var Component=(()=>{var m=Object.create;var a=Object.defineProperty;var p=Object.getOwnPropertyDescriptor;var u=Object.getOwnPropertyNames;var b=Object.getPrototypeOf,g=Object.prototype.hasOwnProperty;var x=(i,n)=>()=>(n||i((n={exports:{}}).exports,n),n.exports),v=(i,n)=>{for(var l in n)a(i,l,{get:n[l],enumerable:!0})},t=(i,n,l,h)=>{if(n&&typeof n=="object"||typeof n=="function")for(let r of u(n))!g.call(i,r)&&r!==l&&a(i,r,{get:()=>n[r],enumerable:!(h=p(n,r))||h.enumerable});return i};var w=(i,n,l)=>(l=i!=null?m(b(i)):{},t(n||!i||!i.__esModule?a(l,"default",{value:i,enumerable:!0}):l,i)),f=i=>t(a({},"__esModule",{value:!0}),i);var c=x((k,d)=>{d.exports=_jsx_runtime});var y={};v(y,{default:()=>o,frontmatter:()=>N});var e=w(c()),N={title:"1. Real Numbers \u5B9E\u6570",date:"2024-12-31",tags:["tutorial","mathematics","real-numbers"],draft:!1,summary:"An introduction to real numbers, covering field axioms, order axioms, and completeness axiom with examples."};function s(i){let n={a:"a",code:"code",em:"em",h1:"h1",h2:"h2",h3:"h3",hr:"hr",li:"li",ol:"ol",p:"p",path:"path",span:"span",strong:"strong",svg:"svg",ul:"ul",...i.components};return(0,e.jsxs)(e.Fragment,{children:[(0,e.jsxs)(n.h1,{className:"content-header",id:"1-real-numbers-\u5B9E\u6570",children:[(0,e.jsx)(n.a,{"aria-hidden":"true",href:"#1-real-numbers-\u5B9E\u6570",tabIndex:"-1",children:(0,e.jsx)(e.Fragment,{children:(0,e.jsx)(n.span,{className:"content-header-link",children:(0,e.jsxs)(n.svg,{className:"h-5 linkicon w-5",fill:"currentColor",viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",children:[(0,e.jsx)(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),(0,e.jsx)(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"1. 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Field Axioms"]}),(0,e.jsxs)(n.p,{children:["The real numbers form a ",(0,e.jsx)(n.strong,{children:"field"}),", satisfying algebraic properties:"]}),(0,e.jsxs)(n.ul,{children:[(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"Commutativity"}),": a + b = b + a, a \xB7 b = b \xB7 a. ",(0,e.jsx)(n.em,{children:"Example"}),": 2 + 3 = 3 + 2, 4 \xB7 5 = 5 \xB7 4."]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"Associativity"}),": (a + b) + c = a + (b + c), (a \xB7 b) \xB7 c = a \xB7 (b \xB7 c). ",(0,e.jsx)(n.em,{children:"Example"}),": (1 + 2) + 3 = 1 + (2 + 3)."]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"Distributivity"}),": a \xB7 (b + c) = a \xB7 b + a \xB7 c. 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",(0,e.jsx)(n.em,{children:"\u4F8B"}),"\uFF1A2 + 3 = 3 + 2, 4 \xB7 5 = 5 \xB7 4\u3002"]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u7ED3\u5408\u5F8B"}),"\uFF1A(a + b) + c = a + (b + c), (a \xB7 b) \xB7 c = a \xB7 (b \xB7 c)\u3002 ",(0,e.jsx)(n.em,{children:"\u4F8B"}),"\uFF1A(1 + 2) + 3 = 1 + (2 + 3)"]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u5206\u914D\u5F8B"}),"\uFF1Aa \xB7 (b + c) = a \xB7 b + a \xB7 c\u3002 ",(0,e.jsx)(n.em,{children:"\u4F8B"}),"\uFF1A3 \xB7 (4 + 5) = 3 \xB7 4 + 3 \xB7 5\u3002"]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u5355\u4F4D\u5143"}),"\uFF1A",(0,e.jsxs)(n.ul,{children:[(0,e.jsx)(n.li,{children:"\u52A0\u6CD5\u5355\u4F4D\u5143\uFF1Aa + 0 = a\u3002"}),(0,e.jsx)(n.li,{children:"\u4E58\u6CD5\u5355\u4F4D\u5143\uFF1Aa \xB7 1 = a\u3002"})]})]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u9006\u5143"}),"\uFF1A",(0,e.jsxs)(n.ul,{children:[(0,e.jsx)(n.li,{children:"\u52A0\u6CD5\u9006\u5143\uFF1A\u5BF9\u4EFB\u610Fa\uFF0C\u5B58\u5728-a\u4F7F\u5F97a + (-a) = 0\u3002"}),(0,e.jsx)(n.li,{children:"\u4E58\u6CD5\u9006\u5143\uFF1A\u5BF9\u4EFB\u610F\u975E\u96F6a\uFF0C\u5B58\u5728a^(-1)\u4F7F\u5F97a \xB7 a^(-1) = 1\u3002"})]})]})]}),(0,e.jsx)(n.hr,{}),(0,e.jsxs)(n.h3,{className:"content-header",id:"2-order-axioms",children:[(0,e.jsx)(n.a,{"aria-hidden":"true",href:"#2-order-axioms",tabIndex:"-1",children:(0,e.jsx)(e.Fragment,{children:(0,e.jsx)(n.span,{className:"content-header-link",children:(0,e.jsxs)(n.svg,{className:"h-5 