[{"data":1,"prerenderedAt":1021},["ShallowReactive",2],{"layer:properties-of-numbers:extend":3},{"layer":4,"contentHash":997,"dependencyHashes":998,"approval":1015,"releaseId":1020},{"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":992,"reviewStatus":993,"authoring":994},1,"properties-of-numbers","en","extend","Properties beyond the whole numbers","Integers, fractions, clocks, computers, puzzles and the problems nobody has solved","Take the properties into new worlds: integers and fractions that repair closure, clock arithmetic, non-commutative everyday actions, rounding inside computers, olympiad puzzles built on parity and the distributive law, projects to try and open questions like Goldbach.",[13,14,15,16,17],"Compare closure for natural numbers, whole numbers, integers and fractions, and explain why negative × negative is positive.","Use clock (modular) arithmetic to solve calendar problems and check which properties it keeps.","Identify real and mathematical operations that are not commutative or not associative.","Solve olympiad-style puzzles with parity, invariants and the distributive law.","Describe open problems such as Goldbach's conjecture and plan a property-based project.",50,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 50 minutes plus projects",{"label":29,"value":30},"Prior knowledge","All earlier layers",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Number-trick lab, hard sprint, families sort",{"label":38,"value":39},"Open problem","Goldbach's conjecture",[41,45,65,71,74,98,101,113,131,137,142,145,172,187,192,195,209,214,223,244,264,269,272,334,339,344,347,352,384,397,402,412,417,427,437,447,457,475,519,524,534,543,565,568,572,640,670,702,707,730,735,740,743,748,754,759,763,768,772,777,825,960,964,977],{"id":42,"type":43,"markdown":44},"intro-extend","prose","The properties you have studied were stated for whole numbers. But mathematicians never stop at one family. What happens on the **left of zero**, with negative numbers? With fractions? On a clock face, where 9 + 5 = 2? Inside a computer, where 0.1 + 0.2 is not quite 0.3? And in everyday actions such as putting on socks and shoes, where order matters a great deal?\n\nThis layer takes the properties on a journey. You will see how the distributive law forces (−3) × (−4) to be +12, why some families gain closure while others lose it, where commutativity breaks in the real world, and how the properties power olympiad puzzles, computer science and cryptography. It ends with projects to try and questions nobody has answered yet.",{"id":46,"type":47,"tone":48,"items":49},"spec-extend","spec","amber",[50,54,58,62],{"label":51,"big":52,"value":53},"New numbers","integers, fractions","Each new family fixes a closure problem: integers fix −, fractions fix ÷ (except by 0).",{"label":55,"big":56,"value":57},"New worlds","clocks, codes","Clock arithmetic keeps all the properties, with a twist: 12 acts like 0.",{"label":59,"big":60,"value":61},"Order matters","socks, turns, words","Many real actions are not commutative; spotting which is a real skill.",{"label":38,"big":63,"value":64},"Goldbach","Is every even number from 4 up a sum of two primes? Checked hugely, never proved.",{"id":66,"type":67,"title":68,"eyebrow":69,"navLabel":70},"ch1","chapter","Left of zero: a glimpse of integers","Chapter 01","1 Integers",{"id":72,"type":43,"markdown":73},"integers-prose","Extend the number line to the left of 0 and mark −1, −2, −3, … The **integers** are … −3, −2, −1, 0, 1, 2, 3, … You already meet them: a temperature of −5 °C in Leh in winter, a lift going to basement level −2, a bank balance that is overdrawn, a cricket team's net run rate, a height below sea level.\n\nWith integers, **subtraction is finally closed**: 3 − 5 = −2 is an integer. In fact subtraction becomes addition of the opposite: 3 − 5 = 3 + (−5). Every integer a has an **additive inverse** −a with a + (−a) = 0. The integers keep closure, commutativity and associativity for + and ×, keep the identities 0 and 1, and keep the distributive law. They are still not closed under division (1 ÷ 2 is not an integer).",{"id":75,"type":76,"caption":77,"columns":78,"rows":84},"closure-families","table","Which families are closed under which operation? (NCERT Class 8 table, extended)",[79,80,81,82,83],"Numbers","+","−","×","÷",[85,90,93,95],[86,87,88,87,89],"Natural (1, 2, 3, …)","✓","✗ (2 − 5)","✗ (2 ÷ 5)",[91,87,88,87,92],"Whole (0, 1, 2, …)","✗ (2 ÷ 5; and ÷ 0)",[94,87,87,87,92],"Integers (… −1, 0, 1 …)",[96,87,87,87,97],"Rational (fractions p\u002Fq)","✓ except division by 0",{"id":99,"type":43,"markdown":100},"integers-props","The table tells a story. Each new family of numbers was invented partly to **repair closure**: integers repair subtraction, fractions (rational numbers) repair division. Division by zero is the one hole that no family of ordinary numbers repairs, for the reasons you saw in Deepen.\n\nIntegers also give subtraction some new behaviour worth knowing. It is still not commutative (5 − 3 = 2 but 3 − 5 = −2; they are opposites) and still not associative: (10 − 4) − 3 = 3 but 10 − (4 − 3) = 9. But rewritten as addition, 10 + (−4) + (−3) can be regrouped freely, because addition of integers is associative. That is the trick behind adding a column of profits and losses in any order.",{"id":102,"type":103,"title":104,"problem":105,"steps":106},"we-neg-times-neg","worked_example","Why (−3) × (−4) must be +12","Use the distributive law to find (−3) × (−4) without any \"rule of signs\".",[107,108,109,110,111,112],"Start with something we know is 0: 4 + (−4) = 0.","Multiply by −3: (−3) × (4 + (−4)) = (−3) × 0 = 0.","Distribute: (−3) × 4 + (−3) × (−4) = 0.","(−3) × 4 means 4 lots of −3 (commutative law), which is −12. So −12 + (−3) × (−4) = 0.","The only number that adds to −12 to make 0 is +12.","So **(−3) × (−4) = +12**. \"Negative times negative is positive\" is not an arbitrary rule. It is forced on us if we want the distributive law to keep working.",{"id":114,"type":115,"prompt":116,"options":117,"explanation":130},"predict-three-negs","prediction","Using \"negative × negative = positive\", what is the sign of **(−2) × (−2) × (−2)**?",[118,121,124,127],{"id":119,"label":120},"a","Positive",{"id":122,"label":123},"b","Negative",{"id":125,"label":126},"c","Zero",{"id":128,"label":129},"d","It depends on the grouping","**Negative: it equals −8.