[{"data":1,"prerenderedAt":1318},["ShallowReactive",2],{"layer:number-system:understand":3},{"layer":4,"contentHash":1297,"dependencyHashes":1298,"approval":1312,"releaseId":1317},{"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":1292,"reviewStatus":1293,"authoring":1294},1,"number-system","en","understand","How place value works, and how to use it","Precise rules for names, commas, comparing, forming, rounding and estimating","Exact rules for place and face value, expanded form, number names and both comma systems, with many worked examples. Then reliable methods for converting, comparing, ordering, forming numbers, rounding, estimating and Roman numerals, plus the mix-ups to avoid.",[13,14,15,16,17],"Find the place value and face value of any digit in a number up to ten crore, and write numbers in expanded form.","Write and read number names correctly in the Indian and International systems and convert between them.","Compare and order large numbers, and form the greatest and smallest numbers from given digits with and without repetition.","Round to the nearest 10, 100, 1,000, 10,000 or lakh and estimate sums, differences and products sensibly.","Read and write Roman numerals using the addition, subtraction and repetition rules.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Lakh, crore, million (Discover)",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Place value, rounding, estimation, sort, match",{"label":38,"value":39},"Key rule","Each place = 10 × the place on its right",[41,45,51,57,60,108,118,126,131,145,150,153,169,178,182,193,198,219,247,256,260,285,290,295,298,329,345,367,375,383,397,420,429,433,444,477,482,487,490,528,536,546,551,554,564,568,588,608,613,616,625,634,642,647,658,663,683,702,710,714,718,734,745,750,753,761,770,778,786,815,844,856,861,866,900,908,916,971,991,1002,1007,1010,1042,1050,1104,1245,1249,1255,1260,1264,1281],{"id":42,"type":43,"markdown":44},"intro-u","prose","In Discover you met big numbers by bundling, reading and comparing them. This layer makes everything **exact**. You will learn the precise rules behind the words (place value, face value, expanded form, period), a reliable method for every common task, and the mistakes that trip people up most often.\n\nThe whole topic rests on one sentence: **in our number system, every place is worth exactly ten times the place to its right.** Almost every method in this lesson is that sentence used cleverly.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-read-u","callout","observation","How to use this layer","Each chapter gives a rule, one or two worked examples, and a quick check. The labs let you practise the method until it is automatic. If a chapter feels hard, go back to the matching chapter in Discover first.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Place value, precisely","Chapter 01","1 Place value",{"id":58,"type":43,"markdown":59},"pv-def","A number is a row of digits. Counting from the **right**, the places are worth 1, 10, 100, 1,000, 10,000 and so on, each ten times the one before. Two definitions follow:\n\n- **Face value** of a digit: the digit itself. It never changes.\n- **Place value** of a digit: face value × value of its place.\n\nSo in **3**,76,215 the digit 3 is in the lakhs place: face value 3, place value 3 × 1,00,000 = 3,00,000.",{"id":61,"type":62,"caption":63,"columns":64,"rows":69},"table-pv-9","table","Place values of every digit in 58,40,27,193 (Indian) = 584,027,193 (International)",[65,66,67,68],"Digit","Indian place","International place","Place value",[70,75,80,85,89,93,97,101,105],[71,72,73,74],"5","ten crores","hundred millions","50,00,00,000",[76,77,78,79],"8","crores","ten millions","8,00,00,000",[81,82,83,84],"4","ten lakhs","millions","40,00,000",[86,87,88,86],"0","lakhs","hundred thousands",[90,91,91,92],"2","ten thousands","20,000",[94,95,95,96],"7","thousands","7,000",[98,99,99,100],"1","hundreds","100",[102,103,103,104],"9","tens","90",[106,107,107,106],"3","ones",{"id":109,"type":110,"title":111,"problem":112,"steps":113},"we-diff-pv","worked_example","Difference between two place values","In 9,48,**7**0**7**, find the difference between the place values of the two 7s.",[114,115,116,117],"The left 7 is in the hundreds place: place value 7 × 100 = 700.","The right 7 is in the ones place: place value 7 × 1 = 7.","Difference: 700 − 7 = **693**.","Their face values are both 7, so the difference of face values is 0. Place value is what makes them different.",{"id":119,"type":110,"title":120,"problem":121,"steps":122},"we-product-pv","Place value times face value","In 3,82,564, what is the product of the place value and the face value of 8?",[123,124,125],"The 8 is in the ten thousands place (count from the right: 4 ones, 6 tens, 5 hundreds, 2 thousands, 8 ten thousands).","Place value: 8 × 10,000 = 80,000. Face value: 8.","Product: 80,000 × 8 = **6,40,000**.",{"id":127,"type":47,"variant":128,"title":129,"markdown":130},"mis-pv-fv","misconception","“The place value of a digit is its position number”","Place value is a **value**, not a position. The 4 in 4,512 is in the thousands place (the fourth place from the right), but its place value is **4,000**, not 4 and not \"fourth\". Always give place value as a number: digit × value of the place.",{"id":132,"type":133,"itemId":134,"prompt":135,"check":136,"hints":140,"feedback":142},"pr-u-pv","practice","number-system.understand-place-value","What is the place value of 5 in **6,05,32,871**?",{"kind":137,"answer":138,"tolerance":139},"number",500000,0,[141],"The commas help: the 5 is the right-hand digit of the lakhs period (05).",{"correct":143,"incorrect":144},"Right: reading from the right, 1 ones, 7 tens, 8 hundreds, 2 thousands, 3 ten thousands, 5 lakhs. So 5 × 1,00,000 = 5,00,000.","Count places from the right: 1 (ones), 7 (tens), 8 (hundreds), 2 (thousands), 3 (ten thousands), 5 (lakhs). Place value = 5 × 1,00,000 = 5,00,000.",{"id":146,"type":53,"title":147,"eyebrow":148,"navLabel":149},"ch02","Expanded form, three ways","Chapter 02","2 Expanded form",{"id":151,"type":43,"markdown":152},"exp-ways","**Expanded form** writes a number as the sum of the place values of its non-zero digits. There are three common styles, and you should be able to move between them.",{"id":154,"type":62,"caption":155,"columns":156,"rows":159},"table-exp","Three ways to expand 70,40,396",[157,158],"Style","Expanded form of 70,40,396",[160,163,166],[161,162],"Sum of place values","70,00,000 + 40,000 + 300 + 90 + 6",[164,165],"Digit × place","7 × 10,00,000 + 4 × 10,000 + 3 × 100 + 9 × 10 + 6 × 1",[167,168],"Standard form (back again)","70,40,396",{"id":170,"type":110,"title":171,"problem":172,"steps":173},"we-exp-back","From expanded form back to a number","Write as a number: 3 × 1,00,00,000 + 6 × 1,00,000 + 2 × 100 + 9 × 1.",[174,175,176,177],"Make a slot for every place from crores down to ones: crores, ten lakhs, lakhs, ten thousands, thousands, hundreds, tens, ones.","Fill the given digits: crores 3, lakhs 6, hundreds 2, ones 9.","Put 0 in every empty