[{"data":1,"prerenderedAt":1132},["ShallowReactive",2],{"layer:number-system:deepen":3},{"layer":4,"contentHash":1109,"dependencyHashes":1110,"approval":1126,"releaseId":1131},{"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":1104,"reviewStatus":1105,"authoring":1106},1,"number-system","en","deepen","Why place value works","Powers of ten, proofs of the rules, error bounds and the Indian story of zero","Powers of ten, and proofs that the rules for comparing, rounding and forming numbers always work. Bound estimate errors, meet Sanskrit names for powers of ten, follow our digits from Brahmi to Aryabhata to Baghdad to Europe, and see metric units as place value.",[13,14,15,16,17],"Write any number in expanded form with powers of ten and explain why 10⁰ = 1.","Give a reasoned argument for the comparison rule and for the formula 9 × 10ⁿ⁻¹ for the number of n-digit numbers.","Solve harder forming problems with conditions, and find upper and lower bounds for estimates.","Describe how the Hindu–Arabic place-value system developed in India and spread to the world, and why zero was essential.","Convert between metric units by treating the prefixes as places.",50,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Go deeper",{"label":26,"value":27},"Reading time","≈ 50 minutes",{"label":29,"value":30},"Prior knowledge","Understand and Investigate layers",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Place value, rounding, estimate sort, prefix match",{"label":38,"value":39},"New notation","Powers of ten: 10³ = 1,000",[41,45,51,54,114,120,130,152,166,171,174,179,194,199,203,208,211,220,230,241,246,256,265,274,283,294,299,302,311,322,327,337,348,352,366,371,374,383,430,435,438,486,490,500,521,532,537,540,585,589,592,603,607,629,632,637,647,650,655,660,663,715,724,756,767,784,789,792,802,811,815,826,835,846,855,864,868,879,922,1050,1054,1060,1065,1069,1074,1090],{"id":42,"type":43,"markdown":44},"intro-d","prose","You can already read, write, compare and round large numbers. This layer asks **why** the methods work, and whether they *always* work. We will prove the comparison rule instead of just trusting it, find the exact worst-case error of an estimate, solve forming puzzles with extra conditions, and trace the long history that gave the whole world the ten digits you use every day.\n\nProofs here are not formal, but they are real arguments: each one explains why something must be true for **every** number, not just the examples we tried.",{"id":46,"type":47,"title":48,"eyebrow":49,"navLabel":50},"ch01","chapter","Powers of ten","Chapter 01","1 Powers of ten",{"id":52,"type":43,"markdown":53},"powers","Multiplying 10 by itself again and again gives the place values. We write this with a small raised number, the **exponent**, that counts how many tens are multiplied:\n\n- 10² = 10 × 10 = 100\n- 10³ = 10 × 10 × 10 = 1,000\n- 10⁵ = 1,00,000 (one lakh), 10⁷ = 1,00,00,000 (one crore), 10⁹ = one billion\n\nThe exponent is also the **number of zeros** after the 1. And each step to the left multiplies by one more 10, which is exactly the rule \"each place is ten times the place to its right\".",{"id":55,"type":56,"caption":57,"columns":58,"rows":63},"table-powers","table","Places as powers of ten",[59,60,61,62],"Power","Value","Indian name","International name",[64,68,72,76,80,84,89,94,99,104,109],[65,66,67,67],"10⁰","1","ones",[69,70,71,71],"10¹","10","tens",[73,74,75,75],"10²","100","hundreds",[77,78,79,79],"10³","1,000","thousands",[81,82,83,83],"10⁴","10,000","ten thousands",[85,86,87,88],"10⁵","1,00,000","lakhs","hundred thousands",[90,91,92,93],"10⁶","10,00,000","ten lakhs","millions",[95,96,97,98],"10⁷","1,00,00,000","crores","ten millions",[100,101,102,103],"10⁸","10,00,00,000","ten crores","hundred millions",[105,106,107,108],"10⁹","1,00,00,00,000","arabs","billions",[110,111,112,113],"10¹⁰","10,00,00,00,000","ten arabs","ten billions",{"id":115,"type":116,"variant":117,"title":118,"markdown":119},"nuance-zero-power","callout","nuance","Why is 10⁰ equal to 1?","Read the powers from the top down: 10³ = 1,000, 10² = 100, 10¹ = 10. Each time the exponent drops by 1 we **divide by 10**. One more step: 10 ÷ 10 = 1, so 10⁰ = 1. This is not a trick; it is the only value that keeps the pattern working, and it makes the ones place fit the same rule as every other place: the ones place is 10⁰.",{"id":121,"type":122,"title":123,"problem":124,"steps":125},"we-pow-exp","worked_example","Expanded form with powers of ten","Write 3,04,05,027 in expanded form using powers of ten.",[126,127,128,129],"Place each non-zero digit: 3 in crores (10⁷), 4 in lakhs (10⁵), 5 in thousands (10³), 2 in tens (10¹), 7 in ones (10⁰).","**3,04,05,027 = 3 × 10⁷ + 4 × 10⁵ + 5 × 10³ + 2 × 10¹ + 7 × 10⁰**.","Check: 3,00,00,000 + 4,00,000 + 5,000 + 20 + 7 = 3,04,05,027. ✓","The exponents that are missing (10⁶, 10⁴, 10²) are the places holding a 0.",{"id":131,"type":132,"items":133},"formulas-pow","formulas",[134,137,140,143,146,149],{"expression":135,"caption":136},"10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ","Multiplying powers of ten adds zeros: 10³ × 10⁵ = 10⁸ (a thousand lakh = ten crore).",{"expression":138,"caption":139},"1 lakh = 10⁵","Five zeros.",{"expression":141,"caption":142},"1 crore = 10⁷","Seven zeros.",{"expression":144,"caption":145},"1 million = 10⁶","Six zeros.",{"expression":147,"caption":148},"1 billion = 10⁹","Nine zeros.",{"expression":150,"caption":151},"1 lakh crore = 10¹²","10⁵ × 10⁷ = 10¹²: one trillion.",{"id":153,"type":154,"itemId":155,"prompt":156,"check":157,"hints":161,"feedback":163},"pr-pow","practice","number-system.deepen-powers","How many zeros are there in 1 crore × 1 lakh when written out in full?",{"kind":158,"answer":159,"tolerance":160},"number",12,0,[162],"Write each as a power of ten first.",{"correct":164,"incorrect":165},"Right: 10⁷ × 10⁵ = 10¹², so 12 zeros. This is one lakh crore, or one trillion.","Crore = 10⁷ (7 zeros), lakh = 10⁵ (5 zeros). Multiplying adds the exponents: 7 + 5 = 12 zeros.",{"id":167,"type":47,"title":168,"eyebrow":169,"navLabel":170},"ch02","One number, one way to write it","Chapter 02","2 Why zero matters",{"id":172,"type":43,"markdown":173},"unique","Here is a quiet but deep fact: **every whole number has exactly one standard way to be written with digits.** Think of the bundling picture. Starting from a heap of sticks, you must make as many bundles of ten as possible, then as many bundles of a hundred as possible, and so on. Each place ends up with between 0 and 9, and there is only one way the bundling can come out. So *four hundred eight* can only be 408.\n\nThis only works because we have a symbol for \"nothing in this place\". Without zero, 48 and 408 and 4,008 would all be written with just a 4 and an 8, perhaps with a gap. Ancient scribes in Babylon used a gap for centuries, and later a special mark, and got confused whenever a gap came at the end of a number.",{"id":175,"type":116,"variant":176,"title":177,"markdown":178},"aha-zero-two","aha","Zero has two jobs","1. **Placeholder:** in 408, the 0 keeps the 4 in the hundreds place. This use appears in several ancient cultures.