** Group the first two: (−2) × (−2) = +4, then 4 × (−2) = −8. Group the last two instead: (−2) × ((−2) × (−2)) = (−2) × 4 = −8. The associative law guarantees the grouping does not matter. The general rule: a product of non-zero numbers is negative exactly when it has an **odd number** of negative factors. It is the even\u002Fodd rule from Discover, reappearing in a new world.",{"id":132,"type":133,"variant":134,"title":135,"markdown":136},"nuance-rules-kept","callout","nuance","Keeping the rules decides the new facts","When mathematicians extend numbers, they ask: *which properties do we want to keep?* Keeping the distributive law forces negative × negative = positive. Keeping the zero property forces \"no division by zero\". The properties are not just facts about old numbers; they are the design rules for new ones. This idea, sometimes called the **principle of permanence**, guided the invention of negative numbers, fractions and, later, complex numbers.",{"id":138,"type":67,"title":139,"eyebrow":140,"navLabel":141},"ch2","Fractions keep the rules","Chapter 02","2 Fractions",{"id":143,"type":43,"markdown":144},"fractions-prose","The rational numbers (fractions such as ½, −¾, 7\u002F3) keep **every** property of + and × you know, and add one more: every number except 0 has a **reciprocal** (multiplicative inverse). ¾ × 4\u002F3 = 1.\n\nThe distributive law is especially handy with fractions: ½ × (18 + 6) = 9 + 3 = 12. And the shortcuts still work: ¼ × 36 × 4 = (¼ × 4) × 36 = 1 × 36 = 36. Here the associative and commutative laws let us bring the reciprocals together to make the identity 1.\n\nOne important change: with fractions, **multiplying does not always make bigger**, and dividing does not always make smaller. 12 × ½ = 6, and 12 ÷ ½ = 24. The claim \"a × b > a\" that was sometimes true for whole numbers is now false for every b between 0 and 1.",{"id":146,"type":76,"caption":147,"columns":148,"rows":152},"fraction-props","Properties with fractions: some examples",[149,150,151],"Property","Example with fractions","Value",[153,157,160,164,168],[154,155,156],"Commutative ×","⅔ × 9 = 9 × ⅔","6",[158,159,156],"Associative ×","(¼ × 8) × 3 = ¼ × (8 × 3)",[161,162,163],"Distributive","½ × (10 + 4) = 5 + 2","7",[165,166,167],"Multiplicative inverse","⅗ × 5\u002F3","1",[169,170,171],"Division by a fraction","12 ÷ ½ = 12 × 2","24",{"id":173,"type":174,"itemId":175,"prompt":176,"check":177,"hints":181,"feedback":184},"pr-fraction-dist","practice","properties-of-numbers.extend-fraction-dist","Use the distributive law: **¾ × 36 + ¾ × 4** = ?",{"kind":178,"answer":179,"tolerance":180},"number",30,0,[182,183],"Take out the common factor ¾.","¾ × (36 + 4) = ¾ × 40.",{"correct":185,"incorrect":186},"Yes: ¾ × 40 = 30.","Take out ¾: ¾ × (36 + 4) = ¾ × 40 = 30.",{"id":188,"type":67,"title":189,"eyebrow":190,"navLabel":191},"ch3","Clock arithmetic: the same rules on a circle","Chapter 03","3 Clock arithmetic",{"id":193,"type":43,"markdown":194},"clock-prose","It is 9 o'clock. What time will it be in 5 hours? Not 14 o'clock on a 12-hour clock face, but **2 o'clock**. On a clock, numbers wrap around. This is called **modular arithmetic**, here \"arithmetic modulo 12\".\n\nDoes clock addition keep the properties?\n\n- **Closed:** adding any two clock times gives a clock time. ✓\n- **Commutative:** 9 + 5 and 5 + 9 both give 2. ✓\n- **Associative:** (9 + 5) + 7 = 2 + 7 = 9, and 9 + (5 + 7) = 9 + 12 = 9. ✓\n- **Identity:** adding 12 hours brings you back to the same time, so **12 behaves like 0**. ✓\n- **Inverses:** from 9, adding 3 hours gets you to 12, the identity. So 3 is the inverse of 9. Every time has one. ✓\n\nThe days of the week are arithmetic modulo 7. If today is Monday, what day is it in 100 days? 100 = 14 × 7 + 2, so 14 full weeks bring you back to Monday and 2 more days give **Wednesday**.",{"id":196,"type":76,"caption":197,"columns":198,"rows":203},"mod5-table","Addition modulo 5: every row and column contains each number exactly once",[80,199,167,200,201,202],"0","2","3","4",[204,205,206,207,208],[199,199,167,200,201,202],[167,167,200,201,202,199],[200,200,201,202,199,167],[201,201,202,199,167,200],[202,202,199,167,200,201],{"id":210,"type":133,"variant":211,"title":212,"markdown":213},"obs-mod-table","observation","Reading the table","The table is symmetric about its main diagonal (top left to bottom right): row 2, column 3 matches row 3, column 2. **Symmetry about the diagonal is what commutativity looks like in a table.** The row for 0 copies the header, showing 0 is the identity. And a 0 appears in every row, showing every number has an inverse (the inverse of 2 is 3, because 2 + 3 leaves remainder 0).",{"id":215,"type":103,"title":216,"problem":217,"steps":218},"we-days","A calendar puzzle","15 August 2026 is a Saturday. What day of the week is 15 August 2027?",[219,220,221,222],"Count the days from 15 August 2026 to 15 August 2027. There is no 29 February between them (2027 is not a leap year), so it is 365 days.","365 = 52 × 7 + 1.","52 full weeks bring us back to Saturday; 1 more day gives **Sunday**.","Rule of thumb: the same date moves forward one weekday each year, or two if a 29 February falls in between.",{"id":224,"type":174,"itemId":225,"prompt":226,"check":227,"hints":239,"feedback":241},"pr-republic-day","properties-of-numbers.extend-republic-day","Republic Day, 26 January 2026, is a Monday. What day of the week is 26 January 2027?",{"kind":228,"options":229,"correct":238},"choice",[230,232,234,236],{"id":119,"label":231},"Monday",{"id":122,"label":233},"Tuesday",{"id":125,"label":235},"Wednesday",{"id":128,"label":237},"Sunday",[122],[240,220],"Is there a 29 February between the two dates?",{"correct":242,"incorrect":243},"Yes: 365 days is 52 weeks and 1 day, so the day moves on by one, to Tuesday.","There is no 29 February between them, so the gap is 365 days = 52 weeks + 1 day. Monday moves on one day to Tuesday.",{"id":245,"type":174,"itemId":246,"prompt":247,"check":248,"hints":258,"feedback":261},"pr-clock","properties-of-numbers.extend-clock","On a 12-hour clock, what is 8 + 7 + 11?",{"kind":228,"options":249,"correct":257},[250,251,253,255],{"id":119,"label":200},{"id":122,"label":252},"26",{"id":125,"label":254},"14",{"id":128,"label":256},"12",[119],[259,260],"Add normally, then take away 12s.","26 − 12 − 12 = 2.",{"correct":262,"incorrect":263},"Right: 8 + 7 + 11 = 26, and 26 − 24 = 2. You could also group (8 + 7) + 11 = 3 + 11 = 2 on the clock.","8 + 7 + 11 = 26 hours after 12 o'clock. Two full turns of 12 hours use 24, leaving 2 o'clock.",{"id":265,"type":67,"title":266,"eyebrow":267,"navLabel":268},"ch4","When order matters: the non-commutative world","Chapter 04","4 Order matters",{"id":270,"type":43,"markdown":271},"order-matters-prose","Outside arithmetic, **most** ways of combining things are not commutative. That is why the commutative property is worth naming at all. Some examples to think about:\n\n- **Words.