place: ten lakhs 0, ten thousands 0, thousands 0, tens 0.","Read off: 3 0 6 0 0 2 0 9 → **3,06,00,209** (three crore six lakh two hundred nine).",{"id":179,"type":47,"variant":128,"title":180,"markdown":181},"mis-exp-order","Jumbled expanded forms","Expanded forms can be given out of order: 40 + 6,000 + 3 + 20,000 is still a number. Do not just write the digits 4, 6, 3, 2 side by side! Place each value in its own place: 20,000 + 6,000 + 40 + 3 = **26,043**. The hundreds place is empty, so it gets a 0.",{"id":183,"type":133,"itemId":184,"prompt":185,"check":186,"hints":188,"feedback":190},"pr-exp","number-system.understand-expanded","Write as a number: 9 × 10,00,000 + 4 × 1,000 + 5 × 10.",{"kind":137,"answer":187,"tolerance":139},9004050,[189],"Ten lakhs is the 7th place from the right.",{"correct":191,"incorrect":192},"Right: nine ten-lakhs, no lakhs, no ten thousands, four thousands, no hundreds, five tens, no ones: 90,04,050.","Make seven slots (ten lakhs to ones) and fill them: 9, 0, 0, 4, 0, 5, 0. That is 90,04,050.",{"id":194,"type":53,"title":195,"eyebrow":196,"navLabel":197},"ch03","Number names: the exact rules","Chapter 03","3 Number names",{"id":199,"type":200,"title":201,"items":202},"steps-names","steps","Writing any number in words",[203,207,210,213,216],{"title":204,"tag":205,"text":206},"Put in the commas","Indian: 3,2,2 · Intl: 3,3,3","Use the system you are asked for. The commas split the number into periods.",{"title":208,"text":209},"Read the leftmost period","Say its number (one or two digits in Indian, up to three in International) followed by the period name: crore, lakh, thousand, or billion, million, thousand.",{"title":211,"text":212},"Move right, period by period","Repeat for each period. Skip any period that is all zeros; do not say \"zero thousand\".",{"title":214,"text":215},"Finish with the ones period","Read the last three digits as an ordinary number up to 999, with no period name.",{"title":217,"text":218},"Tidy the spelling","Hyphenate 21 to 99 (forty-five). Period names stay singular: five lakh, two crore, three million.",{"id":220,"type":62,"caption":221,"columns":222,"rows":226},"table-names","Number names in both systems",[223,224,225],"Number","Indian name","International name",[227,231,235,239,243],[228,229,230],"505050","5,05,050: five lakh five thousand fifty","505,050: five hundred five thousand fifty",[232,233,234],"80080808","8,00,80,808: eight crore eighty thousand eight hundred eight","80,080,808: eighty million eighty thousand eight hundred eight",[236,237,238],"1000001","10,00,001: ten lakh one","1,000,001: one million one",[240,241,242],"99999999","9,99,99,999: nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine","99,999,999: ninety-nine million nine hundred ninety-nine thousand nine hundred ninety-nine",[244,245,246],"123456789","12,34,56,789: twelve crore thirty-four lakh fifty-six thousand seven hundred eighty-nine","123,456,789: one hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine",{"id":248,"type":110,"title":249,"problem":250,"steps":251},"we-name-intl","Words to digits, international","Write in digits: *four hundred six million, seventy thousand, twelve*.",[252,253,254,255],"Set up three-digit periods: millions | thousands | ones.","Millions period: 406. Thousands period: 070 (seventy needs three digits here). Ones period: 012.","Join: **406,070,012**.","In the Indian system this is 40,60,70,012: forty crore sixty lakh seventy thousand twelve.",{"id":257,"type":47,"variant":128,"title":258,"markdown":259},"mis-names","Three favourite mistakes in number names","1. **Forgetting placeholder zeros.** *Seven lakh five* is 7,00,005, not 75 or 7,05.\n2. **Mixing the systems.** *Thirty lakh million* is not a number. Each name belongs to one system: use lakh and crore together, or million and billion together.\n3. **Reading digits one by one in a period.** 45 in the lakhs period is *forty-five lakh*, not *four five lakh* or *four lakh five*.",{"id":261,"type":133,"itemId":262,"prompt":263,"check":264,"hints":280,"feedback":282},"pr-names","number-system.understand-names","Which is the correct International name for **305,000,500**?",{"kind":265,"options":266,"correct":279},"choice",[267,270,273,276],{"id":268,"label":269},"a","Three hundred five million five hundred",{"id":271,"label":272},"b","Thirty crore fifty lakh five hundred",{"id":274,"label":275},"c","Three hundred fifty million five hundred",{"id":277,"label":278},"d","Three hundred five million five thousand",[268],[281],"Split into 305 | 000 | 500.",{"correct":283,"incorrect":284},"Right: 305 million, 000 thousand (skipped), 500.","Read by three-digit periods: 305 | 000 | 500 → three hundred five million, (no thousands), five hundred. Option b is the Indian name of a different number.",{"id":286,"type":47,"variant":287,"title":288,"markdown":289},"ex-cheque","example","Why cheques ask for the amount in words","A bank cheque asks for the amount twice: in figures (₹1,05,000) and in words (*Rupees one lakh five thousand only*). Figures are easy to alter: one extra 0 turns ₹1,05,000 into ₹10,50,000. Words are much harder to tamper with, and the word *only* at the end stops anyone adding more. If the two do not match, the bank will not simply guess, so writing number names correctly really matters.",{"id":291,"type":53,"title":292,"eyebrow":293,"navLabel":294},"ch04","Indian ↔ International","Chapter 04","4 Converting systems",{"id":296,"type":43,"markdown":297},"conv-intro","The two systems write **exactly the same numbers with exactly the same digits**. Only the commas and the names change. Up to the thousands period, they agree completely. They part ways at the sixth digit: the Indian system calls it **lakhs** and starts a new two-place period, while the International system calls it **hundred thousands** and keeps going in threes.",{"id":299,"type":62,"caption":300,"columns":301,"rows":304},"table-charts","The two place-value charts, digit by digit (place 1 = ones)",[302,303,224,225],"Place from right","Value",[305,307,309,311,313,315,318,320,322,324],[98,306,107,107],"1 = 1",[90,308,103,103],"10 = 10",[106,310,99,99],"100 = 100",[81,312,95,95],"1,000 = 1,000",[71,314,91,91],"10,000 = 10,000",[316,317,87,88],"6","1,00,000 = 100,000",[94,319,82,83],"10,00,000 = 1,000,000",[76,321,77,78],"1,00,00,000 = 10,000,000",[102,323,72,73],"10,00,00,000 = 100,000,000",[325,326,327,328],"10","1,00,00,00,000 = 1,000,000,000","arabs","billions",{"id":330,"type":200,"title":331,"items":332},"steps-convert","Converting a number between systems",[333,336,339,342],{"title":334,"text":335},"Remove all commas","Start from the plain digits so old commas cannot confuse you.",{"title":337,"text":338},"Re-insert commas","Indian: first after 3 digits from the right, then every 2. International: every 3.",{"title":340,"text":341},"Read in the new system","Use the new period names. Check with the key facts below.",{"title":343,"text":344},"Check with key facts","10 lakh = 1 million; 1 crore = 10 million; 10 crore = 100 million; 100 crore = 