\n2. **A number in its own right:** 0 is a quantity you can add, subtract and multiply: 5 − 5 = 0, 7 × 0 = 0. Treating zero as a number, with rules for calculating with it, was written down clearly by the Indian mathematician **Brahmagupta** in 628 CE. That second step is what turned a counting notation into modern arithmetic.",{"id":180,"type":181,"prompt":182,"options":183,"explanation":193},"pred-unique","prediction","Can two **different** strings of digits, neither starting with 0, stand for the same whole number?",[184,187,190],{"id":185,"label":186},"a","Yes, for example 0100 and 100",{"id":188,"label":189},"b","Yes, if the digits are rearranged",{"id":191,"label":192},"c","No, never","**c: never.** 0100 is ruled out because it starts with 0, and rearranging digits changes the value (100 and 010 are different, and 010 is not allowed). Here is the argument: if two digit strings differ, look at the first place from the left where they differ. By the comparison argument in the next chapter, the one with the bigger digit there is the bigger number, so they cannot be equal. Different strings always mean different numbers; that is uniqueness.",{"id":195,"type":116,"variant":196,"title":197,"markdown":198},"mis-zero-nothing","misconception","“Adding a zero anywhere makes a number ten times bigger”","Putting a 0 on the **right** end of a whole number multiplies it by 10 (45 → 450), because every digit shifts one place left. Putting a 0 on the **left** (045) changes nothing, and putting one in the **middle** (45 → 405) moves only the digits to its left: 405 is 9 times 45. Zero changes a number only by pushing other digits into new places.",{"id":200,"type":116,"variant":196,"title":201,"markdown":202},"mis-roman-unique","“Every notation has one way to write a number”","Roman numerals do not: 4 has been written both IIII and IV, and 1,999 can be squeezed into odd forms like MIM, which breaks the standard rules but was sometimes carved anyway. Standard rules were agreed later to make each numeral unique. Place value gives uniqueness automatically.",{"id":204,"type":47,"title":205,"eyebrow":206,"navLabel":207},"ch03","Proving the comparison rule","Chapter 03","3 Proof: comparing",{"id":209,"type":43,"markdown":210},"proof-cmp","**Claim 1:** any whole number with n + 1 digits is bigger than any whole number with n digits.\n\n*Argument.* The largest n-digit number is n nines, which equals 10ⁿ − 1 (for example 999 = 1,000 − 1). The smallest (n + 1)-digit number is 10ⁿ. Since 10ⁿ − 1 is less than 10ⁿ, even the smallest longer number beats the largest shorter number. ∎\n\n**Claim 2:** if two numbers have the same number of digits, the first place (from the left) where they differ decides which is bigger.\n\n*Argument.* Suppose they agree on every place to the left of place 10ᵏ, and at place 10ᵏ the first number has a bigger digit. Then at that place the first number is ahead by at least 1 × 10ᵏ. The places to the right can help the second number by at most 99…9 (k nines) = 10ᵏ − 1. That is never enough to catch up: 10ᵏ is more than 10ᵏ − 1. ∎",{"id":212,"type":122,"title":213,"problem":214,"steps":215},"we-proof-eg","Seeing Claim 2 with numbers","Why is 5,30,000 greater than 5,29,999, even though every digit after the first difference is bigger in the second number?",[216,217,218,219],"They agree in the lakhs place (5). The first difference is the ten thousands place: 3 against 2.","So the first number is ahead by at least 10,000 at that place.","The remaining four places (thousands to ones) can add at most 9,999 to the second number.","9,999 is less than 10,000, so the second number can never catch up: 5,30,000 − 5,29,999 = 1, just barely, but always.",{"id":221,"type":122,"title":222,"problem":223,"steps":224},"we-count-proof","Why there are 9 × 10ⁿ⁻¹ numbers with n digits","Show that there are 9 × 10ⁿ⁻¹ whole numbers with exactly n digits, and use it to count 7-digit numbers.",[225,226,227,228,229],"The first digit cannot be 0, so it has 9 choices (1 to 9).","Each of the remaining n − 1 digits has 10 choices (0 to 9).","Each choice gives a different number (by uniqueness), so there are 9 × 10 × 10 × … = 9 × 10ⁿ⁻¹ numbers.","For n = 7: 9 × 10⁶ = **90,00,000** numbers with 7 digits, from 10,00,000 to 99,99,999.","Check the other way: 99,99,999 − 10,00,000 + 1 = 90,00,000. ✓",{"id":231,"type":154,"itemId":232,"prompt":233,"check":234,"hints":236,"feedback":238},"pr-count-rep","number-system.deepen-count-norep","How many 3-digit numbers have **all different digits**?",{"kind":158,"answer":235,"tolerance":160},648,[237],"The second digit may be 0, but it cannot repeat the first.",{"correct":239,"incorrect":240},"Right: 9 choices for the first digit (1–9), 9 for the second (0–9 except the first), 8 for the third: 9 × 9 × 8 = 648.","First digit: 9 choices (not 0). Second: any of 10 digits except the one used, 9 choices. Third: 8 choices. 9 × 9 × 8 = 648.",{"id":242,"type":47,"title":243,"eyebrow":244,"navLabel":245},"ch04","Forming numbers with conditions","Chapter 04","4 Harder forming",{"id":247,"type":122,"title":248,"problem":249,"steps":250},"we-even","Greatest even number","Using each of 1, 4, 7, 0, 5 exactly once, form the greatest 5-digit **even** number.",[251,252,253,254,255],"An even number ends in 0, 2, 4, 6 or 8. From our digits, the last digit must be 0 or 4.","Option 1, end in 0: the other digits in descending order in front: 7, 5, 4, 1 → 75,410.","Option 2, end in 4: 7, 5, 1, 0 in front → 75,104.","Compare: 75,410 > 75,104. **Greatest even number: 75,410**.","Lesson: using the smaller even digit at the end keeps the bigger digits for the high places.",{"id":257,"type":122,"title":258,"problem":259,"steps":260},"we-odd","Smallest odd number","Using each of 0, 2, 4, 5, 7 exactly once, form the smallest 5-digit **odd** number.",[261,262,263,264],"An odd number must end in 5 or 7 here.","The first digit must be the smallest non-zero digit available: 2. Then 0 next.","End in 7: 2, 0, 4, 5 then 7 → 20,457. End in 5: 2, 0, 4, 7 then 5 → 20,475.","**Smallest odd number: 20,457**. Using the larger odd digit at the end lets the smaller digits take the higher places.",{"id":266,"type":122,"title":267,"problem":268,"steps":269},"we-closest","Closest to a target","Using each of 2, 9, 4, 6 once, form the 4-digit number closest to 5,000.",[270,271,272,273],"Candidates must start with 4 (just below 5,000) or 6 (just above).","Starting with 4: make the rest as large as possible → 4,962; distance 5,000 − 4,962 = 38.","Starting with 6: make the rest as small as possible → 6,249; distance 1,249.","**4,962** is closest. A brute-force check of all 24 arrangements agrees.",{"id":275,"type":122,"title":276,"problem":277,"steps":278},"we-mult5-zero","When zero is both the problem and the solution","Using each of 4, 0, 5, 2, 9 exactly once, form the smallest 5-digit multiple of 5.",[279,280,281,282],"A multiple of 5 ends in 0 or 5.","Ending in 0: the 0 is used up at the end, so the front can start with the smallest digit 2: 2, 4, 5, 9 