[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-143746-105":59,"doc-detail-143746-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","topical-mathematics-unit-11-numbers-and-the-four-operations-key-concepts-and-worked-examples","TOPICAL MATHEMATICS - Unit 1.1 Numbers and the Four Operations - Key concepts and worked examples","","Unit 1.1 focuses on foundational number concepts and the four operations through definitions, classifications, and worked calculations. It introduces natural, whole, integer, positive/negative, rational, irrational, and real numbers, then applies operations to find differences and averages from temperature data. The unit develops prime factorisation, highest common factor (HCF), lowest common multiple (LCM), and extends to squares, square roots, cubes, cube roots, reciprocals, and significant figures with rounding examples.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/topical-mathematics-unit-11-numbers-and-the-four-operations-key-concepts-and-worked-examples/143746/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/topical-mathematics-unit-11-numbers-and-the-four-operations-key-concepts-and-worked-examples/143746.png","ImageObject",300,407,{"name":92,"@type":93},"Miles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What number sets are introduced in Unit 1.1?","Question",{"text":113,"@type":114},"The unit introduces natural numbers, whole numbers, integers, positive/negative integers, rational numbers, irrational numbers, and real numbers, along with their set definitions.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do prime factorisation, HCF, and LCM connect in the unit?",{"text":118,"@type":114},"Prime factorisation breaks composite numbers into prime factors; HCF uses the highest common prime-power factors, while LCM uses the smallest set of prime powers that covers all numbers.",{"name":120,"@type":111,"acceptedAnswer":121},"How are significant figures and rounding handled?",{"text":122,"@type":114},"The unit states which digits are significant, including zeros between non-zero digits, and shows how to round a number like 2014 to specified numbers of significant figures.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},143746,1787696566,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145},13056703019404,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","CONTENTS  \nUnit 1.1 Numbers and the Four Operations 1  \nUnit 1.2 Ratio, Rate and Proportion 15  \nUnit 1.3 Percentage 21  \nUnit 1.4 Speed 24  \nUnit 1.5 Algebraic Representation and Formulae 28  \nUnit 1.6 Algebraic Manipulation 33  \nUnit 1.7 Functions and Graphs 41  \nUnit 1.8 Solutions of Equations and Inequalities 51  \nUnit 1.9 Applications of Mathematics in Practical Situations 64  \nUnit 1.10 Set Language and Notation 77  \nUnit 1.11 Matrices 85  \nUnit 2.1 Angles, Triangles and Polygons 91  \nUnit 2.2 Congruence and Similarity 103  \nUnit 2.3 Properties of Circles 113  \nUnit 2.4 Pythagoras’ Theorem and Trigonometry 119  \nUnit 2.5 Mensuration 129  \nUnit 2.6 Coordinate Geometry 138  \nUnit 2.7 Vectors in Two Dimensions 143  \nUnit 3.1 Data Handling 151  \nUnit 3.2 Data Analysis 156  \nUnit 3.3 Probability 167  \nMathematical Formulae 176  \n\n| UNIT\u003Cbr>1.1 | Numbers and\u003Cbr>the Four Operations |\n| --- | --- |\n\nNumbers  \n1. The set of natural numbers, 􀀟 = {1, 2, 3,   }  \n2. The set of whole numbers, W = {0, 1, 2, 3,   }  \n3. The set of integers, Z = {…,–2,–1, 0, 1, 2,…}  \n4. The set of positive integers, Z+ = {1, 2, 3,   }  \n5. The set of negative integers, Z– = {–1,–2,–3,…}  \n6. The set of rational numbers, Q = { ab , a , b 􀀟 Z , b ≠ 0}  \n7. An irrational number is a number which cannot be expressed in the form ab , where a , b are integers and b ≠ 0.  \n8. The set of real numbers R is the set of rational and irrational numbers.  \n9.  \nReal Numbers  \nWhole Numbers  \nNumbers and the Four Operations  1   \nExample 1  \nThe temperature at the bottom of a mountain was 22 °C and the temperature at the top was –7 °C. Find  \n(a) the difference between the two temperatures,  \n(b) the average of the two temperatures.  \nSolution  \n(a) Difference between the temperatures = 22 –(–7)  \n= 22 + 7  \n= 29 °C  \n(b) Average of the temperatures = 22~~ ~~+2(−7)  \n22 − 7  \n= ~~ ~~  \n2  \n15  \n= ~~ ~~  \n2  \n= 7.5 °C  \nPrime Factorisation  \n10. A prime number is a number that can only be divided exactly by 1 and itself.  \nHowever, 1 is not considered as a prime number.  \ne.g. 2, 3, 5, 7, 11, 13,…  \n11. Prime factorisation is the process of expressing a composite number as a product of its prime factors.  \n2  \nUNIT 1.1  \nExample 2  \nExpress 30 as a product of its prime factors.  \nSolution  \nFactors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Of these, 2, 3, 5 are prime factors.  \n2  \n3  \n5  \n\n| 30 |\n| --- |\n| 15 |\n| 5 |\n\n1  \n∴ 30 = 2 × 3 × 5  \nExample 3  \nExpress 220 as a product of its prime factors.  \nSolution  \n220  \n2 110  \n2 55  \n5 11  \n􀀛 220 = 22 × 5 × 11  \nFactors and Multiples  \n12. The highest common factor (HCF) of two or more numbers is the largest factor that is common to all the numbers .  \nNumbers and the Four Operations  3   \nExample 4  \nFind the highest common factor of 18 and 30.  \nSolution  \n18 = 2 × 32  \n30 = 2 × 3 × 5  \nHCF = 2 × 3  \n= 6  \nExample 5  \nFind the highest common factor of 80, 120 and 280.  \nSolution  \nMethod 1  \n2  \n2  \n2  \n5  \n  80, 120, 280    40, 60, 140   20, 30, 70   \n 10, 15, 35   \n2, 3, 7 HCF = 2 × 2 × 2 × 5  \n= 40  \n(Since the three numbers cannot be divided further by a common prime factor, we stop here)  \nMethod 2  \nExpress 80, 120 and 280 as products of their prime factors 80 = 23 × 5  \n120 = 23 × 5  \n280 = 23 × 5 × 7  \nHCF = 23 × 5  \n= 40  \n4  \nUNIT 1.1  \n13. The lowest common multiple (LCM) of two or more numbers is the smallest multiple that is common to all the numbers.  \nExample 6  \nFind the lowest common multiple of 18 and 30.  \nSolution  \n18 = 2 × 32  \n30 = 2 × 3 × 5  \nLCM = 2 × 32 × 5  \n= 90  \nExample 7  \nFind the lowest common multiple of 5, 15 and 30.  \nSolution  \nMethod 1  \n2  \n3  \n5  \n\n| 5, 15, 30 |  |  |\n| --- | --- | --- |\n| 5, | 15, | 15 |\n| 5, | 5, | 5 |\n\n(Continue to divide by the prime factors until 1 is reached)  \n1, 1, 1  \nLCM = 2 × 3 × 5  \n= 30  \nMethod 2  \nExpress 5, 15 and 30 as products of their prime factors.  \n5 = 1 × 5  \n15 = 3 × 5  \n30 = 2 × 3 × 5  \nLCM = 2 × 3 × 5  \n= 30  \nNumbers and the Four Operations  5","cbCaib5Y0QrDtyM9","https://ap.wps.com/l/cbCaib5Y0QrDtyM9","pdf",5554028,178,"English","# Unit 1.1 Numbers and the Four Operations\n## Numbers and number sets\n## The four operations with examples\n## Prime factorisation\n## Factors and multiples (HCF, LCM)\n## Squares, square roots, cubes, cube roots\n## Reciprocal and significant figures","[{\"question\":\"What number sets are introduced in Unit 1.1?\",\"answer\":\"The unit introduces natural numbers, whole numbers, integers, positive/negative integers, rational numbers, irrational numbers, and real numbers, along with their set definitions.\"},{\"question\":\"How do prime factorisation, HCF, and LCM connect in the unit?\",\"answer\":\"Prime factorisation breaks composite numbers into prime factors; HCF uses the highest common prime-power factors, while LCM uses the smallest set of prime powers that covers all numbers.\"},{\"question\":\"How are significant figures and rounding handled?\",\"answer\":\"The unit states which digits are significant, including zeros between non-zero digits, and shows how to round a number like 2014 to specified numbers of significant figures.\"}]","TOPICAL MATHEMATICS - Unit 1.1 Numbers and the Four Operations - Key concepts and worked examples | PDF",449]