1 billion.",{"id":346,"type":347,"items":348},"formulas-conv","formulas",[349,352,355,358,361,364],{"expression":350,"caption":351},"1 lakh = 100 thousand","1,00,000 = 100,000. Five zeros.",{"expression":353,"caption":354},"10 lakh = 1 million","10,00,000 = 1,000,000. Six zeros.",{"expression":356,"caption":357},"1 crore = 10 million","1,00,00,000 = 10,000,000. Seven zeros.",{"expression":359,"caption":360},"10 crore = 100 million","10,00,00,000 = 100,000,000. Eight zeros.",{"expression":362,"caption":363},"100 crore = 1 billion","1,00,00,00,000 = 1,000,000,000. Nine zeros.",{"expression":365,"caption":366},"x million = 10x lakh","Multiply millions by 10 to get lakhs: 2.5 million = 25 lakh.",{"id":368,"type":110,"title":369,"problem":370,"steps":371},"we-conv-1","Crores to millions","A state's budget is ₹2,45,00,00,000 (245 crore). Say it in millions and billions.",[372,373,374],"Remove commas: 2450000000. Regroup in threes: **2,450,000,000**.","Read: two billion four hundred fifty million.","Check with key facts: 1 crore = 10 million, so 245 crore = 2,450 million = 2.45 billion. ✓",{"id":376,"type":110,"title":377,"problem":378,"steps":379},"we-conv-2","Millions to lakhs","A film has 3.6 million views online. How many lakh views is that?",[380,381,382],"1 million = 10 lakh.","3.6 million = 3.6 × 10 lakh = **36 lakh**.","In digits: 3.6 million = 3,600,000 = 36,00,000, which reads thirty-six lakh. ✓",{"id":384,"type":385,"prompt":386,"options":387,"explanation":396},"pred-75lakh","prediction","Which is more: **75 lakh** or **7 million**?",[388,390,392,394],{"id":268,"label":389},"75 lakh",{"id":271,"label":391},"7 million",{"id":274,"label":393},"They are equal",{"id":277,"label":395},"You cannot compare different systems","**a: 75 lakh.** Convert one of them: 7 million = 70 lakh (because 1 million = 10 lakh). And 75 lakh > 70 lakh. In digits: 75,00,000 = 7,500,000, which is 7.5 million, more than 7,000,000. Any two numbers can be compared once they are written in the same system, or simply as plain digits.",{"id":398,"type":399,"component":400,"componentVersion":5,"config":401,"objective":413,"textAlternative":414,"help":415},"lab-pv-intl","interactive","place-value",{"places":402,"system":403,"initial":404,"show":405,"challenges":407},9,"international",3600000,{"names":406,"expanded":406,"neighbours":406,"bothSystems":406},true,[408,409,410,411,412],1000000,25000000,100000000,407050600,999999999,"Build numbers up to 999,999,999 in the International chart and read each one in both systems at once.","The lab has nine columns grouped in threes: Millions (hundred millions, ten millions, millions), Thousands (hundred thousands, ten thousands, thousands) and Ones (hundreds, tens, ones). It starts at 3,600,000, shown as *three million six hundred thousand*; because both systems are switched on, it also shows 36,00,000, *thirty-six lakh*.\n\nChallenges: 1,000,000 (one million = ten lakh); 25,000,000 (twenty-five million = two crore fifty lakh); 100,000,000 (one hundred million = ten crore); 407,050,600 (four hundred seven million fifty thousand six hundred = forty crore seventy lakh fifty thousand six hundred); and 999,999,999, the largest nine-digit number (ninety-nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine).\n\nAs you add counters, notice that the digits never change between systems; only the commas and names do.",{"simplerExplanation":416,"hints":417},"Build the number, then look at both rows of commas. Same digits, different groupings.",[418,419],"25 million: 2 in ten millions and 5 in millions.","100 million needs just one counter.",{"id":421,"type":110,"title":422,"problem":423,"steps":424},"we-conv-3","International to Indian, digit by digit","Write 7,250,000 (7.25 million) with Indian commas and in words.",[425,426,427,428],"Remove commas: 7250000.","Indian commas, from the right: 000, then 50, then 72 → **72,50,000**.","Read: **seventy-two lakh fifty thousand**.","Check with the key fact: 7.25 million × 10 = 72.5 lakh = 72 lakh 50 thousand. ✓",{"id":430,"type":47,"variant":128,"title":431,"markdown":432},"mis-conv-commas","Converting by moving just one comma","A common slip when converting 4,500,000 to the Indian system is to move only the first comma and write **4,50,0000** (a group of four at the end). Every group except the first must be re-made: the correct Indian form is **45,00,000** (forty-five lakh). Always strip **all** the commas and rebuild from the right.",{"id":434,"type":133,"itemId":435,"prompt":436,"check":437,"hints":439,"feedback":441},"pr-conv-lakh","number-system.understand-convert-lakh","A state has a population of **68.5 million**. How many **lakh** is that?",{"kind":137,"answer":438,"tolerance":139},685,[440],"Millions × 10 = lakhs.",{"correct":442,"incorrect":443},"Right: 1 million = 10 lakh, so 68.5 million = 685 lakh (6 crore 85 lakh).","68.5 × 10 = 685 lakh. In digits: 68,500,000 = 6,85,00,000.",{"id":445,"type":399,"component":446,"componentVersion":5,"config":447,"objective":475,"textAlternative":476},"lab-match-equiv","match-pairs",{"prompt":448,"mode":449,"pairs":450},"Match each Indian amount with the same amount in the International system.","memory",[451,454,457,460,463,466,469,472],{"a":452,"b":453},"1 lakh","100 thousand",{"a":455,"b":456},"10 lakh","1 million",{"a":458,"b":459},"1 crore","10 million",{"a":461,"b":462},"25 lakh","2.5 million",{"a":464,"b":465},"10 crore","100 million",{"a":467,"b":468},"100 crore","1 billion",{"a":470,"b":471},"5 crore","50 million",{"a":473,"b":474},"140 crore","1.4 billion","Play a memory game matching Indian and International names for the same amount.","Sixteen face-down cards hide eight pairs; turn two at a time and keep them if they match. The pairs are: 1 lakh and 100 thousand; 10 lakh and 1 million; 1 crore and 10 million; 25 lakh and 2.5 million; 10 crore and 100 million; 100 crore and 1 billion; 5 crore and 50 million; 140 crore and 1.4 billion (roughly India's population).\n\nThe key facts that unlock them all: 10 lakh = 1 million, so lakhs ÷ 10 = millions; and 100 crore = 1 billion, so crores ÷ 100 = billions.",{"id":478,"type":47,"variant":479,"title":480,"markdown":481},"careful-dollar","careful","Watch the units in money news","A headline might say ₹8,000 crore in India and \"$1 billion\" abroad. These are **not** the same amount just because they sound big: first convert the names (8,000 crore = 80 billion), then remember that rupees and dollars are different currencies. Convert the names first, then the money.",{"id":483,"type":53,"title":484,"eyebrow":485,"navLabel":486},"ch05","Neighbours, smallest and largest","Chapter 05","5 Neighbours",{"id":488,"type":43,"markdown":489},"neigh","The **successor** of a whole number n is n + 1 and its **predecessor** is n − 1. Every whole number has a successor, so there is no largest number. Every whole number except 0 has a predecessor.\n\nTwo facts link neighbours to the number of digits:\n\n- The **largest n-digit number** is n nines. Its successor is the **smallest (n + 1)-digit number**: a 1 followed by n zeros.