then 0 → 24,590.","Ending in 5: the 0 is free, but cannot go first, so 2 first, then 0: 2, 0, 4, 9 then 5 → 20,495.","Compare: 20,495 is smaller. **Answer: 20,495**. Keeping the 0 available for the second place beats using it at the end.",{"id":284,"type":154,"itemId":285,"prompt":286,"check":287,"hints":289,"feedback":291},"pr-form-5","number-system.deepen-form-div5","Using the digits 3, 0 and 8 (repetition allowed, and each must appear at least once), what is the greatest 5-digit number that is a multiple of 5?",{"kind":158,"answer":288,"tolerance":160},88830,[290],"What must the last digit be?",{"correct":292,"incorrect":293},"Right: a multiple of 5 ends in 0 or 5, and only 0 is available. Fill the front with 8s, keep a 3: 88,830.","It must end in 0. Then make the front as large as possible while still using a 3: 8, 8, 8, 3 then 0 → 88,830.",{"id":295,"type":47,"title":296,"eyebrow":297,"navLabel":298},"ch05","How wrong can rounding be?","Chapter 05","5 Rounding error",{"id":300,"type":43,"markdown":301},"round-err","When you round to the nearest 100, how far can the rounded number be from the original? The worst case is a number exactly halfway, like 350 → 400: an error of 50. So **rounding to the nearest unit U changes a number by at most half of U**.\n\nThat lets us put a guaranteed limit, called a **bound**, on the error of an estimated sum. If you round 6 numbers each to the nearest 100, each is off by at most 50, so the estimated total is off by at most 6 × 50 = 300. Usually the errors partly cancel and the real error is much smaller, but it can never be more.",{"id":303,"type":122,"title":304,"problem":305,"steps":306},"we-bound","A guaranteed error bound","A shop records six sales: ₹2,349, ₹1,872, ₹4,450, ₹3,017, ₹968, ₹5,321. Estimate the total by rounding each to the nearest hundred, and say how far off the estimate could possibly be.",[307,308,309,310],"Rounded: 2,300, 1,900, 4,500, 3,000, 1,000, 5,300.","Estimated total: ₹18,000.","Each rounding changes a number by at most 50, so the total is off by at most 6 × 50 = ₹300.","Exact total: ₹17,977. Actual error: ₹23, well inside the bound. ✓",{"id":312,"type":154,"itemId":313,"prompt":314,"check":315,"hints":317,"feedback":319},"pr-bound-1000","number-system.deepen-bound","Eight numbers are each rounded to the nearest thousand and then added. At most how far can the estimated total be from the exact total?",{"kind":158,"answer":316,"tolerance":160},4000,[318],"What is the most one number can change?",{"correct":320,"incorrect":321},"Right: each number changes by at most 500, so eight numbers change the total by at most 8 × 500 = 4,000.","Rounding to the nearest thousand changes a number by at most half a thousand, 500. Eight numbers: 8 × 500 = 4,000.",{"id":323,"type":116,"variant":324,"title":325,"markdown":326},"careful-carry","careful","Rounding can add a digit","Rounding 9,960 to the nearest hundred: the hundreds digit is 9 and the deciding digit is 6, so we add 1 hundred. 99 hundreds + 1 hundred = 100 hundreds: **10,000**. The answer has one more digit than the number. Similarly, 9,99,700 to the nearest thousand is 10,00,000. When the rounding place holds a 9 and you round up, carry just as in addition.",{"id":328,"type":122,"title":329,"problem":330,"steps":331},"we-double-proof","Exactly when does double rounding go wrong?","Show that rounding to the nearest ten and then to the nearest hundred gives a different answer from rounding straight to the hundred exactly when the last two digits are 45, 46, 47, 48 or 49.",[332,333,334,335,336],"Rounding straight to the hundred depends only on the last two digits, call them t: round up if t ≥ 50, down if t ≤ 49.","Rounding to the ten first changes t to the nearest multiple of 10. Then the second step rounds up exactly when that new value is 50 or more.","t becomes 50 or more exactly when t ≥ 45 (because 45 to 49 round up to 50).","So the two methods disagree exactly when 45 ≤ t ≤ 49: the direct method rounds down, the two-step method rounds up.","That is 5 values of t out of 100, which is why about 5% of numbers are affected, as the Investigate layer found by computer. ∎",{"id":338,"type":154,"itemId":339,"prompt":340,"check":341,"hints":343,"feedback":345},"pr-carry-lakh","number-system.deepen-round-carry","Round **99,64,300** to the nearest lakh.",{"kind":158,"answer":342,"tolerance":160},10000000,[344],"What is 99 lakh + 1 lakh?",{"correct":346,"incorrect":347},"Right: the lakhs digit is 9 and the deciding digit (ten thousands) is 6, so round up; the carry runs through both 9s: 1,00,00,000, one crore.","Round up because the ten thousands digit is 6. 99 lakh + 1 lakh = 100 lakh = 1 crore = 1,00,00,000. The answer has one more digit.",{"id":349,"type":116,"variant":117,"title":350,"markdown":351},"nuance-half","Round half up is a choice, not a law","Rounding 45 up to 50 is a **convention**. It is simple, but if you round many halfway numbers the up-rounding adds a small upward bias. Some banks and scientists use **round half to even** instead: 45 → 40 but 55 → 60 (always to the even tens digit), so halfway cases go up and down equally often. Computers often use this rule. In school and in this lesson we use round half up unless told otherwise.",{"id":353,"type":354,"component":355,"componentVersion":5,"config":356,"objective":364,"textAlternative":365},"lab-round-big","interactive","rounding-race",{"roundTo":357,"range":360,"rounds":362,"secondsPerRound":363},[358,359],10000,100000,{"min":359,"max":361},99999999,10,20,"Round numbers up to nine crore ninety-nine lakh to the nearest ten thousand or lakh, including tricky carries.","Ten rounds, 20 seconds each. A number between 1,00,000 and 9,99,99,999 appears on a number line; round it to the nearest 10,000 or 1,00,000.\n\nWatch for carries. 4,99,62,000 to the nearest lakh: the lakhs digit is 9 and the deciding digit (ten thousands) is 6, so it rounds up and the carry ripples: 5,00,00,000. To the nearest ten thousand it becomes 4,99,60,000 (deciding digit 2). And 7,34,50,000 to the nearest lakh is exactly halfway, so it rounds up to 7,35,00,000.",{"id":367,"type":47,"title":368,"eyebrow":369,"navLabel":370},"ch06","Upper and lower estimates","Chapter 06","6 Estimate bounds",{"id":372,"type":43,"markdown":373},"bounds","Instead of one estimate, you can find two numbers that the exact answer must lie between. Round **both numbers down** to get a **lower estimate**; round **both up** to get an **upper estimate**. For sums and products of whole numbers, the exact answer is always between them.