"seconds":163},"Which base is hiding in each everyday thing?",[635,638,640,643,646],{"id":636,"label":637},"b2","Base 2",{"id":639,"label":386},"b10",{"id":641,"label":642},"b12","Base 12",{"id":644,"label":645},"b20","Base 20",{"id":647,"label":352},"b60",[649,653,656,660,664,668,672,676,680,684,687,691],{"id":650,"label":651,"bin":647,"why":652},"b1","Seconds in a minute, minutes in an hour","Time is counted in sixties, a legacy of Babylonian base 60.",{"id":636,"label":654,"bin":647,"why":655},"Degrees in a circle: 360","360 = 6 × 60; the Babylonians divided circles in sixties.",{"id":657,"label":658,"bin":636,"why":659},"b3","Computer memory: on\u002Foff switches","A bit is 0 or 1: base 2.",{"id":661,"label":662,"bin":641,"why":663},"b4","A dozen eggs, a gross of pencils","Dozen = 12, gross = 12 × 12 = 144.",{"id":665,"label":666,"bin":639,"why":667},"b5","The ten digits 0–9 on a phone keypad","Our everyday digits are base ten.",{"id":669,"label":670,"bin":639,"why":671},"b6","Rupees and paise: 100 paise = ₹1","100 paise = 10 × 10 paise: base ten.",{"id":673,"label":674,"bin":639,"why":675},"b7","Metric units: 1 km = 1,000 m","Metric prefixes are powers of ten.",{"id":677,"label":678,"bin":636,"why":679},"b8","Barcode stripes, black or white","Two states, like binary digits.",{"id":681,"label":682,"bin":644,"why":683},"b9","Maya calendar counts in twenties","The Maya counted mostly in twenties.",{"id":639,"label":685,"bin":641,"why":686},"Months in a year","Twelve months, a base-twelve habit in calendars.",{"id":688,"label":689,"bin":644,"why":690},"b11","French quatre-vingts (four twenties) for 80","French counts 80 as four twenties, a trace of base 20.",{"id":641,"label":692,"bin":636,"why":693},"Kilobytes: 1,024 = 2¹⁰ bytes in older usage","1,024 = 2¹⁰, a power of two.","Sort everyday examples by the base (2, 10, 12, 20 or 60) that they come from.","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.",{"id":697,"type":47,"title":698,"eyebrow":699,"navLabel":700},"ch07","Olympiad-style puzzles","Chapter 07","7 Puzzles",{"id":702,"type":211,"title":703,"problem":704,"steps":705},"we-ones","How many 1s?","How many times is the digit 1 written when you write every number from 1 to 1,000?",[706,707,708,709],"First count 000 to 999 as three-digit strings (with leading zeros, which do not add any 1s).","Each of the three places shows each digit equally often: 1,000 strings ÷ 10 digits = 100 times per place.","So the digit 1 appears 100 × 3 = 300 times from 1 to 999.","Add the 1 in 1,000: **301**.",{"id":711,"type":211,"title":712,"problem":713,"steps":714},"we-digit-sum","Smallest number with digit sum 30","What is the smallest whole number whose digits add up to 30?",[715,716,717,718],"Fewer digits means a smaller number, so use as few digits as possible. Each digit is at most 9.","Three digits give at most 27, which is too little; so we need four digits.","To make a 4-digit number as small as possible, make the first digit as small as possible and push the big digits to the right: _ 9 9 9.","First digit: 30 − 27 = 3. **Answer: 3,999.**",{"id":720,"type":211,"title":721,"problem":722,"steps":723},"we-palin","Counting palindromes","A palindrome reads the same forwards and backwards, like 4,774. How many palindromes are there from 1 to 99,999?",[724,725,726,727,728],"1 digit: 9 (1–9). 2 digits: 9 (11, 22, …, 99).","3 digits: choose the first digit (9 ways) and the middle digit (10 ways): 90.","4 digits: the first two digits decide the rest: 9 × 10 = 90.","5 digits: the first three digits decide the rest: 9 × 10 × 10 = 900.","Total: 9 + 9 + 90 + 90 + 900 = **1,098**.",{"id":730,"type":157,"itemId":731,"prompt":732,"check":733,"hints":735,"feedback":737},"pr-digsum","number-system.extend-digit-sum","Add up **all the digits** of all the numbers from 1 to 99. (For example, 47 contributes 4 + 7.) What is the total?",{"kind":161,"answer":734,"tolerance":163},900,[736],"1 + 2 + … + 9 = 45.",{"correct":738,"incorrect":739},"Right: each digit 1–9 appears 10 times in the ones place and 10 times in the tens place: 20 × (1 + 2 + … + 9) = 20 × 45 = 900.","Ones place: digits 0–9 each appear 10 times (sum 10 × 45 = 450). Tens place: each of 1–9 appears 10 times (another 450). Total 900.",{"id":741,"type":157,"itemId":742,"prompt":743,"check":744,"hints":755,"feedback":757},"pr-riddle","number-system.extend-riddle","I am a 6-digit number. My lakhs digit is 3 times my ones digit. My ten-thousands digit is 0. My thousands digit is the largest digit. My hundreds and tens digits are both 5. My ones digit is 3. Who am I?",{"kind":610,"options":745,"correct":754},[746,748,750,752],{"id":116,"label":747},"9,05,953",{"id":119,"label":749},"3,09,551",{"id":122,"label":751},"9,09,553",{"id":520,"label":753},"1,09,553",[122],[756],"Start with the ones digit, then use it to find the lakhs digit.",{"correct":758,"incorrect":759},"Right: ones 3, lakhs 3 × 3 = 9, ten thousands 0, thousands 9, hundreds 5, tens 5 → 9,09,553.","Build it place by place: ones 3; lakhs 3 × 3 = 9; ten thousands 0; thousands 9; hundreds 5; tens 5. Reading from lakhs to ones: 9, 0, 9, 5, 5, 3 → 9,09,553.",{"id":761,"type":211,"title":762,"problem":763,"steps":764},"we-1000th","The 1,000th digit","Write the counting numbers in a row: 123456789101112131415… What is the 1,000th digit?",[765,766,767,768,769],"Digits 1 to 9 use up positions 1–9. The numbers 10 to 99 use 90 × 2 = 180 more: positions 10–189.","That leaves 1,000 − 189 = 811 positions, filled by three-digit numbers starting at 100.","811 ÷ 3 = 270 remainder 1. So 270 complete three-digit numbers (100 to 369) fill positions 190–999.","The 1,000th digit is the **first** digit of the next number, 370: **3**.","A quick computer check of the string agrees.",{"id":771,"type":157,"itemId":772,"prompt":773,"check":774,"hints":776,"feedback":779},"pr-pal-4","number-system.extend-palindrome-4","How many **4-digit palindromes** are there between 1,000 and 9,999 that are also multiples of 11? (Hint: test a few first.)",{"kind":161,"answer":775,"tolerance":163},90,[777,778],"Write abba in expanded form.","1,001 = 11 × 91.",{"correct":780,"incorrect":781},"Right: all 90 of them. A 4-digit palindrome abba = 1,001a + 110b = 11 × (91a + 10b), always a multiple of 11.","Try 1,221 = 11 × 111 and 4,884 = 11 × 444. In general abba = 1,000a + 100b + 10b + a = 1,001a + 110b, and both 1,001 and 110 are multiples of 11. So every one of the 90 four-digit palindromes works.",{"id":783,"type":101,"variant":784,"title":785,"markdown":786},"q-challenge","question","Three harder challenges","Try these with a friend. Answers are at the end of the list, but try first.\n\n1. Using each of the digits 1 to 9 exactly once, make two numbers (one with 4 digits and one with 5) whose difference is as small as possible. Hint: it cannot be smaller than the smallest 5-digit number made from the leftover digits minus the largest 4-digit number.\n2. How many whole numbers from 1 to 1,00,000 contain **at least one 5**?\n3. What is the 1,000th digit written when you write 1, 2, 3, 4, … in a row (123456789101112…)?\n\nAnswers: (1) 12,345 − 9,876 = 2469 is the smallest possible; (2) 40951; (3) 3.",{"id":788,"type":47,"title":789,"eyebrow":790,"navLabel":791},"ch08","Fermi estimates: huge numbers from small facts","Chapter 08","8 Fermi estimates",{"id":793,"type":43,"markdown":794},"fermi","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.",{"id":796,"type":211,"title":797,"problem":798,"steps":799},"we-rotis","How many rotis does India eat in a day?","Make a Fermi estimate of the number of rotis (chapatis) eaten in India in one day.",[800,801,802,803,804],"Population: about 140 crore.","Suppose about half of people eat rotis on a typical day: 70 crore people.","Suppose each of them eats about 4 rotis in a day.","70 crore × 4 = **280 crore rotis a day**, or 2,800,000,000: about 2.8 billion.","Is it sensible? The answer depends a lot on the guesses (half? 4 each?), but it is surely more than 10 crore and less than 1,000 crore. Fermi estimates give the right size, not the exact count.",{"id":806,"type":211,"title":807,"problem":808,"steps":809},"we-balls","How 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.",[810,811,812,813,814],"Volume of the room: 8 × 6 × 3 = 144 cubic metres. Round to about 150.","A cricket ball is about 7 cm across. Pretend each takes up a 7 cm cube; about 14 fit along 1 m (100 ÷ 7 ≈ 14).","Balls per cubic metre: about 14 × 14 × 14 ≈ 2,700. Round to 3,000.","Total: 150 × 3,000 = **4,50,000**, roughly 4 to 5 lakh balls (real balls pack a little less tightly than cubes, so perhaps 3 to 4 lakh).","The answer is certainly in lakhs, not thousands or crores: that is what a Fermi estimate promises.",{"id":816,"type":127,"component":817,"componentVersion":5,"config":818,"objective":847,"textAlternative":848},"lab-sprint-real","arith-sprint",{"operations":819,"ranges":822,"rounds":829,"secondsTotal":163,"estimateFirst":134,"wordProblems":830},[820,821],"×","÷",{"a":823,"b":826},{"min":824,"max":825},1000,99999,{"min":827,"max":828},2,99,10,[831,835,839,843],{"prompt":832,"answer":833,"operation":820,"unit":834},"A train carries 1,248 passengers per trip and makes 365 trips in a year. How many passenger journeys is that?",455520,"journeys",{"prompt":836,"answer":837,"operation":821,"unit":838},"A stadium with 1,32,000 seats is divided equally into 24 blocks. How many seats are in each block?",5500,"seats",{"prompt":840,"answer":841,"operation":821,"unit":842},"A school collects ₹2,52,000 from 840 students, each paying the same. How much does each student pay?",300,"₹",{"prompt":844,"answer":845,"operation":820,"unit":846},"A factory makes 7,500 bicycles a month. How many does it make in 12 months?",90000,"bicycles","Estimate, then calculate products and quotients with large numbers, including real-life word problems.","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.\n\nWord 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).",{"id":850,"type":127,"component":851,"componentVersion":5,"config":852,"objective":859,"textAlternative":860},"lab-round-lakh-e","rounding-race",{"roundTo":853,"range":855,"rounds":857,"secondsPerRound":858},[854],100000,{"min":856,"max":139},1000000,8,20,"Round crore-sized numbers to the nearest lakh, the way newspapers report budgets and populations.","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.",{"id":862,"type":47,"title":863,"eyebrow":864,"navLabel":865},"ch09","Projects and people who use big numbers","Chapter 09","9 Projects",{"id":867,"type":868,"title":869,"items":870},"steps-projects","steps","Projects to try",[871,875,879,883,887,891],{"title":872,"tag":873,"text":874},"Newspaper hunt","1 week","Collect 20 large numbers from newspapers or news sites. Rewrite each in both systems and in words, and sort them by size.",{"title":876,"tag":877,"text":878},"Build a lakh","class 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?",{"title":880,"tag":881,"text":882},"Family timeline in seconds","1 evening","Work out how many seconds old each family member is. Who is over a billion seconds (about 31.7 years)?",{"title":884,"tag":885,"text":886},"Binary birthday","30 minutes","Write your birth date and age in binary and base five. Make a binary bracelet with two colours of beads.",{"title":888,"tag":889,"text":890},"Fermi challenge","group","Estimate: how many cricket balls would fill your classroom? How many litres of tea does your town drink a day? Compare answers between groups.",{"title":892,"tag":893,"text":894},"Roman year walk","outing","Find Roman numerals on buildings, clocks and foundation stones. Convert each and note what it labels.",{"id":896,"type":56,"caption":897,"columns":898,"rows":901},"table-careers","Some careers where large numbers matter",[899,900],"Career","Big