[{"data":1,"prerenderedAt":1270},["ShallowReactive",2],{"layer:number-system:discover":3},{"layer":4,"contentHash":1249,"dependencyHashes":1250,"approval":1264,"releaseId":1269},{"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":1244,"reviewStatus":1245,"authoring":1246},1,"number-system","en","discover","Big numbers all around us","Ten digits, a few clever places, and every number you will ever need","Meet place value through bundles of sticks, cricket crowds and rupee notes. Learn to read and write big numbers the Indian way (lakh, crore) and the international way (million, billion), find the number just before and after, compare, round and even read Roman numerals.",[13,14,15,16,17],"Explain how only ten digits can write any whole number, because each place is worth ten of the place to its right.","Read and write numbers up to crores and millions, with commas in the right places in both systems.","Tell the place value of a digit from its face value, and find the predecessor and successor of a number.","Compare two large numbers and round a number to the nearest ten or hundred.","Read simple Roman numerals such as those on clocks and in book chapters.",35,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Discover",{"label":26,"value":27},"Reading time","≈ 35 minutes",{"label":29,"value":30},"Prior knowledge","Counting to 1,000",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Place-value builder, match, sort, rounding race",{"label":38,"value":39},"Big idea","Where a digit sits decides its value",[41,45,51,57,60,81,86,128,157,162,167,170,191,196,201,225,230,235,238,266,276,279,304,317,322,327,347,350,359,363,383,387,398,403,406,428,431,440,444,464,481,485,490,493,517,537,546,554,558,601,604,618,623,626,631,641,668,688,693,696,719,730,742,753,758,778,787,797,815,857,862,865,881,895,899,911,915,926,937,942,945,974,977,982,993,1013,1017,1073,1197,1201,1207,1212,1217,1232],{"id":42,"type":43,"markdown":44},"intro-crowd","prose","Picture the Narendra Modi Stadium in Ahmedabad on the day of a big cricket final. Every seat is full: about **1,32,000** people are cheering. Now picture every one of those fans holding up a card with a number on it, and the numbers go 1, 2, 3, … all the way to the last fan. How would you *say* that last number? How would you *write* it so that someone else can read it at a glance?\n\nNumbers like this are everywhere in India. A train carries more than a thousand passengers. A new car costs several lakh rupees. India's 2011 census counted **1,21,08,54,977** people. This lesson is about how we manage to write all of these with just ten little symbols, 0 to 9.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"q-big","callout","question","Something to wonder about","Why do Indian newspapers write a big number as **12,34,56,789** while an American newspaper writes the same number as **123,456,789**? Are they different numbers? (They are not. By the end of this lesson you will know why the commas move.)",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","A world full of big numbers","Chapter 01","1 Big numbers",{"id":58,"type":43,"markdown":59},"where-big","Small numbers are easy: you can see 3 mangoes or 7 pencils at once. But once numbers grow past a few hundred, nobody can picture them directly. So people invented a *system*: a set of rules for writing numbers so that even huge ones can be read, compared and calculated with.\n\nHere are some real Indian numbers you will be able to read by the end of this lesson. Do not worry if they look scary now.",{"id":61,"type":62,"tone":63,"items":64},"spec-real","spec","blue",[65,69,73,77],{"label":66,"big":67,"value":68},"Cricket crowd","1,32,000","Approximate seating capacity of the Narendra Modi Stadium, Ahmedabad, one of the largest cricket grounds in the world.",{"label":70,"big":71,"value":72},"Everest","8,849 m","Height of Mount Everest: eight thousand eight hundred forty-nine metres.",{"label":74,"big":75,"value":76},"To the Moon","3,84,400 km","Average distance from Earth to the Moon: three lakh eighty-four thousand four hundred kilometres.",{"label":78,"big":79,"value":80},"Census 2011","1,21,08,54,977","India's population in the 2011 census: one hundred twenty-one crore, eight lakh, fifty-four thousand, nine hundred seventy-seven.",{"id":82,"type":47,"variant":83,"title":84,"markdown":85},"ex-shop","example","Big numbers at the kirana shop","Even a small shop deals with big numbers. If a kirana shop sells goods worth ₹4,500 a day, then in a year of 365 days it sells goods worth about 4,500 × 365 = **₹16,42,500**. That is more than sixteen lakh rupees, from a shop that fits in one room.",{"id":87,"type":88,"title":89,"prompt":90,"options":91},"explorer-where","explorer","Where big numbers live","Pick a part of life to see the kind of big numbers it uses.",[92,101,111,119],{"id":93,"label":94,"chain":95,"note":100},"money","Money",[96,97,98,99],"₹10 note","₹500 note","₹1 lakh","₹1 crore","Money is where most Indians meet lakh and crore. Salaries, house prices, loans and government budgets are almost always said in lakhs and crores: \"a ₹12 lakh flat\", \"a ₹500 crore project\". Business news in English often also gives the international form, so the same amount might appear as ₹500 crore and as ₹5 billion.",{"id":102,"label":103,"chain":104,"note":110},"people","People",[105,106,107,108,109],"Your class","Your school","A stadium crowd","A city","India","A class may have 40 students and a school 1,500. A stadium holds more than a lakh of fans. Big cities such as Mumbai and Delhi have more than one crore people each, and India as a whole has more than 140 crore. Populations are counted in a census, which India carries out about once every ten years.",{"id":112,"label":113,"chain":114,"note":118},"distance","Distance",[115,116,117,74],"Your classroom","Delhi to Mumbai","Around Earth","A classroom is about 10 m long. By road, Delhi to Mumbai is roughly 1,400 km. The distance all the way around Earth at the equator is about 40,000 km. The Moon is about 3,84,400 km away on average, and the Sun about 15 crore km. Space agencies such as ISRO work with distances like these every day.",{"id":120,"label":121,"chain":122,"note":127},"time","Time",[123,124,125,126],"A minute","A day","A year","A lifetime","A minute is 60 seconds. A day is 24 × 60 × 60 = 86,400 seconds. A year is 365 × 86,400 = 3,15,36,000 seconds, more than three crore. If you live to 80, your heart will beat roughly 300 crore (3 billion) times.",{"id":129,"type":130,"caption":131,"columns":132,"rows":136},"table-india-daily","table","Big numbers from an ordinary day in India (approximate)",[133,134,135],"What","About how many","Said as",[137,141,145,149,153],[138,139,140],"Passengers on Indian Railways in a day","2,00,00,000","about two crore",[142,143,144],"Babies born in India in a day","60,000 to 70,000","more than sixty thousand",[146,147,148],"Seconds in a day","86,400","eighty-six thousand four hundred",[150,151,152],"Heartbeats of one person in a day (about 70 a minute)","1,00,800","about one lakh",[154,155,156],"Litres of milk produced in India in a day","more than 50,00,00,000","more than fifty crore",{"id":158,"type":47,"variant":159,"title":160,"markdown":161},"careful-approx","careful","Approximate numbers are still useful","The first, second and last rows of this table are rough figures that change every year, so they are written with words like *about* and *more than*. Big real-life numbers are almost never exact, and a good reader notices when a number has been rounded.",{"id":163,"type":53,"title":164,"eyebrow":165,"navLabel":166},"ch02","Ten digits, endless numbers","Chapter 02","2 Ten digits",{"id":168,"type":43,"markdown":169},"ten-digits","Every number you have ever written uses only these ten **digits**: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. There is no special symbol for ten, or for a hundred, or for a crore. So how do we write ten? We write **10**: a 1 and a 0. The trick is *where* the digit sits.