\n\nExample: 438 × 267. Lower: 400 × 200 = 80,000. Upper: 500 × 300 = 1,50,000. That range is wide. Rounding to tens tightens it: 430 × 260 = 1,11,800 and 440 × 270 = 1,18,800. The exact answer, 1,16,946, sits inside both ranges.",{"id":375,"type":122,"title":376,"problem":377,"steps":378},"we-bounds-diff","Bounds for a difference are different","Find a lower and an upper estimate for 7,842 − 3,165 by rounding to thousands.",[379,380,381,382],"For a **difference**, the answer is smallest when the first number is small and the second is big.","Lower estimate: 7,000 − 4,000 = 3,000 (first rounded down, second rounded up).","Upper estimate: 8,000 − 3,000 = 5,000 (first rounded up, second rounded down).","Exact: 4,677, between 3,000 and 5,000. ✓ Rounding \"both down\" would not guarantee a lower bound for a difference.",{"id":384,"type":354,"component":385,"componentVersion":5,"config":386,"objective":428,"textAlternative":429},"lab-sort-bounds","sort-game",{"prompt":387,"bins":388,"items":395,"seconds":160},"Without calculating exactly, is each estimate an underestimate or an overestimate?",[389,392],{"id":390,"label":391},"under","Underestimate",{"id":393,"label":394},"over","Overestimate",[396,400,404,407,410,413,416,419,422,425],{"id":397,"label":398,"bin":390,"why":399},"e1","4,812 + 3,276 ≈ 4,000 + 3,000","Both numbers made smaller (or, for a difference, the answer pushed smaller), so the estimate is too low.",{"id":401,"label":402,"bin":393,"why":403},"e2","4,812 + 3,276 ≈ 5,000 + 4,000","Both numbers made bigger (or, for a difference, the answer pushed bigger), so the estimate is too high.",{"id":405,"label":406,"bin":390,"why":399},"e3","68 × 72 ≈ 60 × 70",{"id":408,"label":409,"bin":393,"why":403},"e4","68 × 72 ≈ 70 × 80",{"id":411,"label":412,"bin":390,"why":399},"e5","9,120 − 2,480 ≈ 9,000 − 3,000",{"id":414,"label":415,"bin":393,"why":403},"e6","9,120 − 2,480 ≈ 10,000 − 2,000",{"id":417,"label":418,"bin":390,"why":399},"e7","149 × 149 ≈ 100 × 100",{"id":420,"label":421,"bin":393,"why":403},"e8","365 × 48 ≈ 400 × 50",{"id":423,"label":424,"bin":390,"why":399},"e9","612 + 387 ≈ 600 + 300",{"id":426,"label":427,"bin":393,"why":403},"e10","7,842 − 3,165 ≈ 8,000 − 3,000","Decide whether each estimate must be too low or too high by looking at the direction of rounding.","Ten estimates to sort into Underestimate or Overestimate, using only the direction of rounding.\n\nUnderestimates: 4,812 + 3,276 ≈ 4,000 + 3,000 (both down); 68 × 72 ≈ 60 × 70 (both down); 9,120 − 2,480 ≈ 9,000 − 3,000 (first down, second up); 149 × 149 ≈ 100 × 100; 612 + 387 ≈ 600 + 300.\n\nOverestimates: 4,812 + 3,276 ≈ 5,000 + 4,000 (both up); 68 × 72 ≈ 70 × 80; 9,120 − 2,480 ≈ 10,000 − 2,000 (first up, second down); 365 × 48 ≈ 400 × 50; 7,842 − 3,165 ≈ 8,000 − 3,000.\n\nFor sums and products: both down gives too low, both up gives too high. For differences: rounding the first number down or the second number up makes the answer too low.",{"id":431,"type":47,"title":432,"eyebrow":433,"navLabel":434},"ch07","Naming powers of ten","Chapter 07","7 Sanskrit names",{"id":436,"type":43,"markdown":437},"sanskrit","Why does the Indian system have a new name every **two** places (lakh, crore) while the International system has one every **three** (million, billion)? The Indian names come from a very old tradition. Sanskrit texts give a **separate name for every power of ten**, going far beyond anything needed for counting cattle or coins. One well-known list, in the Yajurveda, runs up to 10¹²:",{"id":439,"type":56,"caption":440,"columns":441,"rows":444},"table-sanskrit","Sanskrit names for powers of ten (one traditional list; different texts vary)",[59,442,443],"Sanskrit name","Modern Indian name",[445,448,451,454,457,460,463,466,469,472,475,478,482],[65,446,447],"eka","one",[69,449,450],"dasha","ten",[73,452,453],"shata","hundred",[77,455,456],"sahasra","thousand",[81,458,459],"ayuta","ten thousand",[85,461,462],"niyuta \u002F laksha","lakh",[90,464,465],"prayuta","ten lakh",[95,467,468],"arbuda \u002F koti","crore",[100,470,471],"nyarbuda","ten crore",[105,473,474],"samudra","arab (100 crore)",[110,476,477],"madhya","ten arab",[479,480,481],"10¹¹","anta","kharab",[483,484,485],"10¹²","parardha","ten kharab = 1 lakh crore",{"id":487,"type":116,"variant":117,"title":488,"markdown":489},"nuance-names-vary","The old names are not all the same","Different Sanskrit, Buddhist and Jain texts give different lists, and some names (like *arbuda*) are used for different powers in different texts. What they share is the remarkable idea of naming **every** power of ten, which shows how naturally Indian mathematicians thought in place value. Today only lakh and crore are in everyday use, with arab and kharab occasionally used for larger amounts.",{"id":491,"type":122,"title":492,"problem":493,"steps":494},"we-conv-hard","Big conversions using powers","The Union Budget of India is often quoted in \"lakh crore\". Convert ₹50 lakh crore to the International system.",[495,496,497,498,499],"1 lakh crore = 10⁵ × 10⁷ = 10¹².","50 lakh crore = 50 × 10¹² = 5 × 10¹³.","In digits: 50,000,000,000,000.","Since 1 trillion = 10¹², this is **₹50 trillion**.","Shortcut: 1 lakh crore = 1 trillion, so x lakh crore = x trillion.",{"id":501,"type":132,"items":502},"formulas-conv-d",[503,506,509,512,515,518],{"expression":504,"caption":505},"crore → million: × 10","3.2 crore = 32 million.",{"expression":507,"caption":508},"million → crore: ÷ 10","85 million = 8.5 crore.",{"expression":510,"caption":511},"crore → billion: ÷ 100","245 crore = 2.45 billion.",{"expression":513,"caption":514},"billion → crore: × 100","1.4 billion = 140 crore.",{"expression":516,"caption":517},"lakh → million: ÷ 10","36 lakh = 3.6 million.",{"expression":519,"caption":520},"lakh crore = trillion","1 lakh crore = 10¹² = 1 trillion.",{"id":522,"type":154,"itemId":523,"prompt":524,"check":525,"hints":527,"feedback":529},"pr-bn-crore","number-system.deepen-billion-crore","A company is valued at 3.75 billion dollars. How many crore dollars is that?",{"kind":158,"answer":526,"tolerance":160},375,[528],"1 billion = 100 crore.",{"correct":530,"incorrect":531},"Right: 1 billion = 100 crore, so 3.75 billion = 375 crore.","Multiply billions by 100 to get crores: 3.75 × 100 = 375 crore.",{"id":533,"type":47,"title":534,"eyebrow":535,"navLabel":536},"ch08","The journey of our ten digits","Chapter 08","8 History",{"id":538,"type":43,"markdown":539},"history-intro","The digits 0–9 are often called **Arabic numerals**, but mathematicians and historians call them **Hindu–Arabic numerals**, because the system was developed in India and carried to Europe by scholars writing in Arabic. Its story took well over a thousand years.",{"id":541,"type":542,"title":543,"items":544},"timeline-digits","timeline","From Brahmi marks to the world’s digits (dates approximate; some are debated by historians)",[545,549,553,557,561,565,569,573,577,581],{"time":546,"title":547,"text":548},"c. 250 BCE","Brahmi numerals","Inscriptions from the time of Emperor Ashoka use Brahmi numerals, with separate signs for 1–9, for tens and for hundreds. No place value yet and no zero.",{"time":550,"title":551,"text":552},"c. 200 BCE","Pingala","Pingala’s work on Sanskrit poetic metres uses a system of short and long syllables that later scholars recognise as binary counting.",{"time":554,"title":555,"text":556},"100s–600s CE","Place value appears","Indian texts begin writing numbers with nine digits whose value depends on position, and with words for zero such as *shunya* (empty).",{"time":558,"title":559,"text":560},"499 CE","Aryabhata","The