numbers they use",[902,905,908,911,914,917],[903,904],"Census and survey statistician","Counts of crores of people, households and farms; rounding and estimating from samples.",[906,907],"Space scientist or engineer (ISRO)","Distances in lakhs and crores of km, speeds, fuel masses, orbits.",[909,910],"Banker, accountant or economist","Budgets in lakh crore, interest on loans, national income.",[912,913],"Data scientist or software engineer","Billions of records, binary storage, gigabytes and terabytes.",[915,916],"Railway or airline planner","Lakhs of passengers a day, timetables, tickets and seats.",[918,919],"Election official","Tens of crores of voters, lakhs of polling stations, counting and cross-checking totals.",{"id":921,"type":47,"title":922,"eyebrow":923,"navLabel":924},"ch10","Open questions","Chapter 10","10 Open questions",{"id":926,"type":101,"variant":784,"title":927,"markdown":928},"open-questions","Questions to explore","Some of these have no single right answer. Pick one and dig in.\n\n- Should India switch to the International system, or should the world learn lakh and crore? What would be gained and lost?\n- If we had 8 fingers, we might count in base 8. Would arithmetic be easier or harder? What about base 12?\n- Is there a largest number that is *useful*? What is the biggest number you have ever seen used for something real?\n- Why did it take so long, more than a thousand years, for the Indian numerals to replace Roman numerals in Europe?\n- Kaprekar found 6,174 for 4-digit numbers. What happens with 3-digit numbers? With 5-digit numbers?\n- How should a newspaper round a number so that it is honest but easy to read?",{"id":930,"type":931,"prompt":932},"reflect-e","reflection","Choose one number from this layer (a distance, a population, a binary number, a googol…) and explain to a younger child how big it is, using a comparison they can picture.",{"id":934,"type":47,"title":935,"eyebrow":936,"navLabel":937},"ch11","Challenge practice set","Chapter 11","11 Challenge set",{"id":939,"type":43,"markdown":940},"challenge-intro","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).",{"id":942,"type":101,"variant":474,"title":943,"markdown":944},"mis-lakh-lakh","“A lakh lakh is a trillion”","It sounds big enough, but a lakh lakh is 10⁵ × 10⁵ = 10¹⁰ = **10,00,00,00,000**: one thousand crore, or ten billion. A trillion is 10¹², a hundred times more, and equals a **lakh crore**. When two big names are multiplied, add their exponents before you guess.",{"id":946,"type":157,"itemId":947,"prompt":948,"check":949,"hints":950,"feedback":951},"pr-ch-lakhlakh","number-system.extend-lakh-lakh","How many **crore** make one lakh lakh?",{"kind":161,"answer":824,"tolerance":163},[165],{"correct":952,"incorrect":953},"Right: 10¹⁰ ÷ 10⁷ = 10³ = 1,000 crore.","A lakh lakh = 10⁵ × 10⁵ = 10¹⁰. Divide by a crore, 10⁷: 10³ = 1,000.",{"id":955,"type":157,"itemId":956,"prompt":957,"check":958,"hints":959,"feedback":962},"pr-ch-digsum3","number-system.extend-digit-sum-3","How many 4-digit numbers have digits that add up to exactly **3**?",{"kind":161,"answer":829,"tolerance":163},[960,961],"The first digit cannot be 0.","List the numbers starting with 2: 2,100, 2,010, 2,001.",{"correct":963,"incorrect":964},"Right: 10. First digit 3: 3,000 (1 number). First digit 2: the remaining 1 goes in one of 3 places (3 numbers). First digit 1: the remaining 2 is either one digit 2 (3 places) or two 1s (3 ways): 6 numbers. Total 1 + 3 + 6 = 10.","Split by the first digit (1, 2 or 3) and count how the rest of the sum can be spread over the other three places.",{"id":966,"type":211,"title":967,"problem":968,"steps":969},"we-ch-reverse","Reverse and add until a palindrome","Start with 87. Add it to its reverse, and repeat with the answer until you reach a palindrome.",[970,971,972,973,974],"87 + 78 = 165.","165 + 561 = 726.","726 + 627 = 1,353.","1,353 + 3,531 = **4,884**, a