** Join \"rain\" and \"bow\": rainbow. Join \"bow\" and \"rain\": bowrain. Joining text (called *concatenation* in computing) is associative but not commutative.\n- **Turning and flipping.** Take a book lying face up. Turn it a quarter turn clockwise, then flip it over left to right. Now start again: flip first, then turn. The book ends up in a **different** position. Rotations and reflections do not commute.\n- **Cooking.** Boil the rice, then add salt; or add salt, then boil. Similar. Grind the spices, then fry them; or fry them, then grind. Very different.\n- **Rubik's cube.** Twisting the right face then the top face is not the same as top then right. Solving the cube is all about clever orders of moves.",{"id":273,"type":274,"title":275,"prompt":276,"options":277},"explorer-ops","explorer","Is it commutative? Is it associative?","Pick a way of combining things to see which properties it has.",[278,290,303,313,324],{"id":279,"label":280,"chain":281,"badge":286,"note":289},"words","Joining words",[282,283,284,59,285],"\"sun\" then \"flower\"","\"sunflower\"","swap: \"flowersun\"","Grouping does not",{"text":287,"tone":288},"Associative, not commutative","no","Concatenation (joining strings of text) is associative: (\"sun\" + \"flower\") + \"s\" and \"sun\" + (\"flower\" + \"s\") both give \"sunflowers\". But it is not commutative. Computer programs join text millions of times a second, and programmers must always keep the order. It even has an identity: the empty string \"\", which changes nothing when joined.",{"id":291,"label":292,"chain":293,"badge":299,"note":302},"clock","Clock addition",[294,295,296,297,298],"9 o'clock + 5 hours","= 2 o'clock","5 + 9 is also 2","Grouping is free too","12 acts as 0",{"text":300,"tone":301},"Commutative and associative","yes","Clock arithmetic (modulo 12) keeps every property of ordinary addition: closure, commutative, associative, an identity (12, acting as 0) and an inverse for each hour. Modular arithmetic is the basis of modern cryptography, the codes that protect UPI payments and messages.",{"id":304,"label":305,"chain":306,"badge":311,"note":312},"turns","Turn then flip",[307,308,309,310],"Card face up","Quarter turn, then flip","Or flip, then quarter turn","Different results",{"text":287,"tone":288},"Doing one movement after another is associative (doing A, then B, then C is the same however you bracket it) but not commutative. Mathematicians study such movements in a subject called group theory. It explains crystals, the symmetry of rangoli patterns and the Rubik's cube.",{"id":314,"label":315,"chain":316,"badge":321,"note":323},"rps","Rock-paper-scissors",[317,318,319,320],"Rock vs paper → paper","Paper vs rock → paper","(R vs P) vs S → S","R vs (P vs S) → R",{"text":322,"tone":288},"Commutative, not associative","Define \"the winner of a and b\" (with a tie giving that shape). The winner does not depend on who is named first, so it is commutative. But (rock vs paper) vs scissors = paper vs scissors = scissors, while rock vs (paper vs scissors) = rock vs scissors = rock. Different! A knockout tournament of rock, paper and scissors depends on the order of the fixtures.",{"id":325,"label":326,"chain":327,"badge":332,"note":333},"maxop","Taking the larger",[328,329,330,331],"max(3, 8) = 8","max(8, 3) = 8","max of three: any grouping","Identity: 0 for whole numbers",{"text":300,"tone":301},"Choosing the larger of two numbers is both commutative and associative, and 0 is an identity for whole numbers. The highest score in a cricket tournament does not depend on which matches you compare first.",{"id":335,"type":133,"variant":336,"title":337,"markdown":338},"misc-real-life","misconception","\"If maths says order does not matter, it never matters\"","The commutative property is about the **numbers** in an addition or multiplication, not about everything in a problem. 5 packets of 12 biscuits and 12 packets of 5 biscuits contain the same number of biscuits (60), but they are not the same shopping: one is 5 packets and one is 12. When you use a property in a word problem, be clear about what the answer counts.",{"id":340,"type":67,"title":341,"eyebrow":342,"navLabel":343},"ch5","Properties inside computers","Chapter 05","5 Computers",{"id":345,"type":43,"markdown":346},"float-prose","A computer stores most decimal numbers in a fixed number of binary digits, rounding them slightly. Because of this rounding, **computer addition of decimals is not always associative**. In the Python programming language:\n\n- (0.1 + 0.2) + 0.3 gives **0.6000000000000001**\n- 0.1 + (0.2 + 0.3) gives **0.6**\n\nAnd with very large numbers: (10,000,000,000,000,000 + 1) − 10,000,000,000,000,000 gives **0**, because the computer cannot store the \"+ 1\" at that size, while (10,000,000,000,000,000 − 10,000,000,000,000,000) + 1 gives **1**.