Aryabhatiya, by Aryabhata (born 476 CE), describes the decimal places, each ten times the previous: “from place to place, ten times”.",{"time":562,"title":563,"text":564},"628 CE","Brahmagupta","The Brahmasphutasiddhanta gives rules for calculating with zero and negative numbers, treating zero as a number.",{"time":566,"title":567,"text":568},"c. 825 CE","al-Khwarizmi","In Baghdad, al-Khwarizmi writes a book on calculating with the Hindu numerals; its Latin translation spreads the method in Europe.",{"time":570,"title":571,"text":572},"876 CE","Gwalior zero","An inscription at the Chaturbhuj temple in Gwalior shows the number 270 with a small round zero, one of the oldest dated zeros in India.",{"time":574,"title":575,"text":576},"1202 CE","Fibonacci","Leonardo of Pisa (Fibonacci) promotes the Hindu–Arabic numerals in his book *Liber Abaci*, showing merchants how much easier they make calculation.",{"time":578,"title":579,"text":580},"1400s–1500s","Printing","Printed arithmetic books make the ten digits standard across Europe, slowly replacing Roman numerals in accounts.",{"time":582,"title":583,"text":584},"Today","Everywhere","The same ten digits and place-value rules are used on every continent, in every computer and on every phone.",{"id":586,"type":116,"variant":117,"title":587,"markdown":588},"nuance-bakhshali","The Bakhshali manuscript and the date of zero","The Bakhshali manuscript, found in 1881 near Mardan in what is now Pakistan, uses a **dot** as a zero placeholder. Radiocarbon dating announced in 2017 suggested that some of its pages could be as old as the 3rd or 4th century CE, but different pages gave very different dates, scholars criticised both the method and the announcement, and in October 2024 Oxford revised the dating of the manuscript to about 799–1102 CE. What is certain is that a written zero symbol was in use in India well before it reached Europe.",{"id":590,"type":43,"markdown":591},"bhuta","Before written digits were common in texts, Indian astronomers often wrote numbers in **words**, inside verses that were easy to memorise. In the *bhutasankhya* (\"object numbers\") system, familiar things stood for digits: *moon* or *earth* for 1 (there is one of each), *eyes* or *hands* for 2, *fires* for 3 (three sacred fires), *Vedas* for 4, *arrows* of the god of love for 5, *seasons* for 6 and so on, with *sky* or *void* for 0.\n\nThe words were read from the **ones place upward**, so \"sky, eyes, moon\" meant 0 ones, 2 tens, 1 hundred: 120. This only works because the words fill **places**: bhutasankhya is place value written in poetry, and its use of *sky* for an empty place shows zero being treated as a digit.",{"id":593,"type":154,"itemId":594,"prompt":595,"check":596,"hints":598,"feedback":600},"pr-bhuta","number-system.deepen-bhutasankhya","In bhutasankhya, words are read from the ones place upward. If *Vedas* = 4, *sky* = 0, *eyes* = 2 and *moon* = 1, what number is written “Vedas, sky, eyes, moon”?",{"kind":158,"answer":597,"tolerance":160},1204,[599],"The first word is the ones digit.",{"correct":601,"incorrect":602},"Right: ones 4, tens 0, hundreds 2, thousands 1, so the number is 1,204.","Put the words in places starting from the ones: Vedas (4) ones, sky (0) tens, eyes (2) hundreds, moon (1) thousands → 1,204.",{"id":604,"type":116,"variant":196,"title":605,"markdown":606},"mis-arabic","“Arabic numerals were invented by Arabs”","Arab and Persian scholars, especially in Baghdad, learned the numerals from India, used them, improved how they were written and taught them to the world. Arabic writers themselves often called them *Hindi numerals*. Both halves of the name Hindu–Arabic are deserved: India for the invention, the Arabic-speaking world for the spread. Even today the digits used in many Arab countries (٠١٢٣…) look different from the European forms, though the place-value system is the same.",{"id":608,"type":154,"itemId":609,"prompt":610,"check":611,"hints":624,"feedback":626},"pr-history-order","number-system.deepen-history-order","Put these in time order, earliest first: (P) Fibonacci’s *Liber Abaci*, (Q) Aryabhata’s *Aryabhatiya*, (R) Brahmi numerals in Ashoka’s time, (S) al-Khwarizmi’s book on Hindu numerals.",{"kind":612,"options":613,"correct":623},"choice",[614,616,618,620],{"id":185,"label":615},"R, Q, S, P",{"id":188,"label":617},"Q, R, S, P",{"id":191,"label":619},"R, S, Q, P",{"id":621,"label":622},"d","R, Q, P, S",[185],[625],"BCE dates come before all CE dates.",{"correct":627,"incorrect":628},"Right: Brahmi (c. 250 BCE), Aryabhata (499 CE), al-Khwarizmi (c. 825 CE), Fibonacci (1202 CE).","Use the timeline: about 250 BCE, then 499, then about 825, then 1202. That is R, Q, S, P.",{"id":630,"type":43,"markdown":631},"why-powerful","Why did this system win? Compare what it needs with what earlier systems needed:\n\n- **Ten symbols are enough forever.** Roman numerals kept needing new symbols (and bars) for bigger numbers; Brahmi needed separate signs for each ten and each hundred.\n- **Calculation on paper becomes mechanical.** Column addition, carrying, long multiplication and long division all rely on place value. With Roman numerals, merchants used an abacus and wrote only the answer.\n- **Comparing is instant.** Count digits, then scan from the left.\n- **It extends naturally** to decimals (tenths, hundredths) and, as we will see in Extend, to other bases such as binary.",{"id":633,"type":47,"title":634,"eyebrow":635,"navLabel":636},"ch09","Roman numerals under the microscope","Chapter 09","9 Roman limits",{"id":638,"type":122,"title":639,"problem":640,"steps":641},"we-roman-add","Adding without place value","Add MCMXLIV and CCCLXXXIX in Roman numerals, the Roman way.",[642,643,644,645,646],"MCMXLIV = 1,944 and CCCLXXXIX = 389, but a Roman clerk would not convert. First they would undo the subtractions: CM → DCCCC, XL → XXXX, IV → IIII.","MDCCCCXXXXIIII + CCCLXXXVIIII: gather like symbols: M, D, CCCCCCC, L, XXXXXXX, V, IIIIIIII.","Regroup from the bottom: IIIIIIII (8) = V + III; now VV = X; XXXXXXXX (8 Xs) = L + XXX; LL = C; CCCCCCCC (8 Cs) = D + CCC; DD = M.","Result: MM + CCC + XXX + III = MMCCCXXXIII, which is 2333.","Check: 1,944 + 389 = 2,333. ✓ Written in standard form: MMCCCXXXIII.",{"id":648,"type":43,"markdown":649},"vinculum","Standard Roman numerals stop at 3,999 (MMMCMXCIX). For bigger numbers, a bar drawn over a numeral, the **vinculum**, multiplies it by 1,000: a barred V is 5,000 and a barred X is 10,000. Some writers used a box or double bar for hundred-thousands. It works, but every new size needs a new trick, which is exactly the problem place value solves once and for all.",{"id":651,"type":116,"variant":652,"title":653,"markdown":654},"ex-roman-today","example","Where Roman numerals survive","Kings and popes (Louis XIV), film sequels, the Olympics and the Super Bowl, class names in Indian schools, clock faces, book prefaces (pages i, ii, iii…) and copyright years on film credits: the year 1998 appears as MCMXCVIII. They survive as *labels*, not for calculation.",{"id":656,"type":47,"title":657,"eyebrow":658,"navLabel":659},"ch10","Metric