palindrome, after 4 steps.","Most starting numbers reach a palindrome quickly. For **196**, nobody has ever found one, even after computers ran the process for billions of steps; whether one exists is an open question.",{"id":976,"type":211,"title":977,"problem":978,"steps":979},"we-ch-ab","A two-digit puzzle","A 2-digit number AB added to its reverse BA gives 121. How many such numbers are there?",[980,981,982,983],"By place value, AB = 10A + B and BA = 10B + A.","Their sum is 11A + 11B = 11 × (A + B). So 11 × (A + B) = 121, giving A + B = 11.","A and B are digits from 1 to 9 (neither can be 0, or one number would not have 2 digits): (2, 9), (3, 8), … (9, 2).","That gives **8 numbers**: 29, 38, 47, 56, 65, 74, 83, 92.",{"id":985,"type":157,"itemId":986,"prompt":987,"check":988,"hints":990,"feedback":993},"pr-ch-zeros","number-system.extend-count-zeros","How many times is the digit **0** written when writing all numbers from 1 to 1,000?",{"kind":161,"answer":989,"tolerance":163},192,[991,992],"Count zeros in 100 to 199 carefully, then multiply by 9.","Leading zeros are never written.",{"correct":994,"incorrect":995},"Right: 192. Tens and ones digits from 1 to 99: 9 zeros (10, 20, …, 90). From 100 to 999: each hundred has 20 zeros in the tens and ones places (x00 counts two, x01–x09 and x10, x20, …, x90 one each), and there are 9 hundreds: 180. Plus 3 in 1,000: 9 + 180 + 3 = 192.","Count by blocks: 1–99 gives 9; each block of a hundred from 100 to 999 gives 20 (9 hundreds: 180); 1,000 gives 3.",{"id":997,"type":211,"title":998,"problem":999,"steps":1000},"we-ch-moon-drive","Driving 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?",[1001,1002,1003,1004],"Hours: 3,84,400 ÷ 60 ≈ 6,407 hours.","Days: 6,407 ÷ 24 ≈ 267 days, about nine months.","Chandrayaan-3 took about 40 days from launch on 14 July 2023 to landing on 23 August 2023, and most of that time was spent on careful orbits around Earth and the Moon, not on the straight-line distance.","Estimate check: 4,00,000 ÷ 60 ≈ 6,700 hours ≈ 280 days. ✓",{"id":1006,"type":211,"title":1007,"problem":1008,"steps":1009},"we-ch-pop-growth","Estimating 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?",[1010,1011,1012,1013],"Increase: about 146 − 121 = 25 crore over 14 years.","25 crore ÷ 14 ≈ 1.8 crore per year (about 18 million).","Per day: 1.8 crore ÷ 365 ≈ 49,000, so roughly 50,000 more people each day (births minus deaths).","Both inputs are approximate, so give the answer as \"about 50 thousand a day\", not an exact figure.",{"id":1015,"type":157,"itemId":1016,"prompt":1017,"check":1018,"hints":1029,"feedback":1031},"pr-ch-kumbh","number-system.extend-kumbh","For the Maha Kumbh of 2025 at Prayagraj, officials reported about **66 crore** visits over the 45 days of the mela. What is that in the International system?",{"kind":610,"options":1019,"correct":1028},[1020,1022,1024,1026],{"id":116,"label":1021},"66 million",{"id":119,"label":1023},"660 million",{"id":122,"label":1025},"6.6 billion",{"id":520,"label":1027},"66 billion",[119],[1030],"1 crore = 10 million.",{"correct":1032,"incorrect":1033},"Right: 1 crore = 10 million, so 66 crore = 660 million. (Visits, not different people: one person visiting on three days counts three times.)","66 crore × 10 = 660 million. Option c would be more than three-quarters of the whole world population.",{"id":1035,"type":1036,"title":1037,"terms":1038},"gloss-e","glossary","Extend vocabulary",[1039,1042,1045,1048,1051,1053,1057,1060,1063,1067,1071,1074],{"term":1040,"meaning":1041},"Arab","Indian name for 100 crore, 10⁹, equal to one billion.",{"term":1043,"meaning":1044},"Kharab","Indian name for 100 arab, 10¹¹, equal to 100 billion.",{"term":1046,"meaning":1047},"Trillion","In the short scale used today, a million million, 10¹². Equal to 1 lakh crore.",{"term":1049,"meaning":1050},"Short scale \u002F long scale","Two meanings of billion: 10⁹ (short scale, modern English) or 