\n\nThink of bundling sticks. Count out loose sticks. When you reach ten, tie them into one bundle. Now you have *1 bundle and 0 loose sticks*, and that is exactly what 10 means. Keep going: when you have ten bundles, tie them into one big bundle of a hundred. The number 347 means **3 big bundles, 4 bundles and 7 loose sticks**.",{"id":171,"type":172,"title":173,"items":174},"steps-bundles","steps","From loose sticks to big bundles",[175,179,183,187],{"title":176,"tag":177,"text":178},"Ones","1","Single loose sticks. You can have 0 to 9 of them. The tenth one makes a bundle.",{"title":180,"tag":181,"text":182},"Tens","10 ones","A bundle of ten sticks. Ten bundles are tied together into the next size.",{"title":184,"tag":185,"text":186},"Hundreds","10 tens","A big bundle of one hundred sticks.",{"title":188,"tag":189,"text":190},"Thousands","10 hundreds","Ten big bundles, one thousand sticks. The pattern never stops: every new place is worth ten of the place before.",{"id":192,"type":47,"variant":193,"title":194,"markdown":195},"try-bundles","try_it","Bundle it yourself","Collect about 150 matchsticks, straws or ice-cream sticks and some rubber bands.\n\n1. Count out sticks one by one. Each time you reach ten, put a rubber band round them.\n2. When you have ten bundles, band them together into one big bundle.\n3. Write down how many big bundles, bundles and loose sticks you have. Those three digits, in that order, are the number of sticks.\n\nGrab a random handful and try again. Can you say how many there are *without* counting every stick?",{"id":197,"type":47,"variant":198,"title":199,"markdown":200},"aha-where","aha","Where a digit sits decides what it is worth","In **333**, all three digits are 3, but they are worth completely different amounts: the right-hand 3 is three ones, the middle 3 is three tens (thirty) and the left-hand 3 is three hundreds. The digits are identical; only their **places** differ. This one idea, called **place value**, is what makes our number system so powerful.",{"id":202,"type":203,"component":204,"componentVersion":5,"config":205,"objective":218,"textAlternative":219,"help":220},"lab-pv-small","interactive","place-value",{"places":206,"system":207,"initial":208,"show":209,"challenges":212},4,"indian",347,{"names":210,"expanded":210,"neighbours":211,"bothSystems":211},true,false,[213,214,215,216,217],52,408,1000,2305,9990,"Add and remove counters in the ones, tens, hundreds and thousands columns to build numbers, and see each number written in digits, words and expanded form.","The lab shows four columns labelled Thousands, Hundreds, Tens and Ones, with a counter tray under each. It starts at 347: 3 counters in hundreds, 4 in tens and 7 in ones. Below the columns you see the number in digits (347), in words (three hundred forty-seven) and in expanded form (300 + 40 + 7).\n\nAdding one counter to the tens column makes 357: only the tens digit changes. Adding a tenth counter to any column is not allowed; instead ten counters are swapped for one counter in the next column to the left (regrouping). For example, 7 ones plus 3 more ones become 1 ten and 0 ones.\n\nChallenges ask you to build 52, 408, 1,000, 2,305 and 9,990. For 408 you put 4 in hundreds, 0 in tens and 8 in ones: the empty tens column is why 408 needs a 0. For 1,000 you need just one counter, in the thousands column. For 9,990: 9 thousands, 9 hundreds, 9 tens and 0 ones.",{"simplerExplanation":221,"hints":222},"Each column can hold 0 to 9 counters. When you would reach ten, those ten turn into one counter in the column on the left.",[223,224],"Start from the left: how many thousands does the number have?","An empty column still needs a 0 when you write the number.",{"id":226,"type":47,"variant":227,"title":228,"markdown":229},"mis-zero","misconception","“Zero means nothing, so it does not matter”","Zero matters a great deal. **408** and **48** use the same non-zero digits, but 408 is four hundred eight and 48 is forty-eight. The 0 in 408 holds the tens place open, so the 4 stays in the hundreds place. A zero used like this is called a **placeholder**.",{"id":231,"type":53,"title":232,"eyebrow":233,"navLabel":234},"ch03","Ones, tens, hundreds, thousands and beyond","Chapter 03","3 The places",{"id":236,"type":43,"markdown":237},"places-grow","Each place is ten times the place on its right. Going right to left, the places are ones, tens, hundreds, thousands, ten thousands, and then… the names depend on whether you use the Indian system or the International system. We will meet both. For now, here are the first five places, which are the same everywhere.",{"id":239,"type":130,"caption":240,"columns":241,"rows":246},"table-first-places","The first five places. Each is worth ten of the place to its right.",[242,243,244,245],"Place","Worth","Smallest number with this many digits","Largest number with this many digits",[247,250,253,257,261],[176,177,248,249],"1 (1 digit)","9",[180,181,251,252],"10 (2 digits)","99",[184,254,255,256],"10 tens = 100","100 (3 digits)","999",[188,258,259,260],"10 hundreds = 1,000","1,000 (4 digits)","9,999",[262,263,264,265],"Ten thousands","10 thousands = 10,000","10,000 (5 digits)","99,999",{"id":267,"type":268,"title":269,"problem":270,"steps":271},"we-read-4","worked_example","Reading a four-digit number","Read the number 6,025 and say what each digit is worth.",[272,273,274,275],"Put the digits in their places: 6 in thousands, 0 in hundreds, 2 in tens, 5 in ones.","The 6 is worth 6 thousands = 6,000. The 0 means there are no hundreds. The 2 is worth 2 tens = 20. The 5 is worth 5.","Put it together: 6,000 + 20 + 5.","Read it as **six thousand twenty-five**. We skip the empty hundreds place in the name.",{"id":277,"type":43,"markdown":278},"digits-count","The number of **digits** tells you roughly how big a number is. Every 2-digit number is at least 10, every 3-digit number is at least 100 and every 4-digit number is at least 1,000. So a 4-digit number is always bigger than any 3-digit number, however large its digits are: 1,000 is more than 999.",{"id":280,"type":281,"itemId":282,"prompt":283,"check":284,"hints":299,"feedback":301},"pr-hundreds-digit","practice","number-system.discover-hundreds-digit","Which digit is in the **hundreds** place of 9,372?",{"kind":285,"options":286,"correct":298},"choice",[287,289,292,295],{"id":288,"label":249},"a",{"id":290,"label":291},"b","3",{"id":293,"label":294},"c","7",{"id":296,"label":297},"d","2",[290],[300],"Start counting places from the right-hand end.",{"correct":302,"incorrect":303},"Yes: from the right, 2 is ones, 7 is tens, 3 is hundreds and 9 is thousands.","Count from the right: ones (2), tens (7), hundreds (3). The hundreds digit is 3.",{"id":305,"type":281,"itemId":306,"prompt":307,"check":308,"hints":312,"feedback":314},"pr-count-2","number-system.discover-bundles","A teacher has 4 boxes of 1,000 chalk sticks, 7 packets of 100, no bundles of 10, and 3 loose sticks. How many chalk sticks is that altogether?",{"kind":309,"answer":310,"tolerance":311},"number",4703,0,[313],"Write one digit for each place, from thousands down to ones.",{"correct":315,"incorrect":316},"Right: 4 thousands, 7 hundreds, 0 tens and 3 ones make 4,703.","Write a digit for each place: thousands 4, hundreds 7, tens 0, ones 3. That gives 4,703. Do not forget the 0 for the empty tens place.",{"id":318,"type":53,"title":319,"eyebrow":320,"navLabel":321},"ch04","Face value and place value","Chapter 04","4 Face and place",{"id":323,"type":47,"variant":324,"title":325,"markdown":326},"def-fv-pv","definition","Two kinds of value","The **face value** of a digit is simply the digit itself, wherever it sits. The **place value** of a digit is what it is worth in its place: the digit multiplied by the value of the place.\n\nIn 4,**7**52 the face value of 7 is **7**, and its place value is 7 × 100 = **700**.",{"id":328,"type":130,"caption":329,"columns":330,"rows":334},"table-5555","Four fives, four different place values in 5,555",[331,332,242,333],"Digit (from the right)","Face value","Place value",[335,338,341,344],[336,337,176,337],"1st 5","5",[339,337,180,340],"2nd 5","50",[342,337,184,343],"3rd 5","500",[345,337,188,346],"4th 5","5,000",{"id":348,"type":43,"markdown":349},"fv-sum","Add up the four place values in the table: 5,000 + 500 + 50 + 5 = 5,555. That is no accident. **A number is always the sum of the place values of its digits.** Writing a number this way, as a sum, is called its **expanded form**.",{"id":351,"type":268,"title":352,"problem":353,"steps":354},"we-expanded","Writing a number in expanded form","Write 50,307 in expanded form.",[355,356,357,358],"Place each digit: 5 ten thousands, 0 thousands, 3 hundreds, 0 tens, 7 ones.","Place values: 5 × 10,000 = 50,000; 3 × 100 = 300; 7 × 1 = 7. The zeros are worth 0.","Expanded form: **{expanded(50307)}**.","Check by adding: 50,000 + 300 + 7 = 50,307. ✓",{"id":360,"type":47,"variant":227,"title":361,"markdown":362},"mis-face-place","“Face value and place value are the same thing”","A common test slip: asked for the **place value** of 6 in 4,**6**82, a student writes 6. That is the **face value**. The 6 sits in the hundreds place, so its place value is 6 × 100 = **600**.\n\nA quick check: the face value is always a single digit (0 to 9). The place value is that digit followed by as many zeros as there are places to its right. In 4,682 there are two digits to the right of the 6, so its place value is 6 followed by two zeros: 600.",{"id":364,"type":281,"itemId":365,"prompt":366,"check":367,"hints":378,"feedback":380},"pr-fv-pv","number-system.discover-face-vs-place","In **9,254**, which statement is correct?",{"kind":285,"options":368,"correct":377},[369,371,373,375],{"id":288,"label":370},"The face value of 2 is 200",{"id":290,"label":372},"The place value of 2 is 200",{"id":293,"label":374},"The place value of 9 is 9",{"id":296,"label":376},"The face value of 5 is 50",[290],[379],"Face value = the digit. Place value = the digit × its place.",{"correct":381,"incorrect":382},"Right: 2 is in the hundreds place, so its place value is 200 and its face value is 2.","Face values are single digits: 9, 2, 5, 4. Place values: 9,000, 200, 50 and 4. Only b is true.",{"id":384,"type":47,"variant":159,"title":385,"markdown":386},"careful-zero-pv","The place value of 0","The place value of 0 is always 0, wherever it sits: 0 × 1,000 is still 0. That is why expanded forms leave out zeros. The expanded form of 3,080 is 3,000 + 80.",{"id":388,"type":281,"itemId":389,"prompt":390,"check":391,"hints":393,"feedback":395},"pr-pv","number-system.discover-place-value","What is the place value of the digit 8 in **38,461**?",{"kind":309,"answer":392,"tolerance":311},8000,[394],"Which place is the 8 in? Count from the right: ones, tens, hundreds, thousands…",{"correct":396,"incorrect":397},"Yes: 8 is in the thousands place, so its place value is 8 × 1,000 = 8,000. Its face value is just 8.","Count places from the right: 1 is ones, 6 is tens, 4 is hundreds, 8 is thousands. So 8 is worth 8 × 1,000 = 8,000.",{"id":399,"type":53,"title":400,"eyebrow":401,"navLabel":402},"ch05","Lakhs and crores: the Indian way","Chapter 05","5 Lakh and crore",{"id":404,"type":43,"markdown":405},"lakh-intro","What comes after 99,999? Add one more and every 9 rolls over to 0, just like a car's odometer: **99,999 + 1 = 1,00,000**. In India we call this number **one lakh**. It is a hundred thousand.\n\nKeep going. Ten lakh is 10,00,000. A hundred lakh gets a new name: **one crore**, written 1,00,00,000. So in the Indian system, the places after ten thousands are **lakhs, ten lakhs, crores, ten crores**.",{"id":407,"type":130,"caption":408,"columns":409,"rows":413},"table-indian-chart","The Indian place-value chart",[410,411,412],"Period","Places","Value of 1 in the place",[414,418,422,425],[415,416,417],"Crores","Ten crores, crores","10,00,00,000 and 1,00,00,000",[419,420,421],"Lakhs","Ten lakhs, lakhs","10,00,000 and 1,00,000",[188,423,424],"Ten thousands, thousands","10,000 and 1,000",[176,426,427],"Hundreds, tens, ones","100, 10 and 1",{"id":429,"type":43,"markdown":430},"indian-commas","Look at how the chart is split into **periods**. The ones period has three places (hundreds, tens, ones). Every period after that has just **two** places. Commas separate the periods, so an Indian number gets its first comma after 3 digits from the right, and then a comma after every 2 digits.\n\nSo 12345678 is written **1,23,45,678** and read *one crore twenty-three lakh forty-five thousand six hundred seventy-eight*. Each comma tells you the name to say: crore, lakh, thousand.",{"id":432,"type":268,"title":433,"problem":434,"steps":435},"we-read-crore","Reading a crore-sized number","Put commas in 60708090 the Indian way and read it.",[436,437,438,439],"Count 3 digits from the right: 090. Then groups of 2: 08, 70, and 6 is left. So **6,07,08,090**.","Read each period with its name: 6 crore, 07 lakh, 08 thousand, 090.","Name: **six crore seven lakh eight thousand ninety**.","Notice the zeros inside periods: 07 lakh is said as seven lakh, and 090 is just ninety.",{"id":441,"type":47,"variant":227,"title":442,"markdown":443},"mis-commas-lakh","Commas in the wrong places","Two slips appear again and again when students write Indian numbers:\n\n1. **Groups of three all the way**, as in 12,345,678. That is the International way; the Indian form is **1,23,45,678**.\n2. **Counting commas from the left**, as in 12,34,56,78. Commas are always counted from the **right**: first a group of three (678), then groups of two.