prefixes are place value","Chapter 10","10 Metric places",{"id":661,"type":43,"markdown":662},"metric-d","The metric system was designed in France in the 1790s precisely so that units would follow place value. Each prefix is a power of ten, so a length in metres lines up with a place-value chart: kilometres in the thousands place, hectometres in hundreds, decametres in tens, metres in ones, and then decimetres, centimetres and millimetres in the tenths, hundredths and thousandths.",{"id":664,"type":56,"caption":665,"columns":666,"rows":672},"table-prefix","Metric prefixes as places (length shown; the same prefixes work for grams and litres)",[667,668,669,670,671],"Prefix","Symbol","Power of ten","Meaning","Example",[673,678,683,687,691,695,699,704,709],[674,675,105,676,677],"giga","G","billion","1 GB of data ≈ a billion bytes",[679,680,90,681,682],"mega","M","million","1 megawatt = 10,00,000 watts",[684,685,77,456,686],"kilo","k","1 km = 1,000 m",[688,689,73,453,690],"hecto","h","1 hm = 100 m",[692,693,69,450,694],"deca","da","1 dam = 10 m",[696,697,65,447,698],"(none)","—","1 m",[700,621,701,702,703],"deci","one tenth","tenth","10 dm = 1 m",[705,191,706,707,708],"centi","one hundredth","hundredth","100 cm = 1 m",[710,711,712,713,714],"milli","m","one thousandth","thousandth","1,000 mm = 1 m",{"id":716,"type":122,"title":717,"problem":718,"steps":719},"we-metric-d","Unpacking a length","Write 2,345 mm in metres, centimetres and millimetres, and as a place-value chart.",[720,721,722,723],"Since 1,000 mm = 1 m, the thousands digit counts metres: 2 m.","Since 10 mm = 1 cm, the hundreds and tens digits together count centimetres: 34 cm.","The ones digit is millimetres: 5 mm.","**2,345 mm = 2 m 34 cm 5 mm.** In the chart: m 2 | dm 3 | cm 4 | mm 5. Each unit is ten of the next, exactly like place value.",{"id":725,"type":354,"component":726,"componentVersion":5,"config":727,"objective":754,"textAlternative":755},"lab-match-prefix","match-pairs",{"prompt":728,"mode":729,"pairs":730},"Match each measurement with its equal.","connect",[731,734,736,739,742,745,748,751],{"a":732,"b":733},"1 km","1,000 m",{"a":698,"b":735},"100 cm",{"a":737,"b":738},"1 cm","10 mm",{"a":740,"b":741},"1 kg","1,000 g",{"a":743,"b":744},"1 L","1,000 mL",{"a":746,"b":747},"5 km 60 m","5,060 m",{"a":749,"b":750},"3 kg 5 g","3,005 g",{"a":752,"b":753},"1 megawatt","10 lakh watts","Connect metric measurements with their equivalents, treating each prefix as a place value.","Eight pairs to connect: 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm; 1 kg = 1,000 g; 1 L = 1,000 mL; 5 km 60 m = 5,060 m (kilometres go in the thousands place, and the hundreds place is 0); 3 kg 5 g = 3,005 g (two placeholder zeros); 1 megawatt = 10,00,000 watts = ten lakh watts, since mega means a million. The traps are the placeholder zeros: 5 km 60 m is not 560 m, and 3 kg 5 g is not 35 g.",{"id":757,"type":154,"itemId":758,"prompt":759,"check":760,"hints":762,"feedback":764},"pr-metric-d","number-system.deepen-metric","A water tank holds 12 kL (kilolitres). How many litres is that? (Write the number only.)",{"kind":158,"answer":761,"tolerance":160},12000,[763],"What does kilo mean?",{"correct":765,"incorrect":766},"Right: kilo means 1,000, so 12 kL = 12 × 1,000 L = 12,000 L.","Kilo = 1,000. 12 kL = 12 × 1,000 = 12,000 litres.",{"id":768,"type":354,"component":769,"componentVersion":5,"config":770,"objective":782,"textAlternative":783},"lab-pv-deep","place-value",{"places":771,"system":772,"initial":773,"show":774,"challenges":776},9,"indian",30405027,{"names":775,"expanded":775,"neighbours":775,"bothSystems":775},true,[777,778,779,780,781],20020202,909090909,110110011,500000005,123456789,"Build tricky nine-digit numbers full of placeholder zeros and check their names and expanded forms in both systems.","A nine-column Indian chart from ten crores to ones. It starts at 3,04,05,027 (three crore four lakh five thousand twenty-seven; internationally 30,405,027, thirty million four hundred five thousand twenty-seven).\n\nChallenges: 2,00,20,202 (two crore twenty thousand two hundred two); 90,90,90,909 (ninety crore ninety lakh ninety thousand nine hundred nine); 11,01,10,011 (eleven crore one lakh ten thousand eleven); 50,00,00,005 (fifty crore five); 12,34,56,789 (twelve crore thirty-four lakh fifty-six thousand seven hundred eighty-nine; internationally one hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine).",{"id":785,"type":47,"title":786,"eyebrow":787,"navLabel":788},"ch11","A toolkit for hard conversions and forming problems","Chapter 11","11 Toolkit",{"id":790,"type":43,"markdown":791},"toolkit-intro","This chapter collects the hardest routine tasks in the topic, with a method for each that never fails: converting numbers of ten or more digits between the systems, forming numbers under several conditions at once, and bounding real-world estimates. Each method rests on something proved earlier in this layer.",{"id":793,"type":122,"title":794,"problem":795,"steps":796},"we-tk-ind-intl","Ten digits, Indian to International","Convert 3,07,50,04,009 to the International system and read it both ways.",[797,798,799,800,801],"Strip the commas: 3075004009. Ten digits, so it is in the billions (International) and in the hundreds of crores (Indian).","International commas every three digits: **3,075,004,009**.","International name: *three billion seventy-five million four thousand nine*.","Indian name: *three hundred seven crore fifty lakh four thousand nine*. With the larger Indian name, 307 crore = 3 arab 7 crore.","Check with powers: 307 crore = 307 × 10⁷ ≈ 3.07 × 10⁹ = 3.07 billion. ✓",{"id":803,"type":122,"title":804,"problem":805,"steps":806},"we-tk-intl-ind","Eleven digits, International to Indian","Convert 12,040,300,500 to the Indian system and name it using arab.",[807,808,809,810],"Strip the commas: 12040300500. Rebuild from the right: 500, then 00, 03, 04, 20, 1 → **12,04,03,00,500**.","Standard Indian name: *one thousand two hundred four crore three lakh five hundred*.","Using arab (100 crore): 1,204 crore = 12 arab 4 crore, so it is *twelve arab four crore three lakh five hundred*.","International name: *twelve billion forty million three hundred thousand five hundred*.",{"id":812,"type":116,"variant":196,"title":813,"markdown":814},"mis-billion-crore","“A billion is a thousand crore”","Because a billion is a thousand **million**, students sometimes guess it is a thousand **crore** too. It is not: 1 crore = 10 million, so 1 billion = 1,000 million = **100 crore**. A thousand crore is 10¹⁰, ten billion. The safest check is always powers of ten: billion 10⁹, crore 10⁷, so a billion is 10⁹⁻⁷ = 10² = 100 crore.",{"id":816,"type":154,"itemId":817,"prompt":818,"check":819,"hints":821,"feedback":823},"pr-tk-bn","number-system.deepen-billion-to-crore","An airline carried **0.16 billion** passengers in a year. How many **crore** is that?",{"kind":158,"answer":820,"tolerance":160},16,[528,822],"Check with digits: 0.16 billion = 160,000,000.",{"correct":824,"incorrect":825},"Right: 