\n\nThis matters in real life. Weather forecasts, satellite orbits and bank software add millions of numbers. Programmers choose the order of additions carefully, and banks usually store money as whole numbers of paise rather than decimals of rupees, so that whole-number addition, which *is* associative, does the work.",{"id":348,"type":133,"variant":349,"title":350,"markdown":351},"careful-float","careful","The maths is fine; the machine rounds","Real numbers really are associative under addition. The failure above comes from rounding in the machine, not from mathematics. This is a **model limit**: a computer is a model of arithmetic with finitely many digits, and every model has edges where it breaks. Knowing where the edges are is part of being a good programmer or scientist.",{"id":353,"type":76,"caption":354,"columns":355,"rows":359},"careers-table","Where properties of numbers turn up at work",[356,357,358],"Career","Property in action","Example",[360,364,368,372,376,380],[361,362,363],"Software engineer","Associativity lets big sums be split across many computers and recombined","Counting UPI transactions across servers",[365,366,367],"Cryptographer","Modular (clock) arithmetic keeps all the properties","Codes protecting online payments",[369,370,371],"Accountant","Commutative and associative addition; integers for profit and loss","Totalling a ledger in any order",[373,374,375],"Engineer","Distributive law in formulas","Total load = number of beams × (weight + extra load)",[377,378,379],"Game designer","Non-commutative rotations","Turning and moving a character in 3D",[381,382,383],"Data scientist","Averages are not associative","Averaging group averages gives the wrong answer unless groups are equal in size",{"id":385,"type":115,"prompt":386,"options":387,"explanation":396},"predict-average","Class A has 10 students with an average of 80 marks. Class B has 30 students with an average of 60 marks. What is the average for all 40 students?",[388,390,392,394],{"id":119,"label":389},"70",{"id":122,"label":391},"65",{"id":125,"label":393},"75",{"id":128,"label":395},"60","**65.** The tempting answer, 70, averages the two averages as if the classes were the same size. Class B has three times as many students, so it pulls the overall average towards 60. The worked example below shows the calculation.",{"id":398,"type":133,"variant":399,"title":400,"markdown":401},"ex-paise","example","Why banks count in paise","A payment app that stored ₹0.10 as a binary decimal could, after millions of additions, drift by a paisa here and there, and the order of additions could change the total. Storing every amount as a whole number of paise (₹0.10 as 10 paise) keeps all additions in the whole numbers, where addition is exactly commutative and associative. The same idea is used for stock prices, railway fares and electricity bills.",{"id":403,"type":103,"title":404,"problem":405,"steps":406},"we-average-trap","The averaging trap","Class A (10 students) averages 80 marks. Class B (30 students) averages 60. Is the average of all 40 students (80 + 60) ÷ 2 = 70?",[407,408,409,410,411],"Total marks in A: 10 × 80 = 800. In B: 30 × 60 = 1,800.","All 40 students: 800 + 1,800 = 2,600 marks.","True average: 2,600 ÷ 40 = 65, not 70.","Averaging is not associative, so you cannot average the averages unless the groups are the same size.","The fix uses the distributive law: weight each average by its group size before adding.",{"id":413,"type":67,"title":414,"eyebrow":415,"navLabel":416},"ch6","Olympiad-style puzzles","Chapter 06","6 Puzzles",{"id":418,"type":103,"title":419,"problem":420,"steps":421},"we-1001","The abcabc puzzle","Show that every six-digit number of the form abcabc (such as 347,347) is divisible by 7, 11 and 13.",[422,423,424,425,426],"abcabc = abc × 1,000 + abc (place value).","= abc × (1,000 + 1) = abc × 1,001 (distributive law backwards).","And 1,001 = 7 × 11 × 13.","So abcabc = abc × 7 × 11 × 13, divisible by all three.","Check: 347,347 ÷ 1,001 = 347, 347,347 ÷ 7 = 49,621.",{"id":428,"type":103,"title":429,"problem":430,"steps":431},"we-signs","Plus and minus signs","Put a + or − sign in front of each of the numbers 1, 2, 3, …, 10 (for example +1 − 2 − 3 + 4 …). Can the total be 0?",[432,433,434,435,436],"With all + signs, 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, which is odd.","Changing the sign of a number n changes the total by 2 × n (from +n to −n), an even amount.","Changing the total by even amounts never changes its parity. It stays odd whatever signs you choose.","0 is even. So the total can **never** be 0.","What about 1 to 11? That sum is 66 (even), so parity does not forbid 0. Can you find signs that work?",{"id":438,"type":103,"title":439,"problem":440,"steps":441},"we-dominoes","The chessboard and the dominoes","Cut two opposite corner squares off a chessboard, leaving 62 squares. Can 31 dominoes, each covering two neighbouring squares, cover it exactly?",[442,443,444,445,446],"Colour the board in the usual black and white pattern. Opposite corners have the **same** colour, say both white.","After removing them: 32 black and 30 white squares.","Every domino covers one black and one white square (neighbours always differ in colour).","31 dominoes would cover 31 black and 31 white squares. But there are only 30 white squares.","**Impossible.** Counting colours is a parity-style invariant argument, like the cups puzzle in Deepen.",{"id":448,"type":103,"title":449,"problem":450,"steps":451},"we-handshakes","The handshake theorem","At a party, some people shake hands. Prove that the number of people who shook an odd number of hands is even.",[452,453,454,455,456],"Add up everyone's handshake counts. Each handshake is counted twice (once by each person), so the total is 2 × (number of handshakes): **even**.","Split the total into the counts from \"even-shakers\" and from \"odd-shakers\".","The even-shakers' counts add to an even number.","So the odd-shakers' counts must also add to an even number (even − even = even).","A sum of odd numbers is even only when there is an even number of them. So there is an even number of odd-shakers. ∎",{"id":458,"type":459,"component":460,"componentVersion":5,"config":461,"objective":469,"textAlternative":470,"help":471},"lab-orderops-extend","interactive","order-ops",{"expressions":462,"showRuleCard":468},[463,464,465,466,467],"(8 × 2 + 10) ÷ 2 - 8","(23 × 2 + 10) ÷ 2 - 23","(150 × 2 + 10) ÷ 2 - 150","347 × 1001","[(7 + 5) × 3] - [7 × 3 + 5 × 3]",true,"Evaluate a \"think of a number\" trick for several starting numbers and discover why it always ends at 5.","A step-by-step expression lab. The first three expressions are the same trick with different starting numbers: double the number, add 10, halve, subtract the starting number.\n\n(8 × 2 + 10) ÷ 2 − 8 = 26 ÷ 2 − 8 = 13 − 8 = 5.\n(23 × 2 + 10) ÷ 2 − 23 = 56 ÷ 2 − 23 = 28 − 23 = 5.