\n- The **smallest n-digit number** is 1 followed by (n − 1) zeros. Its predecessor is the largest (n − 1)-digit number.",{"id":491,"type":62,"caption":492,"columns":493,"rows":498},"table-extremes","Smallest and largest numbers of each size",[494,495,496,497],"Digits","Smallest","Largest","How many such numbers",[499,501,504,508,512,516,520,524],[90,325,500,104],"99",[106,100,502,503],"999","900",[81,505,506,507],"1,000","9,999","9,000",[71,509,510,511],"10,000","99,999","90,000",[316,513,514,515],"1,00,000","9,99,999","9,00,000",[94,517,518,519],"10,00,000","99,99,999","90,00,000",[76,521,522,523],"1,00,00,000","9,99,99,999","9,00,00,000",[102,525,526,527],"10,00,00,000","99,99,99,999","90,00,00,000",{"id":529,"type":110,"title":530,"problem":531,"steps":532},"we-count-digits","How many 5-digit numbers are there?","Count all the numbers from 10,000 to 99,999.",[533,534,535],"Counting from a to b inclusive gives b − a + 1 numbers.","99,999 − 10,000 + 1 = **90,000**.","Another way: the first digit can be 1 to 9 (9 choices) and each other digit 0 to 9 (10 choices), so 9 × 10 × 10 × 10 × 10 = 90,000. ✓",{"id":537,"type":133,"itemId":538,"prompt":539,"check":540,"hints":541,"feedback":543},"pr-diff-extremes","number-system.understand-extremes","What is the difference between the smallest 7-digit number and the largest 6-digit number?",{"kind":137,"answer":5,"tolerance":139},[542],"Write both numbers out in full.",{"correct":544,"incorrect":545},"Right: 10,00,000 − 9,99,999 = 1. They are neighbours!","Smallest 7-digit number: 10,00,000. Largest 6-digit number: 9,99,999. Their difference is just 1, because one is the successor of the other.",{"id":547,"type":53,"title":548,"eyebrow":549,"navLabel":550},"ch06","Comparing and ordering","Chapter 06","6 Comparing",{"id":552,"type":43,"markdown":553},"cmp-rule","To compare two whole numbers (written without extra zeros in front):\n\n1. **More digits means bigger.** Any 6-digit number is bigger than any 5-digit number, because the smallest 6-digit number, 1,00,000, is bigger than the largest 5-digit number, 99,999.\n2. **Same number of digits:** scan from the left until two digits differ. The number with the bigger digit there is bigger; the digits after that do not matter.\n\n**Ascending order** lists numbers from smallest to largest; **descending order** from largest to smallest. The symbols are > (greater than) and \u003C (less than); the open, wide side always faces the bigger number.",{"id":555,"type":110,"title":556,"problem":557,"steps":558},"we-rivers","Ordering river lengths","Approximate lengths of five Indian rivers: Ganga 2,525 km, Godavari 1,465 km, Yamuna 1,376 km, Narmada 1,312 km, Krishna 1,400 km. Arrange them in ascending order.",[559,560,561,562,563],"All five have 4 digits. Thousands digits: Ganga has 2, the rest have 1. So Ganga is the longest.","Among the four with 1 thousand, compare hundreds: Godavari 4, Yamuna 3, Narmada 3, Krishna 4.","Yamuna and Narmada both have 3 hundreds; compare tens: Yamuna 7, Narmada 1. So Narmada (1,312) is shorter than Yamuna (1,376).","Godavari and Krishna both have 4 hundreds; compare tens: Godavari 6, Krishna 0. So Krishna (1,400) is shorter than Godavari (1,465).","**Ascending: Narmada 1,312 km, Yamuna 1,376 km, Krishna 1,400 km, Godavari 1,465 km, Ganga 2,525 km**, that is 1,312 \u003C 1,376 \u003C 1,400 \u003C 1,465 \u003C 2,525.",{"id":565,"type":47,"variant":128,"title":566,"markdown":567},"mis-longer","“The number with bigger digits is bigger”","99,999 has five 9s and 1,00,000 has one 1 and a lot of zeros, yet 1,00,000 is bigger. **Count the digits first.** Only when two numbers have the same number of digits do you compare digit by digit from the left.",{"id":569,"type":133,"itemId":570,"prompt":571,"check":572,"hints":583,"feedback":585},"pr-cmp","number-system.understand-compare","Which statement is true?",{"kind":265,"options":573,"correct":582},[574,576,578,580],{"id":268,"label":575},"4,05,678 > 4,50,678",{"id":271,"label":577},"12,345 > 1,23,450",{"id":274,"label":579},"9,87,654 \u003C 10,00,000",{"id":277,"label":581},"70,00,000 = 7 crore",[274],[584],"Count the digits of each number before comparing.",{"correct":586,"incorrect":587},"Yes: 9,87,654 has 6 digits and 10,00,000 has 7, so the 7-digit number is bigger.","a is false: 4,05,678 and 4,50,678 first differ at the ten thousands place, where 0 \u003C 5. b is false (5 digits \u003C 6 digits). d is false: 70,00,000 is 70 lakh, not 7 crore (7 crore = 7,00,00,000). c is true.",{"id":589,"type":133,"itemId":590,"prompt":591,"check":592,"hints":603,"feedback":605},"pr-order","number-system.understand-ascending","Which list is in **ascending** order?",{"kind":265,"options":593,"correct":602},[594,596,598,600],{"id":268,"label":595},"4,56,789; 45,678; 4,567",{"id":271,"label":597},"4,567; 45,678; 4,56,789",{"id":274,"label":599},"45,678; 4,567; 4,56,789",{"id":277,"label":601},"4,567; 4,56,789; 45,678",[271],[604],"Ascending = going up, smallest first.",{"correct":606,"incorrect":607},"Right: ascending means smallest first. 4,567 (4 digits), 45,678 (5 digits), 4,56,789 (6 digits).","Count digits: 4,567 has 4, 45,678 has 5 and 4,56,789 has 6. Smallest first: 4,567; 45,678; 4,56,789.",{"id":609,"type":53,"title":610,"eyebrow":611,"navLabel":612},"ch07","Forming the greatest and smallest numbers","Chapter 07","7 Forming numbers",{"id":614,"type":43,"markdown":615},"form-rule","Given some digits, which is the greatest number you can make using each digit once? Put the **biggest digit in the biggest place**: arrange the digits in **descending** order. For the smallest, arrange them in **ascending** order, with one catch: a number cannot start with 0 (then it would have fewer digits). So if 0 is among the digits, put the smallest **non-zero** digit first, then the 0, then the rest in ascending order.",{"id":617,"type":110,"title":618,"problem":619,"steps":620},"we-form","Greatest and smallest from 7, 0, 4, 9, 2","Using each of the digits 7, 0, 4, 9, 2 exactly once, form the greatest and smallest 5-digit numbers, and find their difference.",[621,622,623,624],"Greatest: digits in descending order: **97,420**.","Smallest: ascending order would be 0, 2, 4, 7, 9, but 02,479 is not a 5-digit number.","So put the smallest non-zero digit (2) first, then 0, then the rest: **20,479**.","Difference: 97,420 − 20,479 = **76,941**.",{"id":626,"type":110,"title":627,"problem":628,"steps":629},"we-form-rep","When digits may repeat","Using only the digits 3, 0 and 8, with repetition allowed, form the greatest and smallest 6-digit numbers that use all three digits at least once.",[630,631,632,633],"Repetition is allowed, so to make the number as big as possible fill as many high places as possible with 8.","Greatest: we need a 3 and a 0 somewhere, so push them to the lowest places: 8, 8, 8, 8, 3, 0 → **8,88,830**.","Smallest: the first digit must be the smallest non-zero digit, 3. Then as many 0s as possible, and the 8 must still appear, so put it last: **3,00,008**.","Difference: 8,88,830 − 3,00,008 = 5,88,822.",{"id":635,"type":110,"title":636,"problem":637,"steps":638},"we-form-zero-rep","Zeros, repetition and “at least once”","Using the digits 0, 1 and 2 (repetition allowed, **each at least once**), form the greatest and the smallest 6-digit numbers.",[639,640,641],"Greatest: put 2 in as many high places as possible, then make sure 1 and 0 appear in the lowest places: 2, 2, 2, 2, 1, 0 → **2,22,210**.","Smallest: the first digit cannot be 0, so use 1. Then as many 0s as possible, with the 2 in the ones place: **1,00,002**.","Difference: 2,22,210 − 1,00,002 = 1,22,208.",{"id":643,"type":47,"variant":644,"title":645,"markdown":646},"nuance-rep","nuance","Read the question carefully","\"Using the digits\" can mean *each digit exactly once*, *digits may repeat*, or *each digit at least once*. The answers are different. For 5-digit numbers with the digits 1 and 0 repeated freely, the greatest is 11,111 (0 is not forced to appear) but if *both* must appear it is 11,110.",{"id":648,"type":133,"itemId":649,"prompt":650,"check":651,"hints":653,"feedback":655},"pr-form","number-system.understand-form-smallest","Using each of the digits 5, 0, 1, 8 exactly once, what is the smallest 4-digit number?",{"kind":137,"answer":652,"tolerance":139},1058,[654],"Which digit cannot go first?",{"correct":656,"incorrect":657},"Right: smallest non-zero digit first (1), then 0, then 5 and 8: 1,058.","Do not start with 0. Put 1 first, then 0, then 5, 8 in ascending order: 1,058.",{"id":659,"type":53,"title":660,"eyebrow":661,"navLabel":662},"ch08","Rounding, precisely","Chapter 08","8 Rounding",{"id":664,"type":200,"title":665,"items":666},"steps-round-u","Rounding to any place",[667,670,674,677,680],{"title":668,"text":669},"Choose the rounding place","For example, the thousands place when rounding to the nearest 1,000.",{"title":671,"tag":672,"text":673},"Look one place to the right","the deciding digit","Only this digit decides. The digits after it are ignored.",{"title":675,"text":676},"0 to 4: round down","Keep the rounding digit as it is.",{"title":678,"text":679},"5 to 9: round up","Add 1 to the rounding digit. If it was 9, carry, just like addition.",{"title":681,"text":682},"Zeros after","Every digit to the right of the rounding place becomes 0.",{"id":684,"type":62,"caption":685,"columns":686,"rows":690},"table-round","Rounding 38,69,475 to different places",[687,688,689],"Round to the nearest","Deciding digit","Result",[691,693,695,697,699,701],[325,71,692],"38,69,480",[100,94,694],"38,69,500",[505,81,696],"38,69,000",[509,102,698],"38,70,000",[513,316,700],"39,00,000",[517,76,84],{"id":703,"type":110,"title":704,"problem":705,"steps":706},"we-round-carry","Rounding with a carry","Round 4,99,720 to the nearest thousand.",[707,708,709],"The thousands digit is 9. The deciding digit (hundreds) is 7, so round up.","Adding 1 thousand to 4,99,000 gives 5,00,000: the 9s carry over.","**4,99,720 ≈ 5,00,000**. Check: it is 280 below 5,00,000 and 720 above 4,99,000; 5,00,000 is closer. ✓",{"id":711,"type":47,"variant":128,"title":712,"markdown":713},"mis-chain","“Round the last digit first, then the next…”","Rounding step by step from the right can give the wrong answer. Round 1,449 to the nearest hundred:\n\n- Correct: the deciding digit (tens) is 4, so 1,449 → **1,400**.\n- Wrong chain: 1,449 → 1,450 (to tens) → 1,500 (to hundreds).\n\nAlways round **once**, from the original number, looking only at the one deciding digit.",{"id":715,"type":47,"variant":128,"title":716,"markdown":717},"mis-carry-u","The 9 that carries","Round **3,96,200** to the nearest ten thousand. The ten thousands digit is 9 and the deciding digit (thousands) is 6, so round up. Students often write 3,100,000 or 3,10,00,000 by turning the 9 into “10” in place. The 9 becomes 0 and the 1 carries into the lakhs: **4,00,000**.",{"id":719,"type":399,"component":720,"componentVersion":5,"config":721,"objective":729,"textAlternative":730,"help":731},"lab-round-u","rounding-race",{"roundTo":722,"range":726,"rounds":728,"secondsPerRound":139},[723,724,725],1000,10000,100000,{"min":723,"max":727},9999999,10,"Round numbers up to ninety-nine lakh to the nearest thousand, ten thousand or lakh on a number line.","Each of 10 rounds shows a number between 1,000 and 99,99,999 on a number line with the two nearest multiples marked, for example the nearest thousands, ten thousands or lakhs. You pick the mark it rounds to.\n\nExample: 36,74,512 to the nearest lakh lies between 36,00,000 and 37,00,000; the deciding digit (ten thousands) is 7, so it rounds up to 37,00,000. The same number to the nearest thousand lies between 36,74,000 and 36,75,000; the deciding digit (hundreds) is 5, so it rounds up to 36,75,000. To the nearest ten thousand: the deciding digit (thousands) is 4, so it rounds down to 36,70,000.",{"hints":732},[733],"Find the deciding digit: one place to the right of the place you are rounding to.",{"id":735,"type":133,"itemId":736,"prompt":737,"check":738,"hints":740,"feedback":742},"pr-round-lakh","number-system.understand-round-lakh","Round **12,45,999** to the nearest lakh.",{"kind":137,"answer":739,"tolerance":139},1200000,[741],"Only the ten thousands digit decides.",{"correct":743,"incorrect":744},"Right: the deciding digit (ten thousands) is 4, so round down to 12,00,000.","Rounding to lakhs, look at the ten thousands digit: 4. That means round down: 12,00,000. The 5 and 9s after it do not matter.",{"id":746,"type":53,"title":747,"eyebrow":748,"navLabel":749},"ch09","Estimating sums, differences and products","Chapter 09","9 Estimation",{"id":751,"type":43,"markdown":752},"est-why","An **estimate** is a quick, sensible approximation. We estimate when an exact answer is not needed (roughly how many buses for a school trip?), when we want to **check** an exact answer (is 4,862 × 29 really 1,40,998?), or when exact data does not exist (how many people were at a mela?).\n\nA common method is to **round each number to its greatest place** (its leftmost digit) and then calculate. For sums and differences you can also round every number to the same place, such as the nearest hundred or thousand; choose the place that suits how accurate you need to be.",{"id":754,"type":110,"title":755,"problem":756,"steps":757},"we-est-sum","Estimating a sum","Estimate 5,290 + 17,986.",[758,759,760],"Round to the nearest thousand: 5,290 → 5,000; 17,986 → 18,000.","Estimate: 5,000 + 18,000 = **23,000**.","Exact: 23,276. The estimate is off by only -,276.",{"id":762,"type":110,"title":763,"problem":764,"steps":765},"we-est-prod","Estimating a product (a school trip)","632 students go on a trip and each pays ₹385. Roughly how much is collected?",[766,767,768,769],"Round each number to its greatest place: 632 → 600; 385 → 400.","Estimate: 600 × 400 = **₹2,40,000**.","Exact: 632 × 385 = ₹2,43,320. The estimate is close enough to plan a budget.","Tip: 6 × 4 = 24, then count the zeros: two from 600 and two from 400, so 24 followed by four zeros.",{"id":771,"type":110,"title":772,"problem":773,"steps":774},"we-est-diff","Estimating