1 billion = 100 crore, so 0.16 billion = 16 crore (160,000,000 = 16,00,00,000).","Multiply billions by 100 to get crores: 0.16 × 100 = 16 crore.",{"id":827,"type":122,"title":828,"problem":829,"steps":830},"we-tk-distinct","All digits different","Find the greatest and the smallest 7-digit numbers whose digits are all different, and their difference.",[831,832,833,834],"Greatest: use the largest seven digits in descending order: 9, 8, 7, 6, 5, 4, 3 → **98,76,543**.","Smallest: the first digit must be the smallest non-zero digit, 1; then the smallest remaining digits in ascending order, starting with 0: 1, 0, 2, 3, 4, 5, 6 → **10,23,456**.","Difference: 98,76,543 − 10,23,456 = **88,53,087**.","Notice the zero rule: 0 is used in the second place of the smallest number, not the first, but it is still used, because leaving it out would force a bigger digit into that place.",{"id":836,"type":154,"itemId":837,"prompt":838,"check":839,"hints":841,"feedback":843},"pr-tk-s8","number-system.deepen-smallest-distinct","What is the smallest **8-digit** number whose digits are all different?",{"kind":158,"answer":840,"tolerance":160},10234567,[842],"The first digit cannot be 0, but the second can.",{"correct":844,"incorrect":845},"Right: 1 first, then 0, 2, 3, 4, 5, 6, 7 → 1,02,34,567 (one crore two lakh thirty-four thousand five hundred sixty-seven).","Smallest non-zero digit first (1), then 0, then 2, 3, 4, 5, 6, 7: 1,02,34,567.",{"id":847,"type":122,"title":848,"problem":849,"steps":850},"we-tk-rank","The fifth-greatest arrangement","Using each of 3, 0, 7, 5 once, list the 4-digit numbers in descending order. Which is fifth?",[851,852,853,854],"Numbers starting with 7 come first. With 7 fixed, arrange 5, 3, 0 in descending order: 7,530, 7,503, 7,350, 7,305, 7,053, 7,035.","The fifth in the list is **7,053**.","How many arrangements in total? 3 choices for the first digit (not 0) × 3 × 2 × 1 = 18.","This is how dictionaries and computers order words and numbers: fix the leftmost position, then work rightwards.",{"id":856,"type":122,"title":857,"problem":858,"steps":859},"we-tk-bounds","Bounding a real budget","A district plans 3,48,560 ration kits at ₹1,275 each. Give a lower and an upper estimate of the cost that are guaranteed, then tighten them.",[860,861,862,863],"Coarse bounds: round both down to one significant digit, 3,00,000 × ₹1,000 = ₹30,00,00,000; round both up, 4,00,000 × ₹2,000 = ₹80,00,00,000. True, but far too wide to plan with.","Tighter: 3,48,000 × ₹1,200 = ₹41,76,00,000 and 3,49,000 × ₹1,300 = ₹45,37,00,000.","So the cost is between about ₹41.8 crore and ₹45.4 crore. Exact: ₹44,44,14,000, about ₹44.4 crore.","For a budget, the upper estimate is the one that matters: plan for ₹45.4 crore and you will not run short.",{"id":865,"type":116,"variant":196,"title":866,"markdown":867},"mis-exponent","“10⁵ means 10 × 5”","10⁵ is **not** 50. The exponent counts how many tens are **multiplied together**: 10⁵ = 10 × 10 × 10 × 10 × 10 = 1,00,000, one lakh. In the same way 2 × 10⁵ is two lakh, not 100. A quick check: the exponent is the number of zeros after the 1.",{"id":869,"type":154,"itemId":870,"prompt":871,"check":872,"hints":873,"feedback":876},"pr-tk-pow","number-system.deepen-power-divide","What is 10⁹ ÷ 10⁵? Give the answer as an ordinary number.",{"kind":158,"answer":358,"tolerance":160},[874,875],"Dividing powers of ten subtracts the exponents.","Or cancel five zeros from 1,000,000,000.",{"correct":877,"incorrect":878},"Right: dividing subtracts exponents: 10⁹⁻⁵ = 10⁴ = 10,000. So a billion is ten thousand lakh.","10⁹ ÷ 10⁵ = 10⁴ = 10,000. In words: a billion is ten thousand lakh.",{"id":880,"type":881,"title":882,"terms":883},"gloss-d","glossary","Deeper vocabulary",[884,887,891,894,897,901,904,908,911,914,916,919],{"term":669,"meaning":885,"example":886},"A number made by multiplying 10 by itself; written 10ⁿ, where n is the exponent.","10⁴ = 10,000",{"term":888,"meaning":889,"example":890},"Exponent","The small raised number that says how many times the base is multiplied by itself.","In 10⁷ the exponent is 7.",{"term":892,"meaning":893},"Base","The number each place is worth times the place to its right. Our system uses base ten.",{"term":895,"meaning":896},"Uniqueness","The fact that each whole number has exactly one standard way of being written with digits.",{"term":898,"meaning":899,"example":900},"Bound","A value that a quantity is guaranteed not to go beyond.","The error is at most 300.",{"term":902,"meaning":903},"Upper \u002F lower estimate","Estimates that are guaranteed to be at least \u002F at most the exact answer.",{"term":905,"meaning":906,"example":907},"Round half to even","A rounding convention in which exact halves round to the even neighbour.","45 → 40, 55 → 60",{"term":909,"meaning":910},"Hindu–Arabic numerals","The ten digits 0–9 and their place-value system, developed in India and spread by Arabic-writing scholars.",{"term":912,"meaning":913},"Shunya","Sanskrit for empty or void, the Indian name for zero.",{"term":547,"meaning":915},"Ancient Indian numerals (from about the 3rd century BCE), ancestors of our digit shapes, used without place value.",{"term":917,"meaning":918},"Vinculum","A bar over a Roman numeral multiplying it by 1,000.",{"term":920,"meaning":921},"Metric prefix","A word in front of a unit that multiplies it by a power of ten, such as kilo (10³) or milli (one thousandth).",{"id":923,"type":924,"title":925,"questions":926},"quiz-d","quiz","Go deeper check",[927,939,952,964,977,989,1002,1015,1024,1037],{"itemId":928,"prompt":929,"options":930,"correct":185,"why":938},"number-system.dp-q-pow","10⁵ × 10³ equals…",[931,933,935,936],{"id":185,"label":932},"10⁸, one ten crore",{"id":188,"label":934},"10¹⁵",{"id":191,"label":73},{"id":621,"label":937},"100⁸","Add exponents: 10⁸ = 10,00,00,000 = ten crore = 100 million.",{"itemId":940,"prompt":941,"options":942,"correct":188,"why":951},"number-system.dp-q-zero-power","Why is 10⁰ = 1?",[943,945,947,949],{"id":185,"label":944},"Any number times 0 is 0",{"id":188,"label":946},"Each step down in exponent divides by 10, and 10 ÷ 10 = 1",{"id":191,"label":948},"It is a guess",{"id":621,"label":950},"Because 0 has no value","Keeping the pattern 1,000 → 100 → 10 → 1 forces 10⁰ = 1.",{"itemId":953,"prompt":954,"options":955,"correct":185,"why":963},"number-system.dp-q-count","How many 6-digit numbers are there?",[956,958,960,961],{"id":185,"label":957},"9,00,000",{"id":188,"label":959},"9,99,999",{"id":191,"label":91},{"id":621,"label":962},"90,000","9 × 10⁵ = 9,00,000.",{"itemId":965,"prompt":966,"options":967,"correct":188,"why":976},"number-system.dp-q-proof","Why must 7,00,000 be bigger than 6,99,999?",[968,970,972,974],{"id":185,"label":969},"It has more zeros",{"id":188,"label":971},"The lakhs place gives it an extra 1,00,000, and the lower places of 6,99,999 add at most 99,999",{"id":191,"label":973},"Odd numbers are smaller",{"id":621,"label":975},"It was rounded","The first difference is at the lakhs place; the remaining places can make up at most 99,999, less than 