\n(150 × 2 + 10) ÷ 2 − 150 = 310 ÷ 2 − 150 = 155 − 150 = 5.\n\nAlways 5, because halving distributes: (2n + 10) ÷ 2 = n + 5, and then subtracting n leaves 5.\n\n347 × 1001 = 347,347 (the abcabc pattern). And [(7 + 5) × 3] − [7 × 3 + 5 × 3] = 36 − 36 = 0: the distributive law says the two brackets are always equal.",{"hints":472},[473,474],"Do the brackets first, then the division, then the subtraction.","Compare the final answers. What stays the same?",{"id":476,"type":459,"component":477,"componentVersion":5,"config":478,"objective":513,"textAlternative":514,"help":515},"lab-sprint-extend","arith-sprint",{"operations":479,"ranges":480,"rounds":487,"secondsTotal":488,"estimateFirst":468,"wordProblems":489},[82,83],{"a":481,"b":484},{"min":482,"max":483},12,250,{"min":485,"max":486},2,25,15,180,[490,494,498,502,506,509],{"prompt":491,"answer":492,"operation":82,"unit":493},"A Diwali mela stall sells 125 diyas in each of 8 boxes. How many diyas? (125 × 8 = 1,000.)",1000,"diyas",{"prompt":495,"answer":496,"operation":82,"unit":497},"A school orders 36 sets of books at ₹125 each. Use 125 = 1,000 ÷ 8. Cost in rupees?",4500,"₹",{"prompt":499,"answer":500,"operation":82,"unit":501},"A cinema sells 999 tickets on each of 7 days. Total tickets?",6993,"tickets",{"prompt":503,"answer":504,"operation":82,"unit":505},"A factory packs 1,001 bolts per crate. How many bolts in 64 crates?",64064,"bolts",{"prompt":507,"answer":508,"operation":83,"unit":497},"Share ₹4,848 equally among 48 families. Use (4,800 + 48) ÷ 48. Rupees each?",101,{"prompt":510,"answer":511,"operation":82,"unit":512},"A train travels 88 km each hour for 25 hours. Use 88 × 100 ÷ 4. Distance in km?",2200,"km","A harder mixed sprint: estimate, then multiply and divide exactly, using ×125, ×999, ×1,001 and splitting the dividend.","A 180-second sprint of 15 rounds mixing multiplication and division (division always gives whole answers). Plain questions first ask for an estimate, then the exact answer.\n\nUp to half the rounds are word problems from this list:\n1. 125 diyas × 8 boxes = 1,000 diyas.\n2. 36 sets at ₹125: 36 × 1,000 ÷ 8 = 36,000 ÷ 8 = ₹4,500.\n3. 999 tickets × 7 days: 7,000 − 7 = 6,993 tickets.\n4. 1,001 bolts × 64 crates: 64,000 + 64 = 64,064 bolts.\n5. ₹4,848 ÷ 48 families: 4,800 ÷ 48 + 48 ÷ 48 = 100 + 1 = ₹101 each.\n6. 88 km × 25 hours: 8,800 ÷ 4 = 2,200 km.",{"hints":516},[517,518],"125 × 8 = 1,000 and 25 × 4 = 100 are your best friends.","For division, split the dividend into parts the divisor divides exactly.",{"id":520,"type":67,"title":521,"eyebrow":522,"navLabel":523},"ch7","Curious numbers and quick-fire challenges","Chapter 07","7 Challenges",{"id":525,"type":103,"title":526,"problem":527,"steps":528},"we-last-digit","The last digit of 7 to the power 100","What is the last digit of 7¹⁰⁰ (7 multiplied by itself 100 times)?",[529,530,531,532,533],"Only the last digit matters when multiplying, so track last digits (arithmetic modulo 10).","Powers of 7 end in: 7, 9 (49), 3 (343), 1 (2,401), then 7 again (16,807).","The last digits repeat in a cycle of 4: 7, 9, 3, 1.","100 = 25 × 4, so 7¹⁰⁰ = (7⁴)²⁵ by the associative law.","7⁴ ends in 1, and 1 × 1 × … × 1 ends in 1. So the last digit is **1**.",{"id":535,"type":103,"title":536,"problem":537,"steps":538},"we-big-parity","Even or odd without adding","Is 1 + 2 + 3 + … + 1,000 even or odd?",[539,540,541,542],"Among 1 to 1,000 there are 500 odd numbers.","A sum is odd exactly when it contains an odd number of odd terms. 500 is even.","So the sum is **even**.","Check with the Gauss method: 1,000 × 1,001 ÷ 2 = 500,500, which is even. ✓",{"id":544,"type":76,"caption":545,"columns":546,"rows":549},"cyclic-table","The cyclic number 142857: the same six digits, rotated",[547,548],"Multiply by","Result",[550,552,554,556,558,561,563],[167,551],"142,857",[200,553],"285,714",[201,555],"428,571",[202,557],"571,428",[559,560],"5","714,285",[156,562],"857,142",[163,564],"999,999",{"id":566,"type":43,"markdown":567},"cyclic-prose","Multiplying 142,857 by 1 to 6 just rotates its digits. Then 142,857 × 7 = 999,999. The number comes from the repeating decimal of 1\u002F7 = 0.142857142857…, and the pattern is explained by clock arithmetic modulo 7. It is a lovely example of a pattern hiding a property.\n\nNow try these quick-fire challenges. Each is a property in disguise; the answers are in the quiz and summary.\n\n1. 999 × 999 = 998,001. Can you see why, using (1,000 − 1)²?\n2. What is 25 × 32 × 125? (Hint: 32 = 4 × 8.)\n3. 2,468 × 1,357 is even or odd? What about 2,469 + 1,357?\n4. Is 7,777,777 a perfect square? (Look at the last two digits and divide by 4.)",{"id":569,"type":133,"variant":134,"title":570,"markdown":571},"nuance-777","When the quick test says \"maybe\"","7,777,777 leaves remainder 1 when divided by 4 (77 = 4 × 19 + 1), so the \"remainder 0 or 1\" test from Deepen does **not** rule it out. A test that can only say \"no\" or \"maybe\" is still useful, but here we need more: the last digit of a square can never be 7 (squares end in 0, 1, 4, 5, 6 or 9). So 7,777,777 is **not** a square after all. Two partial tests together settle it.",{"id":573,"type":459,"component":574,"componentVersion":5,"config":575,"objective":634,"textAlternative":635,"help":636},"lab-sort-extend","sort-game",{"prompt":576,"bins":577,"items":593,"seconds":180},"Which is the smallest family (natural, whole, integers, fractions) that is closed for this calculation?",[578,581,584,587,590],{"id":579,"label":580},"n","Natural numbers",{"id":582,"label":583},"w","Whole numbers",{"id":585,"label":586},"z","Integers",{"id":588,"label":589},"q","Fractions (rationals)",{"id":591,"label":592},"none","No family: undefined",[594,598,602,606,610,614,618,622,626,630],{"id":595,"label":596,"bin":579,"why":597},"e1","Adding any two natural numbers","The sum of two counting numbers is a counting number.",{"id":599,"label":600,"bin":579,"why":601},"e2","Multiplying any two natural numbers","The product of two counting numbers is a counting number.",{"id":603,"label":604,"bin":582,"why":605},"e3","Subtracting a natural number from itself","a − a = 0, which needs the whole numbers.",{"id":607,"label":608,"bin":585,"why":609},"e4","Subtracting any integer from any integer","Integers are