\n\nCheck yourself: in the Indian system the last group always has **three** digits and every other group has **two** (the first group may have one).",{"id":445,"type":281,"itemId":446,"prompt":447,"check":448,"hints":459,"feedback":461},"pr-commas","number-system.discover-indian-commas","Which is **4506070** written the Indian way?",{"kind":285,"options":449,"correct":458},[450,452,454,456],{"id":288,"label":451},"4,506,070",{"id":290,"label":453},"45,06,070",{"id":293,"label":455},"4,50,60,70",{"id":296,"label":457},"450,6070",[290],[460],"Count from the right: 3 digits first, then 2.",{"correct":462,"incorrect":463},"Right: three digits from the right (070), then two (06), leaving 45: 45,06,070, forty-five lakh six thousand seventy.","Start from the right: 070, then 06, then 45 → 45,06,070. Option a is the International way.",{"id":465,"type":203,"component":204,"componentVersion":5,"config":466,"objective":475,"textAlternative":476,"help":477},"lab-pv-indian",{"places":467,"system":207,"initial":468,"show":469,"challenges":470},7,250000,{"names":210,"expanded":210,"neighbours":210,"bothSystems":211},[471,472,473,474],100000,1500000,7050300,9999999,"Build numbers up to ninety-nine lakh in the Indian place-value chart and watch the commas and the number name change.","The lab has seven columns: Ten lakhs, Lakhs, Ten thousands, Thousands, Hundreds, Tens and Ones. It starts at 2,50,000: 2 counters in lakhs and 5 in ten thousands. The name reads *two lakh fifty thousand*.\n\nCommas appear after the hundreds place and then after every two places, so as the number grows you see 1,000, then 10,000, then 1,00,000, then 10,00,000. The number just before and just after are also shown: for 2,50,000 they are 2,49,999 and 2,50,001.\n\nChallenges: build 1,00,000 (one counter in lakhs), 15,00,000 (1 in ten lakhs, 5 in lakhs: fifteen lakh), 70,50,300 (7 in ten lakhs, 5 in ten thousands, 3 in hundreds: seventy lakh fifty thousand three hundred) and 99,99,999 (nine in every column: ninety-nine lakh ninety-nine thousand nine hundred ninety-nine, the largest seven-digit number).",{"simplerExplanation":478,"hints":479},"In the Indian system, count 3 digits from the right for the first comma, then put a comma after every 2 digits.",[480],"Fifteen lakh means 15 in the lakhs period: 1 in ten lakhs and 5 in lakhs.",{"id":482,"type":47,"variant":83,"title":483,"markdown":484},"ex-rupees","Lakhs and crores in the news","Indian newspapers, banks and shops use lakhs and crores all the time. A flat might cost ₹85 lakh (₹85,00,000). A film might earn ₹300 crore (₹3,00,00,00,000). A budget of ₹1 lakh crore means 1,00,000 × 1,00,00,000 rupees: a 1 followed by 12 zeros.",{"id":486,"type":53,"title":487,"eyebrow":488,"navLabel":489},"ch06","Millions: the international way","Chapter 06","6 Millions",{"id":491,"type":43,"markdown":492},"intl-intro","Most of the world groups digits in **threes**, all the way. After thousands and ten thousands comes **hundred thousands**, and then a new period: **millions**. A thousand thousands make a million, 1,000,000. A thousand millions make a **billion**, 1,000,000,000.\n\nSo in the International system the periods are **ones, thousands, millions, billions**, and every period has three places. The commas go after every three digits from the right: 12345678 is written **12,345,678** and read *twelve million three hundred forty-five thousand six hundred seventy-eight*.",{"id":494,"type":130,"caption":495,"columns":496,"rows":500},"table-both","The same numbers in both systems",[497,498,499],"Digits","Indian","International",[501,505,509,513],[502,503,504],"100000","1,00,000 (one lakh)","100,000 (one hundred thousand)",[506,507,508],"1000000","10,00,000 (ten lakh)","1,000,000 (one million)",[510,511,512],"10000000","1,00,00,000 (one crore)","10,000,000 (ten million)",[514,515,516],"100000000","10,00,00,000 (ten crore)","100,000,000 (one hundred million)",{"id":518,"type":62,"tone":519,"items":520},"spec-equal","amber",[521,525,529,533],{"label":522,"big":523,"value":524},"1 lakh","100 thousand","1,00,000 = 100,000",{"label":526,"big":527,"value":528},"10 lakh","1 million","10,00,000 = 1,000,000",{"label":530,"big":531,"value":532},"1 crore","10 million","1,00,00,000 = 10,000,000",{"label":534,"big":535,"value":536},"100 crore","1 billion","1,00,00,00,000 = 1,000,000,000",{"id":538,"type":268,"title":539,"problem":540,"steps":541},"we-convert-disc","From crores to millions","A company says it made a profit of ₹4,50,00,000. Write this the international way and say it in millions.",[542,543,544,545],"Remove the commas: 45000000.","Put commas every 3 digits from the right: **45,000,000**.","Read it: forty-five million. In the Indian system it is four crore fifty lakh.","Check with the shortcut: 1 crore = 10 million, so 4.5 crore = 45 million. ✓",{"id":547,"type":268,"title":548,"problem":549,"steps":550},"we-mill-lakh","From millions back to lakhs","A city has 2.5 million people. How many lakh is that?",[551,552,553],"Use the key fact: 1 million = 10 lakh.","2.5 million = 2.5 × 10 lakh = **25 lakh**.","In digits: 2.5 million = 2,500,000 = 25,00,000, which reads twenty-five lakh. ✓",{"id":555,"type":47,"variant":227,"title":556,"markdown":557},"mis-lakh-million","“A lakh is a million”","It is not. A lakh has **5** zeros (1,00,000), and a million has **6** zeros (1,000,000). A million is **ten** lakh. In the same way, a crore is ten million, not one million. Mixing these up makes an answer ten times too big or too small.",{"id":559,"type":560,"title":561,"note":562,"scale":563,"rungs":564},"ladder-sizes","ladder","From one to one billion","Log scale: each rung is ten times the rung below. Indian and international names side by side.","log",[565,567,571,575,578,582,585,589,593,597],{"label":566,"value":5,"display":177},"One",{"label":568,"value":569,"display":570},"Ten",10,"10",{"label":572,"value":573,"display":574},"Hundred",100,"100",{"label":576,"value":215,"display":577},"Thousand","1,000",{"label":579,"value":580,"display":581},"Ten thousand",10000,"10,000",{"label":583,"value":471,"display":584},"1 lakh = 100 thousand","1,00,000",{"label":586,"value":587,"display":588},"10 lakh = 1 million",1000000,"10,00,000",{"label":590,"value":591,"display":592},"1 crore = 10 million",10000000,"1,00,00,000",{"label":594,"value":595,"display":596},"10 crore = 100 million",100000000,"10,00,00,000",{"label":598,"value":599,"display":600},"100 crore = 1 billion",1000000000,"1,00,00,00,000",{"id":602,"type":43,"markdown":603},"why-commas","Why bother with commas at all? Try reading 100000000 quickly. Now try 10,00,00,000. The commas chop a long row of digits into small groups your eyes can count at a glance, and each comma is a signpost that tells you which name to say next. Both systems do the same job; they just put the signposts in different places.",{"id":605,"type":606,"prompt":607,"options":608,"explanation":617},"pred-commas","prediction","The number 50000000 has no commas. Which of these is the **Indian** way of writing it?",[609,611,613,615],{"id":288,"label":610},"50,000,000",{"id":290,"label":612},"5,00,00,000",{"id":293,"label":614},"500,00,000",{"id":296,"label":616},"5,000,0000","**b: 5,00,00,000.** First comma after three digits from the right (000), then after every two digits (00, 00), leaving 5 at the front. It reads *five crore*. The international way is **a: 50,000,000**, read *fifty million*. Same number, same digits, different commas: 5 crore = 50 million.",{"id":619,"type":53,"title":620,"eyebrow":621,"navLabel":622},"ch07","Saying and writing number names","Chapter 07","7 Number names",{"id":624,"type":43,"markdown":625},"names-rule","To read a big number, read it **one period at a time**, from the left, and say the period's name at each comma. Periods that are all zeros are skipped.