a difference","A town had 1,83,275 voters; 97,640 of them voted. Roughly how many did not vote?",[775,776,777],"Round both to the nearest ten thousand: 1,83,275 → 1,80,000; 97,640 → 1,00,000.","Estimate: 1,80,000 − 1,00,000 = **80,000**.","Exact: 85,635. Close.",{"id":779,"type":110,"title":780,"problem":781,"steps":782},"we-est-rain","Estimating in a real context: a monsoon water tank","A school roof collects rainwater. In June it collected 38,460 litres, in July 72,915 litres, in August 64,280 litres and in September 29,740 litres. Estimate the total to the nearest ten thousand litres to plan a new tank.",[783,784,785],"Round each to the nearest ten thousand: 40,000; 70,000; 60,000; 30,000.","Estimate: 40,000 + 70,000 + 60,000 + 30,000 = **2,00,000 litres** (two lakh).","Exact: 2,05,395 litres. The estimate is off by only 5,395 litres, plenty accurate for choosing a tank size.",{"id":787,"type":399,"component":788,"componentVersion":5,"config":789,"objective":813,"textAlternative":814},"lab-sprint-est","arith-sprint",{"operations":790,"ranges":794,"rounds":728,"secondsTotal":139,"estimateFirst":406,"wordProblems":800},[791,792,793],"+","-","×",{"a":795,"b":798},{"min":796,"max":797},100,9999,{"min":728,"max":799},999,[801,805,809],{"prompt":802,"answer":803,"operation":793,"unit":804},"A train has 18 coaches and each carries 72 passengers. How many passengers can it carry?",1296,"passengers",{"prompt":806,"answer":807,"operation":791,"unit":808},"A library had 4,856 books and bought 1,275 more. How many books does it have now?",6131,"books",{"prompt":810,"answer":811,"operation":792,"unit":812},"A farmer harvested 7,250 kg of rice and sold 3,875 kg. How much is left?",3375,"kg","Estimate first, then calculate exactly, and see how close your estimate was.","Each of 10 rounds shows a sum, difference or product with a first number from 100 to 9,999 and a second from 10 to 999. You first enter an estimate (round each number, then calculate), then the exact answer. The game shows how far your estimate was from the exact answer. There is no timer.\n\nSome rounds are word problems, which ask only for the exact answer, so estimate in your head first: a train with 18 coaches of 72 passengers (about 20 × 70 = 1,400; exact 1,296); a library with 4,856 books buying 1,275 more (about 5,000 + 1,000 = 6,000; exact 6,131); a farmer with 7,250 kg of rice selling 3,875 kg (about 7,000 − 4,000 = 3,000; exact 3,375).\n\nGood estimates use numbers you can calculate in your head. They should be close enough to catch a mistake such as a missing zero.",{"id":816,"type":62,"caption":817,"columns":818,"rows":822},"table-when-est","Estimate or calculate exactly?",[819,820,821],"Situation","Estimate or exact?","Why",[823,827,831,835,838,841],[824,825,826],"How many buses for 632 students, 50 per bus?","Estimate first, then check exactly","About 12 or 13 buses; the exact count (13, because 12 buses seat only 600) decides the booking.",[828,829,830],"Bill at the kirana shop","Exact","Money changes hands; every rupee counts. An estimate helps you check the total.",[832,833,834],"Crowd at a mela","Estimate","Nobody can count every person; a rounded figure is all that is possible.",[836,829,837],"Medicine dose","A small error can be harmful.",[839,833,840],"Is my calculator answer sensible?","A rough check catches missing zeros and pressed-wrong keys.",[842,833,843],"Distance to plan a road trip","Rounded kilometres are enough to plan fuel and time.",{"id":845,"type":133,"itemId":846,"prompt":847,"check":848,"hints":850,"feedback":853},"pr-est-kirana","number-system.understand-estimate-kirana","A kirana shop sells about ₹4,850 of goods a day. Estimate its sales for a 31-day month by rounding each number to its greatest place. (Give the estimate in rupees.)",{"kind":137,"answer":849,"tolerance":139},150000,[851,852],"Round both numbers first, then multiply.","5 × 3 = 15; count the zeros.",{"correct":854,"incorrect":855},"Right: ₹4,850 ≈ ₹5,000 and 31 ≈ 30, so about ₹1,50,000 (one and a half lakh). Exact: 4,850 × 31 = ₹1,50,350.","Round 4,850 to 5,000 and 31 to 30. Then 5,000 × 30 = 1,50,000.",{"id":857,"type":47,"variant":858,"title":859,"markdown":860},"aha-check","aha","Estimation catches big mistakes","Riya calculates 4,862 × 29 and gets **14,098**. Estimate: 5,000 × 30 = 1,50,000. Her answer is about ten times too small, so she must have lost a zero. The exact answer is 1,40,998. An estimate will not tell you the exact answer, but it tells you when an answer is impossible.",{"id":862,"type":53,"title":863,"eyebrow":864,"navLabel":865},"ch10","Roman numerals: all the rules","Chapter 10","10 Roman numerals",{"id":867,"type":62,"caption":868,"columns":869,"rows":873},"table-roman-rules","The rules of Roman numerals",[870,871,872],"Rule","Examples","Not allowed",[874,878,882,886,889,893,897],[875,876,877],"Symbols: I 1, V 5, X 10, L 50, C 100, D 500, M 1,000","MDCLXVI = 1,666","—",[879,880,881],"Repeating a symbol adds it; at most three times in a row","III = 3, XXX = 30, CCC = 300","IIII, XXXX",[883,884,885],"V, L and D are never repeated","X (not VV), C (not LL), M (not DD)","VV, LL, DD",[887,888,877],"Smaller after bigger: add","VI = 6, LX = 60, MC = 1,100",[890,891,892],"Smaller before bigger: subtract, only for these six pairs","IV 4, IX 9, XL 40, XC 90, CD 400, CM 900","IL, IC, XD, VX",[894,895,896],"V, L and D are never subtracted","45 is XLV","VL for 45",[898,899,877],"A bar over a symbol multiplies it by 1,000","V with a bar = 5,000",{"id":901,"type":110,"title":902,"problem":903,"steps":904},"we-to-roman","Writing 1,994 in Roman numerals","Write 1,994 in Roman numerals.",[905,906,907],"Split by place value: 1,994 = 1,000 + 900 + 90 + 4.","1,000 = M; 900 = CM; 90 = XC; 4 = IV.","Join them: **MCMXCIV**.",{"id":909,"type":110,"title":910,"problem":911,"steps":912},"we-from-roman","Reading CDXLVIII","What number is CDXLVIII?",[913,914,915],"Split into groups: CD | XL | VIII.","CD = 500 − 100 = 400; XL = 50 − 10 = 40; VIII = 5 + 3 = 8.","Total: 400 + 40 + 8 = **448**.",{"id":917,"type":399,"component":918,"componentVersion":5,"config":919,"objective":969,"textAlternative":970},"lab-sort-roman","sort-game",{"prompt":920,"bins":921,"items":928,"seconds":139},"Is each Roman numeral written correctly by the standard rules?",[922,925],{"id":923,"label":924},"valid","Correct",{"id":926,"label":927},"invalid","Breaks a rule",[929,933,937,941,945,949,953,957,961,965],{"id":930,"label":931,"bin":923,"why":932},"r0","XC","XC = 90: X may be subtracted from C.",{"id":934,"label":935,"bin":926,"why":936},"r1","IC","I may be subtracted only from V and X. 99 is XCIX.",{"id":938,"label":939,"bin":926,"why":940},"r2","XXXX","X cannot be repeated four times. 40 is XL.",{"id":942,"label":943,"bin":923,"why":944},"r3","XLIV","XLIV = 40 + 4 = 44.",{"id":946,"label":947,"bin":926,"why":948},"r4","VX","V is never subtracted. 5 is just V.",{"id":950,"label":951,"bin":923,"why":952},"r5","MCM","MCM = 1,000 + 900 = 1,900.",{"id":954,"label":955,"bin":926,"why":956},"r6","LL","L is never repeated. 