1,00,000.",{"itemId":978,"prompt":979,"options":980,"correct":191,"why":988},"number-system.dp-q-err","You round 8 numbers to the nearest 10 and add them. The error in the total is at most…",[981,983,984,986],{"id":185,"label":982},"5",{"id":188,"label":70},{"id":191,"label":985},"40",{"id":621,"label":987},"80","Each number changes by at most 5, so at most 8 × 5 = 40.",{"itemId":990,"prompt":991,"options":992,"correct":188,"why":1001},"number-system.dp-q-even","The greatest 4-digit even number using 3, 8, 5, 1 once each is…",[993,995,997,999],{"id":185,"label":994},"8,531",{"id":188,"label":996},"5,318",{"id":191,"label":998},"8,513",{"id":621,"label":1000},"1,358","The only even digit is 8, so it must go last: 5, 3, 1 in front → 5,318.",{"itemId":1003,"prompt":1004,"options":1005,"correct":188,"why":1014},"number-system.dp-q-diff-bound","A guaranteed upper estimate for 6,210 − 2,870 (rounding to thousands) is…",[1006,1008,1010,1012],{"id":185,"label":1007},"6,000 − 2,000 = 4,000",{"id":188,"label":1009},"7,000 − 2,000 = 5,000",{"id":191,"label":1011},"6,000 − 3,000 = 3,000",{"id":621,"label":1013},"7,000 − 3,000 = 4,000","Round the first up and the second down: 7,000 − 2,000 = 5,000. Exact: 3,340.",{"itemId":1016,"prompt":1017,"options":1018,"correct":188,"why":1023},"number-system.dp-q-history","Who wrote rules for calculating with zero in 628 CE?",[1019,1020,1021,1022],{"id":185,"label":559},{"id":188,"label":563},{"id":191,"label":575},{"id":621,"label":567},"Brahmagupta’s Brahmasphutasiddhanta (628 CE) treats zero as a number with its own rules.",{"itemId":1025,"prompt":1026,"options":1027,"correct":191,"why":1036},"number-system.dp-q-trillion","1 lakh crore is…",[1028,1030,1032,1034],{"id":185,"label":1029},"1 billion",{"id":188,"label":1031},"100 billion",{"id":191,"label":1033},"1 trillion",{"id":621,"label":1035},"10 trillion","10⁵ × 10⁷ = 10¹² = 1 trillion.",{"itemId":1038,"prompt":1039,"options":1040,"correct":191,"why":1049},"number-system.dp-q-metric","4 m 7 mm in millimetres is…",[1041,1043,1045,1047],{"id":185,"label":1042},"47 mm",{"id":188,"label":1044},"407 mm",{"id":191,"label":1046},"4,007 mm",{"id":621,"label":1048},"4,070 mm","4 m = 4,000 mm; add 7 mm → 4,007 mm (two placeholder zeros).",{"id":1051,"type":1052,"prompt":1053},"reflect-d","reflection","Imagine a world that never invented zero. Describe two everyday things (a price tag, a bus number, a phone number, a cricket score…) that would be confusing, and how people might have worked around it.",{"id":1055,"type":1056,"conceptId":1057,"relation":1058,"explanation":1059},"conn-d-ops","connection","four-operations","helps_understand","Long multiplication and division are place value in action: each partial product is shifted by a power of ten.",{"id":1061,"type":1056,"conceptId":1062,"relation":1063,"explanation":1064},"conn-d-prop","properties-of-numbers","related_to","Zero’s rules (a + 0 = a, a × 0 = 0) are the properties that Brahmagupta first wrote down.",{"id":1066,"type":1056,"conceptId":1067,"relation":1063,"explanation":1068},"conn-d-hcf","hcf-and-lcm","Powers of ten factor as 2s and 5s (10³ = 2³ × 5³), which is why prime factorisation and place value fit together.",{"id":1070,"type":1056,"conceptId":1071,"relation":1072,"explanation":1073},"conn-d-elec","electricity","applied_in","Electricity uses metric prefixes as place value all the time: kilowatts, megawatts, milliamps and kilovolts.",{"id":1075,"type":1076,"title":1077,"points":1078},"cheat-d","summary","Cheat sheet",[1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089],"**Powers of ten:** 10ⁿ is 1 followed by n zeros; 10⁰ = 1; 10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ.","**Expanded form with powers:** 3,04,05,027 = 3 × 10⁷ + 4 × 10⁵ + 5 × 10³ + 2 × 10¹ + 7 × 10⁰.","**Uniqueness:** each whole number has one standard form, because bundling into tens is forced; zero makes this possible.","**Comparison proof:** 10ⁿ beats 10ⁿ − 1; a lead of 10ᵏ at the first difference beats at most 10ᵏ − 1 from lower places.","**Counting:** 9 × 10ⁿ⁻¹ numbers with n digits; 648 three-digit numbers with all digits different.","**Forming with conditions:** fix the forced digit first (last digit for even, odd, multiple of 5), then order the rest.","**Rounding error:** at most half the rounding unit per number; add these up to bound the error of a sum.","**Bounds:** both down gives a lower estimate and both up an upper estimate for sums and products; for differences, round in opposite directions.","**Names:** Sanskrit named every power of ten; lakh 10⁵, crore 10⁷, arab 10⁹, kharab 10¹¹; 1 lakh crore = 1 trillion.","**History:** Brahmi (c. 250 BCE) → Indian place value and zero (Aryabhata 499, Brahmagupta 628) → al-Khwarizmi (c. 825) → Fibonacci (1202) → the world.","**Metric prefixes** are powers of ten: giga, mega, kilo, hecto, deca, deci, centi, milli.",{"id":1091,"type":1092,"sourceIds":1093},"sources-d","sources",[1094,1095,1096,1097,1098,1099,1100,1101,1102,1103],"number-system-britannica-hindu-arabic","number-system-wiki-brahmi","number-system-wiki-aryabhata","number-system-wiki-indian-numbering","number-system-bipm-si-prefixes","number-system-mathsisfun-roman","number-system-ncert-class7-large-numbers","number-system-wiki-zero","number-system-wiki-large-numbers","number-system-wiki-bhutasankhya",[1094,1095,1096,1097,1098,1099,1100,1101,1102,1103],"needs_review",{"generatedBy":1107,"notes":1108},"claude-code","Draft generated with Python-checked numbers; pending owner review.","316400122e89316a0b6cb1173c7953d757c5541d4034a7d56806959a61330b80",{"logic:practice":1111,"component:rounding-race@1":1112,"component:sort-game@1":1113,"component:match-pairs@1":1114,"component:place-value@1":1115,"source:number-system-bipm-si-prefixes":1116,"source:number-system-britannica-hindu-arabic":1117,"source:number-system-mathsisfun-roman":1118,"source:number-system-ncert-class7-large-numbers":1119,"source:number-system-wiki-aryabhata":1120,"source:number-system-wiki-bhutasankhya":1121,"source:number-system-wiki-brahmi":1122,"source:number-system-wiki-indian-numbering":1123,"source:number-system-wiki-large-numbers":1124,"source:number-system-wiki-zero":1125},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","cae1e81c06fcfcc0152eb325ac8fb01f568e21a19ad76a75d79194e17231ae72","4b1ef23b043f9642cf2f3613a60f4e05e4eb4a7c356f1100a1808edc05a4ecb3","a63f6c9a1c311ff55526f628c2aa874f9c1a5449b9b395e4d1a775681172821c","ef0f2abd2b3d7e393f1878aa51359c7a18194adf57596671bc0dff39cd1aa2f5","9dc5f1741ed9fad68db69882c1cc09c8d4b3cf5902d5d0273891bad1da1d10c0","06b14f89ca316f2027233e015c1b44e7d6f0439b80b8026e3fed4a8454f2808b","8b4bd727c8b79d60c35fd2f913222373f3d49b8aa4745b0777346e52e19ad1e7","ceef45990d3415628548568ccb465ff0c200063821e866eadbd672cd1ee4cb05","045032c28b7bfe33f8bc73c5c09ee8dd8076293c90dbe922d8a461e059edd6e3","a5a78ee61f6a2c6167acebe3b47f1b5643280f121e751e4e1ae14d3d2d8fa6bb","8f31cd97e5e3383d1f40dba76e52c45514feae1844aacb5d7c7078492704a72a",{"state":1127,"reviewer":1128,"selfReview":775,"reviewedAt":1129,"method":1130},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598158]