10¹² (long scale, older British and many European languages).",{"term":108,"meaning":1052},"The number 10¹⁰⁰, a 1 followed by 100 zeros.",{"term":1054,"meaning":1055,"example":1056},"Base","The number of digits a place-value system uses, and how many times bigger each place is than the next.","Base 2, base 10, base 60",{"term":420,"meaning":1058,"example":1059},"Base 2: place values 1, 2, 4, 8, 16…, digits 0 and 1.","101 = 5",{"term":1061,"meaning":1062},"Bit","A single binary digit, 0 or 1.",{"term":1064,"meaning":1065,"example":1066},"Palindrome","A number that reads the same forwards and backwards.","12,321",{"term":1068,"meaning":1069,"example":1070},"Digit sum","The sum of a number’s digits.","The digit sum of 3,999 is 30.",{"term":1072,"meaning":1073},"Fermi estimate","A rough estimate of a large quantity built from simple, sensible guesses multiplied together.",{"term":1075,"meaning":1076},"Light-year","The distance light travels in a year, about 9.46 lakh crore km (9.46 trillion km).",{"id":1078,"type":1079,"title":1080,"questions":1081},"quiz-e","quiz","Extend check",[1082,1095,1108,1119,1132,1145,1155,1166,1179,1188],{"itemId":1083,"prompt":1084,"options":1085,"correct":119,"why":1094},"number-system.ext-q-arab","One arab is equal to…",[1086,1088,1090,1092],{"id":116,"label":1087},"10 crore",{"id":119,"label":1089},"100 crore",{"id":122,"label":1091},"1,000 crore",{"id":520,"label":1093},"1 lakh crore","Arab = 10⁹ = 100 crore = 1 billion.",{"itemId":1096,"prompt":1097,"options":1098,"correct":116,"why":1107},"number-system.ext-q-trillion","India’s economy is sometimes described in trillions of dollars. 4 trillion is…",[1099,1101,1103,1105],{"id":116,"label":1100},"4 lakh crore",{"id":119,"label":1102},"40 lakh crore",{"id":122,"label":1104},"4 crore crore",{"id":520,"label":1106},"400 crore","1 trillion = 10¹² = 1 lakh crore, so 4 trillion = 4 lakh crore.",{"itemId":1109,"prompt":1110,"options":1111,"correct":116,"why":1118},"number-system.ext-q-binary","Binary 11011 equals…",[1112,1114,1116,1117],{"id":116,"label":1113},"27",{"id":119,"label":1115},"11,011",{"id":122,"label":499},{"id":520,"label":502},"16 + 8 + 2 + 1 = 27.",{"itemId":1120,"prompt":1121,"options":1122,"correct":116,"why":1131},"number-system.ext-q-binary2","What is 20 in binary?",[1123,1125,1127,1129],{"id":116,"label":1124},"10100",{"id":119,"label":1126},"11000",{"id":122,"label":1128},"10010",{"id":520,"label":1130},"1100","20 = 16 + 4 → 10100.",{"itemId":1133,"prompt":1134,"options":1135,"correct":119,"why":1144},"number-system.ext-q-base60","Which everyday measurement uses base 60?",[1136,1138,1140,1142],{"id":116,"label":1137},"Money",{"id":119,"label":1139},"Minutes and seconds",{"id":122,"label":1141},"Metres",{"id":520,"label":1143},"Litres","60 seconds = 1 minute and 60 minutes = 1 hour, a Babylonian legacy.",{"itemId":1146,"prompt":1147,"options":1148,"correct":122,"why":1154},"number-system.ext-q-maya","Which civilisation independently invented a zero symbol (a shell) in a base-20 system?",[1149,1150,1151,1152],{"id":116,"label":373},{"id":119,"label":336},{"id":122,"label":361},{"id":520,"label":1153},"Greek","The Maya of Central America used a shell glyph for zero.",{"itemId":1156,"prompt":1157,"options":1158,"correct":119,"why":1165},"number-system.ext-q-pal","How many 3-digit palindromes are there?",[1159,1160,1162,1164],{"id":116,"label":493},{"id":119,"label":1161},"90",{"id":122,"label":1163},"99",{"id":520,"label":435},"First digit 9 ways, middle digit 10 ways, last digit forced: 90.",{"itemId":1167,"prompt":1168,"options":1169,"correct":119,"why":1178},"number-system.ext-q-sun","The Sun is about 15 crore km away. In the International system that is…",[1170,1172,1174,1176],{"id":116,"label":1171},"15 million km",{"id":119,"label":1173},"150 million km",{"id":122,"label":1175},"1.5 billion km",{"id":520,"label":1177},"1.5 million km","1 crore = 10 million, so 15 crore = 