\n\n- Indian: 4,05,00,309 → *four crore* | *five lakh* | (no thousands) | *three hundred nine*.\n- International: 40,500,309 → *forty million* | *five hundred thousand* | *three hundred nine*.\n\nIn number names we say *lakh*, *crore* and *million* without an **s**: *five lakh*, not *five lakhs*. (When we name a **place**, like the lakhs place, we do use the plural.)",{"id":627,"type":47,"variant":628,"title":629,"markdown":630},"nuance-and","nuance","To “and” or not to “and”","People often say \"one hundred **and** five\". In this lesson we write number names without \"and\" (one hundred five), which is common in Indian textbooks, but both are understood. What matters is that each period and its name are clear. On a cheque, the amount in words is written so carefully because it must be impossible to change.",{"id":632,"type":268,"title":633,"problem":634,"steps":635},"we-names","Reading one number both ways","Write the number 3060025 with commas and in words in both systems.",[636,637,638,639,640],"Indian commas: first after 3 digits, then every 2 → **30,60,025**.","Read period by period: 30 lakh, 60 thousand, 25 → *thirty lakh sixty thousand twenty-five*.","International commas: every 3 digits → **3,060,025**.","Read: 3 million, 60 thousand, 25 → *three million sixty thousand twenty-five*.","Check: 30 lakh = 3 million (because 10 lakh = 1 million). ✓",{"id":642,"type":203,"component":643,"componentVersion":5,"config":644,"objective":666,"textAlternative":667},"lab-match-names","match-pairs",{"prompt":645,"mode":646,"pairs":647},"Match each number to its name.","connect",[648,651,654,657,660,663],{"a":649,"b":650},"70,05,000","Seventy lakh five thousand",{"a":652,"b":653},"2,50,00,000","Two crore fifty lakh",{"a":655,"b":656},"44,00,000","Forty-four lakh",{"a":658,"b":659},"1,000,000","One million",{"a":661,"b":662},"100,000,000","One hundred million",{"a":664,"b":665},"60,600","Sixty thousand six hundred","Match numbers written with commas to their names in the Indian and International systems.","The lab shows six numbers on the left and six number names on the right, shuffled. Tap a number and then its name to connect them.\n\nThe correct pairs are: 70,05,000 is *seventy lakh five thousand*; 2,50,00,000 is *two crore fifty lakh*; 44,00,000 is *forty-four lakh*; 1,000,000 is *one million*; 100,000,000 is *one hundred million*; 60,600 is *sixty thousand six hundred*.\n\nThe trick is to read the commas: in Indian numbers the commas mark thousand, lakh and crore; in international numbers they mark thousand, million and billion. Watch out for zeros: in 70,05,000 the 0 in the ten lakhs place and the 0 in ten thousands make the name *seventy lakh five thousand*, not *seven lakh fifty thousand*.",{"id":669,"type":281,"itemId":670,"prompt":671,"check":672,"hints":683,"feedback":685},"pr-name-digits","number-system.discover-name-to-digits","Which number is **two crore thirty lakh six hundred**?",{"kind":285,"options":673,"correct":682},[674,676,678,680],{"id":288,"label":675},"2,30,00,600",{"id":290,"label":677},"2,03,00,600",{"id":293,"label":679},"23,00,600",{"id":296,"label":681},"2,30,06,000",[288],[684],"Write the crore, lakh, thousand and ones periods separately, then join them.",{"correct":686,"incorrect":687},"Right: 2 crore, then 30 lakh, then 00 thousand, then 600: 2,30,00,600.","Fill each period: crore 2, lakh 30, thousand 00, ones period 600. That gives 2,30,00,600.",{"id":689,"type":53,"title":690,"eyebrow":691,"navLabel":692},"ch08","One before, one after","Chapter 08","8 Before and after",{"id":694,"type":43,"markdown":695},"pred-succ","The number that comes **just after** a number is its **successor**: add 1. The number that comes **just before** is its **predecessor**: subtract 1.\n\n- The successor of 4,599 is 4,600.\n- The predecessor of 4,600 is 4,599.\n- The predecessor of 1 is 0. Among whole numbers, 0 has no predecessor.\n\nWatch out when a number ends in 9s or 0s: adding or taking away 1 can change several digits at once, like a car's odometer rolling over.",{"id":697,"type":130,"caption":698,"columns":699,"rows":703},"table-neighbours","Predecessors and successors, including some tricky rollovers",[700,701,702],"Predecessor (− 1)","Number","Successor (+ 1)",[704,708,710,712,716],[705,706,707],"4,599","4,600","4,601",[260,581,709],"10,001",[711,265,584],"99,998",[713,714,715],"2,49,999","2,50,000","2,50,001",[717,588,718],"9,99,999","10,00,001",{"id":720,"type":281,"itemId":721,"prompt":722,"check":723,"hints":725,"feedback":727},"pr-succ-roll","number-system.discover-successor-rollover","What is the **successor** of 1,09,999?",{"kind":309,"answer":724,"tolerance":311},110000,[726],"Which digit is the first one that is not a 9, counting from the right?",{"correct":728,"incorrect":729},"Right: 1,09,999 + 1 = 1,10,000. The four 9s roll over and the 0 in the ten thousands place becomes 1.","Add 1 to the ones: 9 + 1 makes ten, which carries through all the 9s to the ten thousands place: 1,10,000.",{"id":731,"type":606,"prompt":732,"options":733,"explanation":741},"pred-roll","What is the successor of **9,99,999**?",[734,736,737,739],{"id":288,"label":735},"9,99,1000",{"id":290,"label":588},{"id":293,"label":738},"99,99,999",{"id":296,"label":740},"9,99,998","**b: 10,00,000 (ten lakh).** Adding 1 to the ones place makes ten ones, which become one ten; that makes ten tens, which become one hundred… and the rollover runs all the way to the left. The largest six-digit number plus 1 is the **smallest seven-digit number**. In the international system: 999,999 + 1 = 1,000,000, one million.",{"id":743,"type":281,"itemId":744,"prompt":745,"check":746,"hints":748,"feedback":750},"pr-pred","number-system.discover-predecessor","What is the **predecessor** of 3,00,000?",{"kind":309,"answer":747,"tolerance":311},299999,[749],"Take away 1. All the zeros borrow and turn into 9s.",{"correct":751,"incorrect":752},"Right: 3,00,000 − 1 = 2,99,999. Every 0 turned into a 9 and the 3 became 2.","Predecessor means one less: 3,00,000 − 1 = 2,99,999 (two lakh ninety-nine thousand nine hundred ninety-nine).",{"id":754,"type":53,"title":755,"eyebrow":756,"navLabel":757},"ch09","Which number is bigger?","Chapter 09","9 Comparing",{"id":759,"type":172,"title":760,"items":761},"steps-compare","Comparing two whole numbers",[762,766,770,774],{"title":763,"tag":764,"text":765},"Count the digits","step 1","The number with more digits is bigger. 