closed under subtraction; whole numbers are not (2 − 5).",{"id":611,"label":612,"bin":588,"why":613},"e5","Dividing any integer by any non-zero integer","Answers like 2 ÷ 5 need fractions.",{"id":615,"label":616,"bin":591,"why":617},"e6","Dividing any number by 0","Division by zero is undefined in every family of ordinary numbers.",{"id":619,"label":620,"bin":582,"why":621},"e7","Multiplying 0 by any whole number","The answer is 0, a whole number but not a natural number.",{"id":623,"label":624,"bin":585,"why":625},"e8","Finding the additive inverse of 5","The inverse is −5, an integer.",{"id":627,"label":628,"bin":588,"why":629},"e9","Finding the reciprocal of 5","The reciprocal is ⅕, a fraction.",{"id":631,"label":632,"bin":591,"why":633},"e10","Finding the reciprocal of 0","No number times 0 gives 1.","Choose the smallest family of numbers in which each kind of calculation always stays, or recognise when it is undefined.","A sorting game with five bins: natural numbers, whole numbers, integers, fractions (rationals), and \"No family: undefined\".\n\nNatural numbers: adding two natural numbers; multiplying two natural numbers.\nWhole numbers: a − a (gives 0); 0 × a (gives 0).\nIntegers: subtracting any integer from any integer; the additive inverse of 5 (−5).\nFractions: dividing an integer by a non-zero integer (2 ÷ 5); the reciprocal of 5 (⅕).\nUndefined: dividing any number by 0; the reciprocal of 0.\n\nThe idea: each bigger family was built to repair a closure failure, but division by zero is never repaired.",{"hints":637},[638,639],"Does the answer need 0? negatives? fractions?","Dividing by 0 never has an answer.",{"id":641,"type":76,"caption":642,"columns":643,"rows":649},"worlds-table","Which properties survive in which world?",[644,645,646,647,648],"World","Commutative","Associative","Identity","Inverses",[650,653,656,659,662,664,666],[651,301,301,199,652],"Whole numbers, +","no (needs negatives)",[654,301,301,199,655],"Integers, +","yes: −a",[657,301,301,167,658],"Fractions, ×","yes, except 0",[660,301,301,661,301],"Clock (mod 12), +","12 (acts as 0)",[280,288,301,663,288],"the empty word",[665,301,288,591,288],"Rock-paper-scissors winner",[667,288,301,668,669],"Turns and flips of a card","do nothing","yes: undo the move",{"id":671,"type":459,"component":672,"componentVersion":5,"config":673,"objective":696,"textAlternative":697,"help":698},"lab-match-worlds","match-pairs",{"prompt":674,"mode":675,"pairs":676},"Match each world to the fact about its properties.","connect",[677,680,683,686,688,689,690,693],{"a":678,"b":679},"Adding integers","Every number has an inverse",{"a":681,"b":682},"Subtracting integers","Closed, but not commutative",{"a":684,"b":685},"Dividing non-zero fractions","Closed, but not associative",{"a":292,"b":687},"12 acts as the identity",{"a":280,"b":287},{"a":665,"b":322},{"a":691,"b":692},"Adding decimals on a computer","Rounding can break associativity",{"a":694,"b":695},"Reciprocal of 0","Does not exist","Connect each number world or everyday operation with the property fact that describes it.","A matching game with eight worlds and eight property facts.\n\nAdding integers → every number has an inverse (−a).\nSubtracting integers → closed, but not commutative (5 − 3 = 2, 3 − 5 = −2).\nDividing non-zero fractions → closed, but not associative.\nClock addition → 12 acts as the identity.\nJoining words → associative, not commutative (\"rainbow\" vs \"bowrain\").\nRock-paper-scissors winner → commutative, not associative.\nAdding decimals on a computer → rounding can break associativity.\nReciprocal of 0 → does not exist, because nothing times 0 is 1.",{"hints":699},[700,701],"Think about which worlds allow swapping and which allow regrouping.","Clock arithmetic wraps round at 12.",{"id":703,"type":67,"title":704,"eyebrow":705,"navLabel":706},"ch8","Projects to try","Chapter 08","8 Projects",{"id":708,"type":709,"title":710,"items":711},"steps-projects","steps","Five projects, from one afternoon to a whole week",[712,716,719,723,727],{"title":713,"tag":714,"text":715},"Kirana detective","1 afternoon","Watch a shopkeeper total a bill. Which pairs does he add first? Write down three bills and show which property each shortcut uses.",{"title":717,"tag":714,"text":718},"Chair planner","Your school needs seating for 60, 72 and 100 people in rectangles. List every arrangement, spot the turn-around pairs and pick the best one for a stage.",{"title":720,"tag":721,"text":722},"Invent an operation","2–3 days","Make up a rule for combining two numbers. Test commutative, associative, identity and closure. Present your operation like a scientist, with counterexamples.",{"title":724,"tag":725,"text":726},"Mental maths league","1 week","Design a 10-question contest where every question has a property shortcut. Run it with friends; time the ones who use shortcuts versus those who do not.",{"title":728,"tag":725,"text":729},"Order-matters diary","Each day, note one everyday process where order matters and one where it does not. Which is more common?",{"id":731,"type":133,"variant":732,"title":733,"markdown":734},"try-invent","try_it","Starter for \"invent an operation\"","Try **a ★ b = a + b − 1** on whole numbers from 1 upward.\n\n1. Is it commutative? (Compare 3 ★ 5 and 5 ★ 3.)\n2. Is it associative? (Compare (2 ★ 3) ★ 4 with 2 ★ (3 ★ 4).)\n3. Is there an identity? (Find e with a ★ e = a.)\n4. Does every number have an inverse? (Find b with a ★ b = e.)\n\nSpoiler: it is commutative and associative, with identity 1; the inverse of a would be 2 − a, which exists in the whole numbers only for a = 1 or 2 (and 0, if you allow it). So your operation behaves like addition wearing a disguise.",{"id":736,"type":67,"title":737,"eyebrow":738,"navLabel":739},"ch9","Open questions and wider horizons","Chapter 09","9 Open questions",{"id":741,"type":43,"markdown":742},"goldbach-prose","Even and odd numbers look simple, yet they hide questions no one has answered.\n\n**Goldbach's conjecture (1742).** Every even number from 4 upward seems to be the sum of two primes: 4 = 2 + 2, 10 = 3 + 7 = 5 + 5, 100 = 3 + 97. Computers have checked every even number up to 4 × 10¹⁸ (four billion billion) without finding a counterexample — a distributed search completed in 2013. But nobody has a proof, so it is still a **conjecture**. This is exactly the difference between \"true for many examples\" and \"always true\" that you met in Investigate, at the highest level.