150 million.",{"itemId":1180,"prompt":1181,"options":1182,"correct":122,"why":1187},"number-system.ext-q-scale","In older British usage (long scale), a billion meant…",[1183,1184,1185,1186],{"id":116,"label":550},{"id":119,"label":65},{"id":122,"label":80},{"id":520,"label":90},"Long scale: billion = a million million = 10¹². Modern English uses 10⁹.",{"itemId":1189,"prompt":1190,"options":1191,"correct":119,"why":1199},"number-system.ext-q-ones","How many times is the digit 1 written from 1 to 1,000?",[1192,1194,1196,1198],{"id":116,"label":1193},"300",{"id":119,"label":1195},"301",{"id":122,"label":1197},"271",{"id":520,"label":615},"100 times in each of three places for 1–999, plus one more in 1,000.",{"id":1201,"type":1202,"conceptId":1203,"relation":1204,"explanation":1205},"conn-e-patterns","connection","patterns","related_to","Binary counting, palindromes and Kaprekar-style routines are rich sources of number patterns.",{"id":1207,"type":1202,"conceptId":1208,"relation":1209,"explanation":1210},"conn-e-data","data-handling","applied_in","Census data, election results and budgets are large-number data sets that data handling organises and summarises.",{"id":1212,"type":1202,"conceptId":1213,"relation":1204,"explanation":1214},"conn-e-ops","order-of-operations","Fermi estimates chain several multiplications and divisions, where the order of operations matters.",{"id":1216,"type":1202,"conceptId":1217,"relation":1204,"explanation":1218},"conn-e-angles","angles","Degrees in a full turn (360) and minutes of arc come from Babylonian base 60.",{"id":1220,"type":1221,"title":1222,"points":1223},"cheat-e","summary","Cheat sheet",[1224,1225,1226,1227,1228,1229,1230,1231,1232,1233],"**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.",{"id":1235,"type":1236,"sourceIds":1237},"sources-e","sources",[1238,1239,1240,1241,1242,1243,1244,1245,1246],"number-system-ncert-class7-large-numbers","number-system-wiki-indian-numbering","number-system-wiki-binary","number-system-britannica-hindu-arabic","number-system-bipm-si-prefixes","number-system-ncert-class6-knowing-numbers","number-system-wiki-mars-orbiter","number-system-wiki-modi-stadium","number-system-wiki-indian-railways",[1238,1239,1240,1241,1242,1243,1244,1245,1246],"needs_review",{"generatedBy":1250,"notes":1251},"claude-code","Draft generated with Python-checked numbers; pending owner review.","bd90a2d5519af5cba3545d8b662b3ccdf097a5bc284e8cc40a398cb25b9192d8",{"component:place-value@1":1254,"logic:practice":1255,"component:match-pairs@1":1256,"component:sort-game@1":1257,"component:arith-sprint@1":1258,"component:rounding-race@1":1259,"source:number-system-bipm-si-prefixes":1260,"source:number-system-britannica-hindu-arabic":1261,"source:number-system-ncert-class6-knowing-numbers":1262,"source:number-system-ncert-class7-large-numbers":1263,"source:number-system-wiki-binary":1264,"source:number-system-wiki-indian-numbering":1265,"source:number-system-wiki-indian-railways":1266,"source:number-system-wiki-mars-orbiter":1267,"source:number-system-wiki-modi-stadium":1268},"cae1e81c06fcfcc0152eb325ac8fb01f568e21a19ad76a75d79194e17231ae72","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","4b1ef23b043f9642cf2f3613a60f4e05e4eb4a7c356f1100a1808edc05a4ecb3","a63f6c9a1c311ff55526f628c2aa874f9c1a5449b9b395e4d1a775681172821c","4e646d96c29f6b98469c9d11ee30da4e543f849b673fd067975aa685d8fef8a3","9dc5f1741ed9fad68db69882c1cc09c8d4b3cf5902d5d0273891bad1da1d10c0","e045e4dd865a933a76a0ef14e78db76641c9a727b1d34259d746cef789c5b001","045032c28b7bfe33f8bc73c5c09ee8dd8076293c90dbe922d8a461e059edd6e3","acfe2624fdde75e26fde9ffd53d3508825fe265b41a08a9787cc09866a9e2848","2d52d92be5022af4a35158dc5286cab1fad9e04133d70b8b08b5022b44b76e9a","372871233991a7b89e62e12c142d512f875d5a982e82368724e7199f5c4c6547",{"state":1270,"reviewer":1271,"selfReview":134,"reviewedAt":1272,"method":1273},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597799]