1,00,000 (6 digits) is bigger than 99,999 (5 digits).",{"title":767,"tag":768,"text":769},"Same count? Look left","step 2","Compare the leftmost digits. Bigger leftmost digit, bigger number: 5,20,000 is more than 4,99,999.",{"title":771,"tag":772,"text":773},"Still equal? Move right","step 3","If the leftmost digits match, compare the next digits, and so on until they differ.",{"title":775,"tag":776,"text":777},"Write the sign","step 4","Use the symbols: 72,418 > 72,398 (greater than), 45,000 = 45,000, 38,200 \u003C 38,290 (less than).",{"id":779,"type":268,"title":780,"problem":781,"steps":782},"we-compare","Which city is bigger?","Town A has 2,45,806 people and Town B has 2,45,860 people. Which has more?",[783,784,785,786],"Both have 6 digits, so step 1 does not decide.","Compare place by place from the left: 2 = 2, 4 = 4, 5 = 5, 8 = 8.","Tens place: 0 in Town A, 6 in Town B. 6 is bigger.","So **2,45,860 > 2,45,806**: Town B has more people, by 2,45,860 − 2,45,806 = 54.",{"id":788,"type":268,"title":789,"problem":790,"steps":791},"we-order","Putting mountains in order","Arrange these heights of Himalayan peaks from greatest to smallest (descending order): Nanda Devi 7,816 m, Kangchenjunga 8,586 m, Everest 8,849 m, Makalu 8,485 m.",[792,793,794,795,796],"All four have 4 digits, so compare the thousands digits: 7, 8, 8, 8. Nanda Devi (7,816) is the smallest.","For the three with 8 thousands, compare hundreds: Kangchenjunga 5, Everest 8, Makalu 4.","So Everest (8 hundreds) is greatest, then Kangchenjunga (5), then Makalu (4).","**Descending order: 8,849 > 8,586 > 8,485 > 7,816**, that is Everest, Kangchenjunga, Makalu, Nanda Devi.","Ascending order (smallest first) is the same list backwards.",{"id":798,"type":281,"itemId":799,"prompt":800,"check":801,"hints":810,"feedback":812},"pr-compare-disc","number-system.discover-compare","Which sign goes in the box? **3,45,678 ☐ 3,45,687**",{"kind":285,"options":802,"correct":809},[803,805,807],{"id":288,"label":804},">",{"id":290,"label":806},"\u003C",{"id":293,"label":808},"=",[290],[811],"Find the first place where the digits differ.",{"correct":813,"incorrect":814},"Right: the numbers agree until the tens place, where 7 is less than 8, so 3,45,678 \u003C 3,45,687.","Compare from the left: 3, 4, 5, 6 all match. Tens: 7 against 8. So the first number is smaller: \u003C.",{"id":816,"type":203,"component":817,"componentVersion":5,"config":818,"objective":855,"textAlternative":856},"lab-sort-lakh","sort-game",{"prompt":819,"bins":820,"items":827,"seconds":311},"Sort each number: is it less than one lakh, or one lakh or more?",[821,824],{"id":822,"label":823},"small","Less than 1 lakh",{"id":825,"label":826},"big","1 lakh or more",[828,831,834,838,842,845,848,851],{"id":829,"label":265,"bin":822,"why":830},"s1","99,999 has 5 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is less.",{"id":832,"label":584,"bin":825,"why":833},"s2","1,00,000 has 6 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is at least one lakh.",{"id":835,"label":836,"bin":822,"why":837},"s3","98,765","98,765 has 5 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is less.",{"id":839,"label":840,"bin":825,"why":841},"s4","1,00,001","1,00,001 has 6 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is at least one lakh.",{"id":843,"label":714,"bin":825,"why":844},"s5","2,50,000 has 6 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is at least one lakh.",{"id":846,"label":260,"bin":822,"why":847},"s6","9,999 has 4 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is less.",{"id":849,"label":588,"bin":825,"why":850},"s7","10,00,000 has 7 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is at least one lakh.",{"id":852,"label":853,"bin":822,"why":854},"s8","90,000","90,000 has 5 digits; one lakh (1,00,000) is the smallest 6-digit number, so this is less.","Sort numbers into “less than one lakh” and “one lakh or more” by counting digits.","The game shows eight number cards and two bins: *Less than 1 lakh* and *1 lakh or more*. The quick rule is to count digits: 1,00,000 is the smallest six-digit number, so every number with 5 or fewer digits is less than one lakh, and every number with 6 or more digits is at least one lakh.\n\nLess than 1 lakh: 99,999, 98,765, 9,999 and 90,000. One lakh or more: 1,00,000, 1,00,001, 2,50,000 and 10,00,000. The tricky pair is 99,999 and 1,00,000: they differ by just 1, but one has five digits and the other six.",{"id":858,"type":53,"title":859,"eyebrow":860,"navLabel":861},"ch10","About how many? Rounding","Chapter 10","10 Rounding",{"id":863,"type":43,"markdown":864},"round-intro","Often we do not need an exact number. \"About 1,30,000 people watched the final\" is easier to remember than 1,32,014. Choosing a nearby, simpler number is called **rounding**.\n\nTo round 47 to the nearest ten, ask: is 47 closer to 40 or to 50? It is 7 away from 40 and 3 away from 50, so it rounds to **50**. A number exactly in the middle, like 45, is rounded **up** to 50 by the usual school rule.",{"id":866,"type":172,"title":867,"items":868},"steps-round","Rounding to the nearest ten or hundred",[869,872,875,878],{"title":870,"text":871},"Find the two neighbours","For 368 to the nearest hundred: it lies between 300 and 400.",{"title":873,"text":874},"Look at the next digit to the right","For hundreds, look at the tens digit. In 368 it is 6.",{"title":876,"text":877},"0–4: round down; 5–9: round up","The tens digit 6 is 5 or more, so round up to 400.",{"title":879,"text":880},"Check on a number line","368 is 68 away from 300 but only 32 away from 400. 400 is closer. ✓",{"id":882,"type":203,"component":883,"componentVersion":5,"config":884,"objective":889,"textAlternative":890,"help":891},"lab-round-disc","rounding-race",{"roundTo":885,"range":886,"rounds":888,"secondsPerRound":311},[569,573],{"min":569,"max":887},999,8,"Round numbers up to 999 to the nearest ten or hundred by picking the closer mark on a number line.","Each round shows a number between 10 and 999 on a number line, with the two nearest tens (or hundreds) marked at the ends. You tap the mark it rounds to. There is no timer; each correct answer scores points and builds a streak.\n\nExample: 263 to the nearest ten sits between 260 and 270; the ones digit 3 is less than 5, so it rounds down to 260. The same 263 to the nearest hundred sits between 200 and 300; the tens digit 6 is 5 or more, so it rounds up to 300. A number such as 750 is exactly halfway between 700 and 800; by the usual rule it rounds up to 800.",{"hints":892},[893,894],"Look only at the digit just to the right of the place you are rounding to.","5 or more: round up. 4 or less: round down.",{"id":896,"type":47,"variant":83,"title":897,"markdown":898},"ex-headline","Rounding in the headlines","A news report says \"Nearly 1.3 lakh fans packed the stadium\". The real ticket count might have been 1,28,742. Rounding to the nearest thousand gives 1,29,000, and to the nearest ten thousand gives 1,30,000, which reporters say as \"1.3 lakh\". Rounded numbers are easier to read and remember; exact numbers matter when you are selling tickets.",{"id":900,"type":606,"prompt":901,"options":902,"explanation":910},"pred-45","Round **45** to the nearest ten. It is exactly halfway between 40 and 50. What does the usual school rule say?",[903,905,906,908],{"id":288,"label":904},"40",{"id":290,"label":340},{"id":293,"label":907},"It cannot be rounded",{"id":296,"label":909},"45 stays 45","**b: 50.