\n\n**Odd perfect numbers.** A perfect number equals the sum of its factors other than itself: 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14. Every perfect number found so far is **even**. Is there an odd one? No one knows, after more than 2,000 years of searching.",{"id":744,"type":133,"variant":745,"title":746,"markdown":747},"open-questions","question","Questions to explore","Some of these are unsolved; others are just good questions to think about.\n\n- Goldbach's conjecture: is every even number from 4 up the sum of two primes?\n- Is there an odd perfect number?\n- Are there operations on whole numbers that are associative but have no identity and are not commutative? Invent one.\n- Which rows × columns arrangements of chairs make the best view of a stage? Is there a rule?\n- If computers cannot add decimals associatively, how do scientists trust huge calculations like climate models?\n- Why did it take humans so long to accept zero and negative numbers, when the properties seem so natural now?",{"id":749,"type":750,"conceptId":751,"relation":752,"explanation":753},"conn-prime","connection","prime-and-composite","related_to","Goldbach's conjecture joins even numbers and primes; chair arrangements are factor pairs.",{"id":755,"type":750,"conceptId":756,"relation":757,"explanation":758},"conn-number-system","number-system","part_of","Integers and fractions extend the number system; each family repairs a closure failure of the one before.",{"id":760,"type":750,"conceptId":761,"relation":752,"explanation":762},"conn-patterns","patterns","Cyclic numbers such as 142857 and calendar cycles are patterns explained by clock arithmetic.",{"id":764,"type":750,"conceptId":765,"relation":766,"explanation":767},"conn-data","data-handling","applied_in","The averaging trap: the mean of group means is wrong unless groups are equal in size, because averaging is not associative.",{"id":769,"type":750,"conceptId":770,"relation":752,"explanation":771},"conn-shape","shape-and-space","Turning and flipping shapes do not commute, the start of the study of symmetry.",{"id":773,"type":67,"title":774,"eyebrow":775,"navLabel":776},"ch10","Review","Chapter 10","10 Review",{"id":778,"type":779,"title":780,"terms":781},"glossary-extend","glossary","Wider vocabulary",[782,785,789,793,797,801,804,807,810,814,818,822],{"term":586,"meaning":783,"example":784},"The whole numbers together with their negatives: … −2, −1, 0, 1, 2, …","Temperature −5 °C",{"term":786,"meaning":787,"example":788},"Rational numbers","Numbers that can be written as a fraction p\u002Fq of integers with q ≠ 0.","¾, −2, 7\u002F3",{"term":790,"meaning":791,"example":792},"Additive inverse","The opposite of a number: a + (−a) = 0.","The inverse of 9 is −9.",{"term":794,"meaning":795,"example":796},"Modular arithmetic","Arithmetic in which numbers wrap round after reaching a fixed number, as on a clock.","9 + 5 = 2 (mod 12)",{"term":798,"meaning":799,"example":800},"Concatenation","Joining pieces of text end to end. Associative but not commutative.","\"rain\" + \"bow\" = \"rainbow\"",{"term":802,"meaning":803},"Floating-point number","The rounded way computers store decimals, which can make addition non-associative.",{"term":805,"meaning":806},"Principle of permanence","The idea that new kinds of numbers should keep the old properties where possible.",{"term":808,"meaning":809,"example":39},"Conjecture","A statement believed true but not yet proved.",{"term":811,"meaning":812,"example":813},"Perfect number","A number equal to the sum of its factors other than itself.","6 = 1 + 2 + 3",{"term":815,"meaning":816,"example":817},"Cyclic number","A number whose multiples by 1, 2, … are rotations of its digits.","142857",{"term":819,"meaning":820,"example":821},"Invariant","A quantity that does not change during a process.","The parity of a sum when signs are flipped.",{"term":823,"meaning":824},"Weighted average","An average where each value is multiplied by how often it occurs before dividing.",{"id":826,"type":827,"title":828,"questions":829},"quiz-extend","quiz","Extend challenge",[830,840,853,862,874,887,900,913,926,937,950],{"itemId":831,"prompt":832,"options":833,"correct":125,"why":839},"properties-of-numbers.extend-q-integers","Which family is closed under subtraction?",[834,835,836,837],{"id":119,"label":580},{"id":122,"label":583},{"id":125,"label":586},{"id":128,"label":838},"Odd numbers","Any integer minus any integer is an integer. 2 − 5 = −3 escapes the whole numbers.",{"itemId":841,"prompt":842,"options":843,"correct":122,"why":852},"properties-of-numbers.extend-q-negneg","(−6) × (−5) =",[844,846,848,850],{"id":119,"label":845},"−30",{"id":122,"label":847},"30",{"id":125,"label":849},"−11",{"id":128,"label":851},"11","Keeping the distributive law forces negative × negative to be positive: 30.",{"itemId":854,"prompt":855,"options":856,"correct":122,"why":861},"properties-of-numbers.extend-q-clock","On a 12-hour clock, 10 + 5 + 9 =",[857,858,859,860],{"id":119,"label":171},{"id":122,"label":256},{"id":125,"label":199},{"id":128,"label":200},"10 + 5 + 9 = 24, two full turns, back to 12 o'clock. On a clock 12 plays the role of 0.",{"itemId":863,"prompt":864,"options":865,"correct":122,"why":873},"properties-of-numbers.extend-q-days","If today is Thursday, what day is it in 50 days?",[866,868,870,872],{"id":119,"label":867},"Thursday",{"id":122,"label":869},"Friday",{"id":125,"label":871},"Saturday",{"id":128,"label":237},"50 = 7 × 7 + 1: 7 weeks and 1 day, so Friday.",{"itemId":875,"prompt":876,"options":877,"correct":122,"why":886},"properties-of-numbers.extend-q-words","Joining words (\"sun\" + \"set\") is…",[878,880,882,884],{"id":119,"label":879},"commutative and associative",{"id":122,"label":881},"associative but not commutative",{"id":125,"label":883},"commutative but not associative",{"id":128,"label":885},"neither","\"sunset\" ≠ \"setsun\", but joining three words gives the same result however you group them.",{"itemId":888,"prompt":889,"options":890,"correct":122,"why":899},"properties-of-numbers.extend-q-float","A