100 is C.",{"id":958,"label":959,"bin":923,"why":960},"r7","CCCXC","CCCXC = 300 + 90 = 390.",{"id":962,"label":963,"bin":926,"why":964},"r8","IL","I cannot be subtracted from L. 49 is XLIX.",{"id":966,"label":967,"bin":923,"why":968},"r9","DCCC","DCCC = 500 + 300 = 800.","Sort Roman numerals into correctly written ones and ones that break a rule.","Ten cards show Roman numerals; sort each into *Correct* or *Breaks a rule*.\n\nCorrect: XC (90), XLIV (44), MCM (1,900), CCCXC (390) and DCCC (800).\n\nBreaks a rule: IC (I can only be subtracted from V and X; 99 is XCIX), XXXX (no symbol four times; 40 is XL), VX (V is never subtracted), LL (L is never repeated; 100 is C) and IL (49 is XLIX, not IL).",{"id":972,"type":133,"itemId":973,"prompt":974,"check":975,"hints":986,"feedback":988},"pr-roman-1444","number-system.understand-roman-write","Which is **1,444** in Roman numerals?",{"kind":265,"options":976,"correct":985},[977,979,981,983],{"id":268,"label":978},"MCDXLIV",{"id":271,"label":980},"MCCCCXXXXIIII",{"id":274,"label":982},"MIVXLIV",{"id":277,"label":984},"MCDLIV",[268],[987],"Write each place separately.",{"correct":989,"incorrect":990},"Right: 1,000 + 400 + 40 + 4 = M + CD + XL + IV = MCDXLIV.","Split by place value: 1,000 = M, 400 = CD, 40 = XL, 4 = IV → MCDXLIV. Option b repeats symbols four times; d gives 1,454.",{"id":992,"type":133,"itemId":993,"prompt":994,"check":995,"hints":997,"feedback":999},"pr-roman-u","number-system.understand-roman","What number is **MMXLIX**?",{"kind":137,"answer":996,"tolerance":139},2049,[998],"Look for subtraction pairs first: XL and IX.",{"correct":1000,"incorrect":1001},"Right: MM = 2,000, XL = 40, IX = 9. Total 2,049.","Split: MM | XL | IX = 2,000 + 40 + 9 = 2,049.",{"id":1003,"type":53,"title":1004,"eyebrow":1005,"navLabel":1006},"ch11","Place value in measurement","Chapter 11","11 Units",{"id":1008,"type":43,"markdown":1009},"units-pv","The metric system is place value in disguise. Its units go up and down in steps of ten, and the prefixes tell you the place:\n\n- **kilo** means 1,000 (1 km = 1,000 m; 1 kg = 1,000 g)\n- **centi** means one hundredth (1 m = 100 cm)\n- **milli** means one thousandth (1 m = 1,000 mm; 1 L = 1,000 mL)\n\nSo converting between metric units is just moving digits between places, exactly like converting thousands to ones.",{"id":1011,"type":62,"caption":1012,"columns":1013,"rows":1018},"table-units","Everyday metric conversions",[1014,1015,1016,1017],"Measure","Big unit","Small unit","Example",[1019,1024,1028,1032,1037],[1020,1021,1022,1023],"Length","1 km","1,000 m","3 km 45 m = 3,000 + 45 = 3,045 m",[1020,1025,1026,1027],"1 m","100 cm","2 m 7 cm = 207 cm",[1020,1029,1030,1031],"1 cm","10 mm","15 cm 4 mm = 154 mm",[1033,1034,1035,1036],"Mass","1 kg","1,000 g","5 kg 250 g = 5,250 g",[1038,1039,1040,1041],"Capacity","1 L","1,000 mL","A 2 L bottle holds 2,000 mL",{"id":1043,"type":110,"title":1044,"problem":1045,"steps":1046},"we-units","Metres on a railway","A train journey is 1,284 km. How many metres is that? Write it both ways.",[1047,1048,1049],"1 km = 1,000 m, so multiply by 1,000: every digit moves three places to the left.","1,284 × 1,000 = 1,284,000 m = **12,84,000 m**.","In words: twelve lakh eighty-four thousand metres, or one million two hundred eighty-four thousand metres.",{"id":1051,"type":1052,"title":1053,"terms":1054},"gloss-u","glossary","Vocabulary: precise meanings",[1055,1058,1061,1064,1068,1072,1076,1080,1084,1088,1092,1095,1097,1101],{"term":1056,"meaning":1057},"Numeral","A symbol or group of symbols that stands for a number, such as 47, XLVII or the word forty-seven.",{"term":1059,"meaning":1060},"Place-value system","A way of writing numbers in which a digit's value depends on its position. Ours is a base-ten place-value system.",{"term":1062,"meaning":1063},"Standard form","A number written in the usual way with digits (and commas), such as 7,04,396.",{"term":1065,"meaning":1066,"example":1067},"Expanded form","A number written as the sum of the place values of its non-zero digits.","7,04,396 = 7,00,000 + 4,000 + 300 + 90 + 6",{"term":1069,"meaning":1070,"example":1071},"Period","A group of places marked off by commas and read with one name.","Lakhs period, millions period",{"term":1073,"meaning":1074,"example":1075},"Indian system","Place-value naming with periods ones (3 places), thousands, lakhs, crores (2 places each).","1,23,45,678",{"term":1077,"meaning":1078,"example":1079},"International system","Place-value naming with periods of three places: ones, thousands, millions, billions.","12,345,678",{"term":1081,"meaning":1082,"example":1083},"Ascending order","Arranged from smallest to greatest.","12, 45, 300",{"term":1085,"meaning":1086,"example":1087},"Descending order","Arranged from greatest to smallest.","300, 45, 12",{"term":1089,"meaning":1090,"example":1091},"Greater than (>) \u002F less than (\u003C)","Symbols comparing two numbers; the open side faces the bigger number.","8 > 3 and 3 \u003C 8",{"term":833,"meaning":1093,"example":1094},"A sensible approximate value, usually found by rounding before calculating.","632 × 385 ≈ 600 × 400",{"term":688,"meaning":1096},"When rounding, the digit one place to the right of the rounding place. 0–4 round down, 5–9 round up.",{"term":1098,"meaning":1099,"example":1100},"Kilo, centi, milli","Metric prefixes meaning 1,000, one hundredth and one thousandth.","1 km = 1,000 m",{"term":1102,"meaning":1103},"Vinculum","A bar over a Roman numeral that multiplies its value by 1,000.",{"id":1105,"type":1106,"title":1107,"questions":1108},"quiz-u","quiz","Understand check",[1109,1121,1133,1146,1159,1169,1182,1194,1207,1219,1232],{"itemId":1110,"prompt":1111,"options":1112,"correct":274,"why":1120},"number-system.und-q-pv","In 8,03,56,742, what is the place value of 3?",[1113,1114,1116,1118],{"id":268,"label":106},{"id":271,"label":1115},"30,000",{"id":274,"label":1117},"3,00,000",{"id":277,"label":1119},"30,00,000","From the right: 2 ones, 4 tens, 7 hundreds, 6 thousands, 5 ten thousands, 3 lakhs, 0 ten lakhs, 8 crores. So the 3 is in the lakhs place: 3 × 1,00,000 = 3,00,000.",{"itemId":1122,"prompt":1123,"options":1124,"correct":271,"why":1132},"number-system.und-q-conv","5 crore in the International system is…",[1125,1127,1128,1130],{"id":268,"label":1126},"5 million",{"id":271,"label":471},{"id":274,"label":1129},"500 million",{"id":277,"label":1131},"5 billion","1 crore = 10 million, so 5 crore = 50 million.",{"itemId":1134,"prompt":1135,"options":1136,"correct":268,"why":1145},"number-system.und-q-names","Which is 20,02,020 in words?",[1137,1139,1141,1143],{"id":268,"label":1138},"Twenty lakh two thousand twenty",{"id":271,"label":1140},"Two lakh two thousand twenty",{"id":274,"label":1142},"Twenty lakh twenty thousand twenty",{"id":277,"label":1144},"Two million two thousand twenty","Periods: 20 lakh | 02 thousand | 020 → twenty lakh two thousand twenty.",{"itemId":1147,"prompt":1148,"options":1149,"correct":274,"why":1158},"number-system.und-q-exp","6 × 10,000 + 5 × 100 + 4 equals…",[1150,1152,1154,1156],{"id":268,"label":1151},"654",{"id":271,"label":1153},"6,504",{"id":274,"label":1155},"60,504",{"id":277,"label":1157},"65,004","Places: ten thousands 6, thousands 0, hundreds 5, tens 0, ones 4 → 60,504.",{"itemId":1160,"prompt":1161,"options":1162,"correct":271,"why":1168},"number-system.und-q-count","How many 3-digit numbers are there?",[1163,1164,1165,1167],{"id":268,"label":502},{"id":271,"label":503},{"id":274,"label":1166},"899",{"id":277,"label":505},"From 100 to 999: 999 − 100 + 1 = 900.",{"itemId":1170,"prompt":1171,"options":1172,"correct":268,"why":1181},"number-system.und-q-form","The greatest 4-digit number using 3, 0, 6, 1 once each is…",[1173,1175,1177,1179],{"id":268,"label":1174},"6,310",{"id":271,"label":1176},"6,301",{"id":274,"label":1178},"1,036",{"id":277,"label":1180},"3,610","Descending order of digits: 6, 3, 1, 0 → 6,310.",{"itemId":1183,"prompt":1184,"options":1185,"correct":271,"why":1193},"number-system.und-q-round","9,96,500 rounded to the nearest thousand is…",[1186,1188,1190,1191],{"id":268,"label":1187},"9,96,000",{"id":271,"label":1189},"9,97,000",{"id":274,"label":517},{"id":277,"label":1192},"9,90,000","Thousands digit 6, deciding digit (hundreds) 5 → round up to 9,97,000.",{"itemId":1195,"prompt":1196,"options":1197,"correct":271,"why":1206},"number-system.und-q-est","The best estimate of 689 × 42 is…",[1198,1200,1202,1204],{"id":268,"label":1199},"2,800",{"id":271,"label":1201},"28,000",{"id":274,"label":1203},"2,80,000",{"id":277,"label":1205},"24,000","689 ≈ 700, 42 ≈ 40, and 700 × 40 = 28,000. (Exact: 28,938.)",{"itemId":1208,"prompt":1209,"options":1210,"correct":271,"why":1218},"number-system.und-q-roman","Which Roman numeral is 99?",[1211,1212,1214,1216],{"id":268,"label":935},{"id":271,"label":1213},"XCIX",{"id":274,"label":1215},"LXXXXIX",{"id":277,"label":1217},"IXC","99 = 90 + 9 = XC + IX = XCIX. IC breaks the rule that I is subtracted only from V and X.",{"itemId":1220,"prompt":1221,"options":1222,"correct":274,"why":1231},"number-system.und-q-units","4 km 60 m in metres is…",[1223,1225,1227,1229],{"id":268,"label":1224},"460 m",{"id":271,"label":1226},"4,600 m",{"id":274,"label":1228},"4,060 m",{"id":277,"label":1230},"40,060 m","4 km = 4,000 m; add 60 m: 4,060 m.",{"itemId":1233,"prompt":1234,"options":1235,"correct":271,"why":1244},"number-system.und-q-cmp","Which list is in descending order?",[1236,1238,1240,1242],{"id":268,"label":1237},"1,00,001; 99,999; 1,00,100",{"id":271,"label":1239},"1,00,100; 1,00,001; 99,999",{"id":274,"label":1241},"99,999; 1,00,001; 1,00,100",{"id":277,"label":1243},"1,00,001; 1,00,100; 99,999","Descending means largest first: 1,00,100 > 1,00,001 > 99,999.",{"id":1246,"type":1247,"prompt":1248},"reflect-u","reflection","The two systems write the same numbers with the same digits. Which system do you find easier for very big numbers, and why? Think of one situation in which it would be dangerous to confuse a lakh with a million.",{"id":1250,"type":1251,"conceptId":1252,"relation":1253,"explanation":1254},"conn-u-ops","connection","four-operations","helps_understand","Estimating before multiplying or dividing uses rounding to the greatest place, and carrying in column methods is regrouping between places.",{"id":1256,"type":1251,"conceptId":1257,"relation":1258,"explanation":1259},"conn-u-prop","properties-of-numbers","related_to","Whole numbers, successors and predecessors lead straight into properties such as closure and the role of 0 and 1.",{"id":1261,"type":1251,"conceptId":1262,"relation":1258,"explanation":1263},"conn-u-angles","constructing-angles","Reading measurement scales, like a protractor or a ruler in mm and cm, relies on the same place-value thinking as metric units.",{"id":1265,"type":1266,"title":1267,"points":1268},"cheat-u","summary","Cheat sheet",[1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280],"**Place value** = face value × value of the place. **Face value** = the digit. Place value of 0 is 0.","**Expanded form:** 70,40,396 = 70,00,000 + 40,000 + 300 + 90 + 6 = 7 × 10,00,000 + 4 × 10,000 + 3 × 100 + 9 × 10 + 6 × 1.","**Names:** read period by period; skip all-zero periods; lakh, crore, million stay singular.","**Commas:** Indian 3 then 2s (12,34,56,789); International 3s (123,456,789).","**Key facts:** 10 lakh = 1 million; 1 crore = 10 million; 10 crore = 100 million; 100 crore = 1 billion.","**n-digit numbers:** smallest 10ⁿ⁻¹ written as 1 then (n − 1) zeros; largest is n nines; there are 9 × 10ⁿ⁻¹ of them.","**Compare:** more digits wins; else first differing digit from the left.","**Forming:** greatest = digits in descending order; smallest = ascending, but never start with 0.","**Rounding:** look only at the deciding digit; 5 or more rounds up; round once from the original number.","**Estimating:** round each number (often to its greatest place), then calculate; use it to plan and to check.","**Roman:** I V X L C D M; repeat at most 3 times; V, L, D never repeated or subtracted; only IV IX XL XC CD CM subtract.","**Metric prefixes:** kilo 1,000; centi one hundredth; milli one thousandth.",{"id":1282,"type":1283,"sourceIds":1284},"sources-u","sources",[1285,1286,1287,1288,1289,1290,1291],"number-system-ncert-class6-knowing-numbers","number-system-ncert-class7-large-numbers","number-system-khan-place-value","number-system-wiki-indian-numbering","number-system-mathsisfun-roman","number-system-bipm-si-prefixes","number-system-mathsisfun-place-value",[1285,1286,1287,1288,1289,1290,1291],"needs_review",{"generatedBy":1295,"notes":1296},"claude-code","Draft generated with Python-checked numbers; pending owner review.","0e76cfe5bf5544e057f833a1b88dff1ace2c2b8a601a9e9b6f6094026c2edcdd",{"logic:practice":1299,"component:place-value@1":1300,"component:match-pairs@1":1301,"component:rounding-race@1":1302,"component:arith-sprint@1":1303,"component:sort-game@1":1304,"source:number-system-bipm-si-prefixes":1305,"source:number-system-khan-place-value":1306,"source:number-system-mathsisfun-place-value":1307,"source:number-system-mathsisfun-roman":1308,"source:number-system-ncert-class6-knowing-numbers":1309,"source:number-system-ncert-class7-large-numbers":1310,"source:number-system-wiki-indian-numbering":1311},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","cae1e81c06fcfcc0152eb325ac8fb01f568e21a19ad76a75d79194e17231ae72","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","4b1ef23b043f9642cf2f3613a60f4e05e4eb4a7c356f1100a1808edc05a4ecb3","8d110718a4c12de7f84b3e0653db1fd8b1147917a7dc31c48861b97c25731e0d","6a40a6a059b316a1b3d0621b353f1e4e944ed797310ad87c02f106fc0469f729","ef0f2abd2b3d7e393f1878aa51359c7a18194adf57596671bc0dff39cd1aa2f5","4e646d96c29f6b98469c9d11ee30da4e543f849b673fd067975aa685d8fef8a3","9dc5f1741ed9fad68db69882c1cc09c8d4b3cf5902d5d0273891bad1da1d10c0","045032c28b7bfe33f8bc73c5c09ee8dd8076293c90dbe922d8a461e059edd6e3",{"state":1313,"reviewer":1314,"selfReview":406,"reviewedAt":1315,"method":1316},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597895]