** When the digit to the right is 5, we round **up**. This is only an agreement (a convention), because 45 is equally close to both. Everyone using the same rule means everyone gets the same answer.",{"id":912,"type":47,"variant":227,"title":913,"markdown":914},"mis-carry-9","Rounding numbers full of 9s","Round **996** to the nearest ten. The ones digit is 6, so we round **up**: 99 tens + 1 ten = 100 tens = **1000**. Students often write 9,100 or 990 here. When the digit you are rounding is a 9 and you round up, it becomes 0 and you **carry** 1 to the next place, just as in addition. The answer can even gain a digit.",{"id":916,"type":281,"itemId":917,"prompt":918,"check":919,"hints":921,"feedback":923},"pr-round-ten","number-system.discover-round-ten","Round **2,398** to the nearest ten.",{"kind":309,"answer":920,"tolerance":311},2400,[922],"Round up, and carry if the tens digit is 9.",{"correct":924,"incorrect":925},"Right: the ones digit is 8, so round up. 239 tens + 1 ten = 240 tens = 2,400.","The ones digit 8 means round up. The tens digit 9 becomes 0 and carries: 2,398 → 2,400.",{"id":927,"type":281,"itemId":928,"prompt":929,"check":930,"hints":932,"feedback":934},"pr-round","number-system.discover-round-hundred","Round **7,451** to the nearest hundred.",{"kind":309,"answer":931,"tolerance":311},7500,[933],"Which two hundreds does 7,451 lie between?",{"correct":935,"incorrect":936},"Yes: 7,451 lies between 7,400 and 7,500. The tens digit is 5, so round up to 7,500.","Look at the tens digit, 5. That means round up: 7,451 → 7,500.",{"id":938,"type":53,"title":939,"eyebrow":940,"navLabel":941},"ch11","Numbers the Romans wrote","Chapter 11","11 Roman numerals",{"id":943,"type":43,"markdown":944},"roman-intro","Look at an old clock face, or the chapter numbers in some books, or \"Class VI\" on a school notebook. These are **Roman numerals**, used in ancient Rome about 2,000 years ago. They use letters instead of digits.",{"id":946,"type":130,"caption":947,"columns":948,"rows":952},"table-roman","The seven Roman symbols",[949,950,951],"Symbol","Value","A way to remember it",[953,956,959,962,965,968,971],[954,177,955],"I","One finger held up",[957,337,958],"V","The V between thumb and fingers of an open hand",[960,570,961],"X","Two Vs, one on top of the other",[963,340,964],"L","Half of a hundred",[966,574,967],"C","Latin *centum*, a hundred (as in century)",[969,343,970],"D","Half of a thousand",[972,577,973],"M","Latin *mille*, a thousand (as in millimetre)",{"id":975,"type":43,"markdown":976},"roman-rules","The rules are simple to start with:\n\n- **Add** symbols written from biggest to smallest: VI = 5 + 1 = 6, XXV = 10 + 10 + 5 = 25, CLX = 100 + 50 + 10 = 160.\n- **Subtract** a smaller symbol written just before a bigger one: IV = 5 − 1 = 4, IX = 10 − 1 = 9, XL = 50 − 10 = 40.\n- A symbol is not repeated more than three times: 3 is III, but 4 is IV, not IIII.\n\nSo the classes at school are I, II, III, IV, V, VI, VII, VIII, IX, X. And this year, 2026, is **MMXXVI**.",{"id":978,"type":47,"variant":979,"title":980,"markdown":981},"obs-clock","observation","A clock surprise","Many clock faces show 4 o'clock as **IIII** instead of IV. Clockmakers have done this for centuries; nobody is completely sure why. One common explanation is that IIII balances VIII on the other side of the dial. In school Roman numerals, 4 is written IV.",{"id":983,"type":281,"itemId":984,"prompt":985,"check":986,"hints":988,"feedback":990},"pr-roman","number-system.discover-roman","What number is the Roman numeral **XIV**?",{"kind":309,"answer":987,"tolerance":311},14,[989],"A small symbol before a bigger one is subtracted.",{"correct":991,"incorrect":992},"Right: X = 10, then IV = 4, so XIV = 14.","Split it: X is 10. Then I is before V, so IV = 5 − 1 = 4. Total 10 + 4 = 14.",{"id":994,"type":281,"itemId":995,"prompt":996,"check":997,"hints":1008,"feedback":1010},"pr-roman-write","number-system.discover-roman-write","How is **19** written in Roman numerals?",{"kind":285,"options":998,"correct":1007},[999,1001,1003,1005],{"id":288,"label":1000},"XVIIII",{"id":290,"label":1002},"IXX",{"id":293,"label":1004},"XIX",{"id":296,"label":1006},"XVIV",[293],[1009],"Split 19 into 10 + 9.",{"correct":1011,"incorrect":1012},"Yes: 19 = 10 + 9 = X + IX = XIX.","19 is 10 + 9. Ten is X and nine is IX, so 19 is XIX. IXX is not allowed, and XVIIII repeats I too often.",{"id":1014,"type":47,"variant":628,"title":1015,"markdown":1016},"mis-roman-zero","Why we do not use Roman numerals for sums","Roman numerals have **no zero** and **no place value**: an X is worth ten wherever it stands. That makes big numbers long (3,888 is MMMDCCCLXXXVIII, fifteen symbols) and column addition impossible. Our digits, invented in India, won because place value makes both writing and calculating easy.",{"id":1018,"type":1019,"title":1020,"terms":1021},"gloss-discover","glossary","Words for big numbers",[1022,1026,1029,1032,1035,1039,1041,1044,1047,1050,1053,1057,1061,1065,1069],{"term":1023,"meaning":1024,"example":1025},"Digit","One of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 used to write numbers.","472 has three digits: 4, 7 and 2.",{"term":242,"meaning":1027,"example":1028},"A position in a number, such as ones, tens or hundreds. Each place is worth ten times the place to its right.","In 472, the 4 is in the hundreds place.",{"term":333,"meaning":1030,"example":1031},"What a digit is worth because of its place: the digit times the value of the place.","In 472 the place value of 4 is 400.",{"term":332,"meaning":1033,"example":1034},"The value of the digit itself, wherever it is.","In 472 the face value of 4 is 4.",{"term":1036,"meaning":1037,"example":1038},"Expanded form","A number written as the sum of the place values of its digits.","472 = 400 + 70 + 2",{"term":410,"meaning":1040},"A group of places separated by a comma: ones, thousands, lakhs, crores (Indian) or ones, thousands, millions, billions (International).",{"term":1042,"meaning":1043},"Lakh","One hundred thousand, 1,00,000. Ten lakh make one million.",{"term":1045,"meaning":1046},"Crore","One hundred lakh, 1,00,00,000. Equal to ten million.",{"term":1048,"meaning":1049},"Million","One thousand thousand, 1,000,000. Equal to ten lakh.",{"term":1051,"meaning":1052},"Billion","One thousand million, 1,000,000,000. Equal to one hundred crore.",{"term":1054,"meaning":1055,"example":1056},"Successor","The number just after a number: add 1.","The successor of 999 is 1,000.",{"term":1058,"meaning":1059,"example":1060},"Predecessor","The number just before a number: subtract 1.","The predecessor of 1,000 is 999.",{"term":1062,"meaning":1063,"example":1064},"Rounding","Replacing a number with a nearby simpler number, such as the nearest ten or hundred.","47 rounds to 50.",{"term":1066,"meaning":1067,"example":1068},"Roman numerals","The Roman way of writing numbers with the letters I, V, X, L, C, D and M.","XII = 12",{"term":1070,"meaning":1071,"example":1072},"Placeholder","A zero that keeps other digits in their correct places.","The 0 in 408.",{"id":1074,"type":1075,"title":1076,"questions":1077},"quiz-discover","quiz","Check your big-number sense",[1078,1091,1102,1112,1122,1135,1148,1160,1173,1184],{"itemId":1079,"prompt":1080,"options":1081,"correct":293,"why":1090},"number-system.disc-q-pv","What is the place value of 6 in 2,61,450?",[1082,1084,1086,1088],{"id":288,"label":1083},"6",{"id":290,"label":1085},"6,000",{"id":293,"label":1087},"60,000",{"id":296,"label":1089},"6,00,000","Places from the right: 0 ones, 5 tens, 4 hundreds, 1 thousands, 6 ten thousands. So 6 × 10,000 = 60,000.",{"itemId":1092,"prompt":1093,"options":1094,"correct":288,"why":1101},"number-system.disc-q-fv","What