computer gives (0.1 + 0.2) + 0.3 = 0.6000000000000001 but 0.1 + (0.2 + 0.3) = 0.6. Why?",[891,893,895,897],{"id":119,"label":892},"Addition is not associative",{"id":122,"label":894},"The computer rounds decimals stored in binary",{"id":125,"label":896},"The computer is broken",{"id":128,"label":898},"0.1 is a negative number","Real addition is associative. Rounding in the machine causes the tiny difference: a model limit.",{"itemId":901,"prompt":902,"options":903,"correct":119,"why":912},"properties-of-numbers.extend-q-1001","538,538 ÷ 13 =",[904,906,908,910],{"id":119,"label":905},"41,426",{"id":122,"label":907},"41,462",{"id":125,"label":909},"4,142",{"id":128,"label":911},"538","538,538 = 538 × 1,001 = 538 × 7 × 11 × 13, so 538,538 ÷ 13 = 41,426.",{"itemId":914,"prompt":915,"options":916,"correct":122,"why":925},"properties-of-numbers.extend-q-signs","Signs + or − before each of 1 to 10. Possible totals are always…",[917,919,921,923],{"id":119,"label":918},"even",{"id":122,"label":920},"odd",{"id":125,"label":922},"multiples of 5",{"id":128,"label":924},"zero","All + gives 55 (odd); each sign change moves the total by an even amount, so it stays odd.",{"itemId":927,"prompt":928,"options":929,"correct":119,"why":936},"properties-of-numbers.extend-q-avg","Group A: 20 students, average 50. Group B: 20 students, average 70. The overall average is…",[930,931,932,934],{"id":119,"label":395},{"id":122,"label":391},{"id":125,"label":933},"55",{"id":128,"label":935},"cannot tell","Equal group sizes, so averaging the averages works here: (20 × 50 + 20 × 70) ÷ 40 = 60.",{"itemId":938,"prompt":939,"options":940,"correct":119,"why":949},"properties-of-numbers.extend-q-999","999 × 999 =",[941,943,945,947],{"id":119,"label":942},"998,001",{"id":122,"label":944},"999,001",{"id":125,"label":946},"998,991",{"id":128,"label":948},"1,000,001","(1,000 − 1)² = 1,000,000 − 2,000 + 1 = 998,001.",{"itemId":951,"prompt":952,"options":953,"correct":122,"why":959},"properties-of-numbers.extend-q-rps","In rock-paper-scissors, \"the winner of a and b\" is…",[954,956,957,958],{"id":119,"label":955},"associative",{"id":122,"label":883},{"id":125,"label":885},{"id":128,"label":881},"Who is named first does not matter, but (rock vs paper) vs scissors = scissors while rock vs (paper vs scissors) = rock.",{"id":961,"type":962,"prompt":963},"reflect-extend","reflection","Pick one new \"world\" from this layer (integers, clocks, words, computer decimals, rock-paper-scissors). Which properties survive there and which break? Explain one surprise you found, and why you think it happens.",{"id":965,"type":966,"title":967,"points":968},"cheat-extend","summary","Cheat sheet",[969,970,971,972,973,974,975,976],"**Integers** repair subtraction (closed under −) and give every number an additive inverse; **fractions** repair division, except by 0.","**Negative × negative = positive** is forced by the distributive law: (−3)(4 + (−4)) = 0 gives (−3)(−4) = 12.","**Clock arithmetic** (modulo 12, modulo 7 for weekdays) keeps closure, commutativity, associativity, identity and inverses.","**Order often matters in real life:** words, turns and flips, cooking, the Rubik's cube. Joining words is associative but not commutative.","**Rock-paper-scissors and averages** are commutative but not associative; average group averages only when groups are equal.","**Computers** round decimals, so (0.1 + 0.2) + 0.3 ≠ 0.1 + (0.2 + 0.3) on a machine: a model limit, not a maths failure.","**Puzzles:** abcabc = abc × 7 × 11 × 13; parity forbids ±1 ± 2 … ± 10 = 0; colour counting forbids the domino tiling.","**Open:** Goldbach (checked to 4 × 10¹⁸, unproved) and odd perfect numbers (none known).",{"id":978,"type":979,"sourceIds":980},"sources-extend","sources",[981,982,983,984,985,986,987,988,989,990,991],"properties-of-numbers-ncert-class6-whole-numbers","properties-of-numbers-ncert-class7-integers","properties-of-numbers-ncert-class8-rational-numbers","properties-of-numbers-mathsisfun-properties","properties-of-numbers-mathsisfun-divide-by-zero","properties-of-numbers-wiki-commutative","properties-of-numbers-wiki-distributive","properties-of-numbers-wiki-division-by-zero","properties-of-numbers-wiki-parity","properties-of-numbers-wiki-brahmagupta","properties-of-numbers-wiki-goldbach",[981,982,983,984,985,986,987,988,989,990,991],"needs_review",{"generatedBy":995,"notes":996},"claude-code","Draft generated with Python generators; every stated number computed and asserted. Pending owner review.","bffb598ba577b7c1cbf00cd48ffac18cc7cb721f043aa188d00419ff063f8dec",{"logic:practice":999,"component:order-ops@1":1000,"component:arith-sprint@1":1001,"component:sort-game@1":1002,"component:match-pairs@1":1003,"source:properties-of-numbers-mathsisfun-divide-by-zero":1004,"source:properties-of-numbers-mathsisfun-properties":1005,"source:properties-of-numbers-ncert-class6-whole-numbers":1006,"source:properties-of-numbers-ncert-class7-integers":1007,"source:properties-of-numbers-ncert-class8-rational-numbers":1008,"source:properties-of-numbers-wiki-brahmagupta":1009,"source:properties-of-numbers-wiki-commutative":1010,"source:properties-of-numbers-wiki-distributive":1011,"source:properties-of-numbers-wiki-division-by-zero":1012,"source:properties-of-numbers-wiki-goldbach":1013,"source:properties-of-numbers-wiki-parity":1014},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","f8a42fd8c82c266f4310450d74f1ae0449b3dba177afcf000b24aafa2839b629","a37b85a36742e06f34cdc083710eacdd662e2f7493395766e4b8c5632bfade3a","56b97dbf2988a58e1aa10f693ec9dd1039e1de14e795b1bdfa19980b6b18c1fa","aab8fe53a32660ef6307b4dbd6f5af02319957d9aecf3c0396895bc01ab46e2e","acb8b92cad8387724d72528e70ef68195351e5b2f03b580c5da191fde82dc67a","32e7279091a5877afe54d6da75cdd06a04f2ff8677338777578592d6dd98a338","a3e0fb47a2115f5962c9795b138bf38cd9c2c4874bc66e1b834717b9a01908c1","befe26bd7ec4e1c3911549d33a91844ae3eb5bd63388711f61822d398d3d576f","046256cc76cdb1b7cca72a1b391bef686855afdfb6a68d18b45a38723d1fe93d","93d67dbe81864df7a775ac02db3ccfbb2af796cfbeafa91ff5325a5d51d7ca11","01c5bcf03886aec6f9c2c18c9ec5333b19ce1d2ca6bfb4f8544c4ca9f1578e27",{"state":1016,"reviewer":1017,"selfReview":468,"reviewedAt":1018,"method":1019},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597155]