is the face value of 6 in 2,61,450?",[1095,1096,1098,1099],{"id":288,"label":1083},{"id":290,"label":1097},"60",{"id":293,"label":1087},{"id":296,"label":1100},"0","Face value is just the digit itself: 6.",{"itemId":1103,"prompt":1104,"options":1105,"correct":290,"why":1111},"number-system.disc-q-lakh","How many zeros does one lakh have when written in digits?",[1106,1108,1109,1110],{"id":288,"label":1107},"4",{"id":290,"label":337},{"id":293,"label":1083},{"id":296,"label":294},"One lakh is 1,00,000: a 1 followed by five zeros.",{"itemId":1113,"prompt":1114,"options":1115,"correct":290,"why":1121},"number-system.disc-q-million","One million is equal to…",[1116,1117,1118,1119],{"id":288,"label":522},{"id":290,"label":526},{"id":293,"label":530},{"id":296,"label":1120},"100 lakh","10 lakh = 10,00,000 = 1,000,000 = one million.",{"itemId":1123,"prompt":1124,"options":1125,"correct":290,"why":1134},"number-system.disc-q-commas","Which is the Indian way to write 7654321?",[1126,1128,1130,1132],{"id":288,"label":1127},"7,654,321",{"id":290,"label":1129},"76,54,321",{"id":293,"label":1131},"765,4321",{"id":296,"label":1133},"7,65,43,21","First comma after three digits (321), then every two digits: 76,54,321.",{"itemId":1136,"prompt":1137,"options":1138,"correct":290,"why":1147},"number-system.disc-q-succ","What is the successor of 49,999?",[1139,1141,1143,1145],{"id":288,"label":1140},"49,998",{"id":290,"label":1142},"50,000",{"id":293,"label":1144},"40,000",{"id":296,"label":1146},"59,999","49,999 + 1 = 50,000. The 9s roll over to 0s.",{"itemId":1149,"prompt":1150,"options":1151,"correct":290,"why":1159},"number-system.disc-q-compare","Which number is the greatest?",[1152,1154,1156,1158],{"id":288,"label":1153},"98,999",{"id":290,"label":1155},"1,00,100",{"id":293,"label":1157},"1,00,010",{"id":296,"label":265},"The two six-digit numbers are bigger than the five-digit ones. Comparing 1,00,100 and 1,00,010: the hundreds digits are 1 and 0, so 1,00,100 is greater.",{"itemId":1161,"prompt":1162,"options":1163,"correct":288,"why":1172},"number-system.disc-q-round","Round 849 to the nearest hundred.",[1164,1166,1168,1170],{"id":288,"label":1165},"800",{"id":290,"label":1167},"850",{"id":293,"label":1169},"900",{"id":296,"label":1171},"840","The tens digit is 4, which is less than 5, so round down to 800. (849 is 49 from 800 and 51 from 900.)",{"itemId":1174,"prompt":1175,"options":1176,"correct":290,"why":1183},"number-system.disc-q-roman","What is XL in our numerals?",[1177,1178,1179,1181],{"id":288,"label":1097},{"id":290,"label":904},{"id":293,"label":1180},"15",{"id":296,"label":1182},"110","X (10) before L (50) is subtracted: 50 − 10 = 40.",{"itemId":1185,"prompt":1186,"options":1187,"correct":288,"why":1196},"number-system.disc-q-name","Which name matches 3,05,000?",[1188,1190,1192,1194],{"id":288,"label":1189},"Three lakh five thousand",{"id":290,"label":1191},"Thirty lakh fifty thousand",{"id":293,"label":1193},"Three lakh fifty thousand",{"id":296,"label":1195},"Three crore five thousand","Read by periods: 3 lakh, 05 thousand, 000. So three lakh five thousand.",{"id":1198,"type":1199,"prompt":1200},"reflect-disc","reflection","Find three big numbers in a newspaper, on a food packet or on a signboard. Write each one in words, and say whether it uses Indian or international commas. Which was hardest to read, and why?",{"id":1202,"type":1203,"conceptId":1204,"relation":1205,"explanation":1206},"conn-ops","connection","four-operations","helps_understand","Column addition, subtraction and carrying all work because of place value: carrying is just swapping ten of one place for one of the next.",{"id":1208,"type":1203,"conceptId":1209,"relation":1210,"explanation":1211},"conn-patterns","patterns","related_to","Counting in tens, hundreds and thousands, and the 9s rolling over to 0s, are number patterns you can predict.",{"id":1213,"type":1203,"conceptId":1214,"relation":1215,"explanation":1216},"conn-data","data-handling","applied_in","Tables of real data, such as populations or cricket scores, are full of large numbers that need to be read, compared and rounded.",{"id":1218,"type":1219,"title":1220,"points":1221},"cheat-disc","summary","Cheat sheet",[1222,1223,1224,1225,1226,1227,1228,1229,1230,1231],"**Ten digits** (0–9) write every whole number because **place** decides value: each place is worth ten of the place to its right.","**Place value** = digit × value of its place. **Face value** = the digit itself. The place value of 0 is always 0.","**Expanded form** writes a number as the sum of its place values: 6,025 = 6,000 + 20 + 5.","**Indian system:** ones, thousands, lakhs, crores. First comma after 3 digits, then every 2: 1,23,45,678.","**International system:** ones, thousands, millions, billions. Commas every 3 digits: 12,345,678.","**1 lakh = 100 thousand; 10 lakh = 1 million; 1 crore = 10 million; 100 crore = 1 billion.**","**Successor** = number + 1; **predecessor** = number − 1. Watch the 9s and 0s roll over.","**Comparing:** more digits wins; if equal, compare from the leftmost digit.","**Rounding:** look at the next digit to the right; 0–4 round down, 5–9 round up.","**Roman numerals:** I 1, V 5, X 10, L 50, C 100, D 500, M 1,000. Smaller before bigger means subtract.",{"id":1233,"type":1234,"sourceIds":1235},"sources-disc","sources",[1236,1237,1238,1239,1240,1241,1242,1243],"number-system-ncert-class6-knowing-numbers","number-system-khan-place-value","number-system-wiki-indian-numbering","number-system-mathsisfun-roman","number-system-britannica-hindu-arabic","number-system-wiki-modi-stadium","number-system-wiki-indian-railways","number-system-mathsisfun-place-value",[1236,1237,1238,1239,1240,1241,1242,1243],"needs_review",{"generatedBy":1247,"notes":1248},"claude-code","Draft generated with Python-checked numbers; pending owner review.","27d20e42ac5a6e8ae630fe15018f0e026f351a300d56df3304f22d0049e1d453",{"component:place-value@1":1251,"logic:practice":1252,"component:match-pairs@1":1253,"component:sort-game@1":1254,"component:rounding-race@1":1255,"source:number-system-britannica-hindu-arabic":1256,"source:number-system-khan-place-value":1257,"source:number-system-mathsisfun-place-value":1258,"source:number-system-mathsisfun-roman":1259,"source:number-system-ncert-class6-knowing-numbers":1260,"source:number-system-wiki-indian-numbering":1261,"source:number-system-wiki-indian-railways":1262,"source:number-system-wiki-modi-stadium":1263},"cae1e81c06fcfcc0152eb325ac8fb01f568e21a19ad76a75d79194e17231ae72","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","a63f6c9a1c311ff55526f628c2aa874f9c1a5449b9b395e4d1a775681172821c","8d110718a4c12de7f84b3e0653db1fd8b1147917a7dc31c48861b97c25731e0d","6a40a6a059b316a1b3d0621b353f1e4e944ed797310ad87c02f106fc0469f729","ef0f2abd2b3d7e393f1878aa51359c7a18194adf57596671bc0dff39cd1aa2f5","4e646d96c29f6b98469c9d11ee30da4e543f849b673fd067975aa685d8fef8a3","045032c28b7bfe33f8bc73c5c09ee8dd8076293c90dbe922d8a461e059edd6e3","acfe2624fdde75e26fde9ffd53d3508825fe265b41a08a9787cc09866a9e2848","372871233991a7b89e62e12c142d512f875d5a982e82368724e7199f5c4c6547",{"state":1265,"reviewer":1266,"selfReview":210,"reviewedAt":1267,"method":1268},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598273]