linkicon w-5",fill:"currentColor",viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",children:[(0,e.jsx)(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),(0,e.jsx)(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"2. Order Axioms"]}),(0,e.jsxs)(n.p,{children:["R is a ",(0,e.jsx)(n.strong,{children:"totally ordered field"})," with:"]}),(0,e.jsxs)(n.ul,{children:[(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"Transitivity"}),": If a < b and b < c, then a < c. ",(0,e.jsx)(n.em,{children:"Example"}),": 1 < 2 < 3 implies 1 < 3."]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"Trichotomy"}),": For any a, b, exactly one holds: a < b, a = b, or a > b."]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"Compatibility with Operations"}),":",(0,e.jsxs)(n.ul,{children:[(0,e.jsx)(n.li,{children:"If a < b, then a + c < b + c."}),(0,e.jsxs)(n.li,{children:["If a < b and c > 0, then a \xB7 c < b \xB7 c. ",(0,e.jsx)(n.em,{children:"Example"}),": 2 < 5 implies 2 + 3 < 5 + 3, 2 \xB7 4 < 5 \xB7 4."]})]})]})]}),(0,e.jsxs)(n.h3,{className:"content-header",id:"2-\u5E8F\u516C\u7406",children:[(0,e.jsx)(n.a,{"aria-hidden":"true",href:"#2-\u5E8F\u516C\u7406",tabIndex:"-1",children:(0,e.jsx)(e.Fragment,{children:(0,e.jsx)(n.span,{className:"content-header-link",children:(0,e.jsxs)(n.svg,{className:"h-5 linkicon w-5",fill:"currentColor",viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",children:[(0,e.jsx)(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),(0,e.jsx)(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"2. \u5E8F\u516C\u7406"]}),(0,e.jsxs)(n.p,{children:["R\u662F\u4E00\u4E2A",(0,e.jsx)(n.strong,{children:"\u5168\u5E8F\u57DF"}),"\uFF0C\u6EE1\u8DB3\uFF1A"]}),(0,e.jsxs)(n.ul,{children:[(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u4F20\u9012\u6027"}),"\uFF1A\u82E5a < b\u4E14b < c\uFF0C\u5219a < c\u3002 ",(0,e.jsx)(n.em,{children:"\u4F8B"}),"\uFF1A1 < 2 < 3 implies 1 < 3\u3002"]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u4E09\u5206\u6027"}),"\uFF1A\u5BF9\u4EFB\u610Fa, b\uFF0C\u6709\u4E14\u4EC5\u6709\u4E00\u79CD\u60C5\u51B5\u6210\u7ACB\uFF1Aa < b\uFF0Ca = b\uFF0C\u6216a > b\u3002"]}),(0,e.jsxs)(n.li,{children:[(0,e.jsx)(n.strong,{children:"\u4E0E\u8FD0\u7B97\u517C\u5BB9"}),"\uFF1A",(0,e.jsxs)(n.ul,{children:[(0,e.jsx)(n.li,{children:"\u82E5a < b\uFF0C\u5219a + c < b + c\u3002"}),(0,e.jsxs)(n.li,{children:["\u82E5a < b\u4E14c > 0\uFF0C\u5219a \xB7 c < b \xB7 c\u3002 ",(0,e.jsx)(n.em,{children:"\u4F8B"}),"\uFF1A2 < 5 implies 2 + 3 < 5 + 3\uFF0C2 \xB7 4 < 5 \xB7 4\u3002"]})]})]})]}),(0,e.jsx)(n.hr,{}),(0,e.jsxs)(n.h3,{className:"content-header",id:"3-completeness-axiom",children:[(0,e.jsx)(n.a,{"aria-hidden":"true",href:"#3-completeness-axiom",tabIndex:"-1",children:(0,e.jsx)(e.Fragment,{children:(0,e.jsx)(n.span,{className:"content-header-link",children:(0,e.jsxs)(n.svg,{className:"h-5 linkicon w-5",fill:"currentColor",viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",children:[(0,e.jsx)(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),(0,e.jsx)(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"3. Completeness Axiom"]}),(0,e.jsxs)(n.p,{children:["The ",(0,e.jsx)(n.strong,{children:"key property"})," distinguishing R from Q:"]}),(0,e.jsx)(n.ul,{children:(0,e.jsxs)(n.li,{children:["Every non-empty subset of R that is bounded above has a ",(0,e.jsx)(n.strong,{children:"least upper bound (supremum)"})," in R. ",(0,e.jsx)(n.em,{children:"Example"}),": Let S = ",(0,e.jsx)(n.code,{children:"{x in Q | x^2 < 2}"}),". In Q, S has no supremum (since sqrt(2) is not in Q), but in R, sup(S) = sqrt(2)."]})}),(0,e.jsxs)(n.h3,{className:"content-header",id:"3-\u5B8C\u5907\u6027\u516C\u7406",children:[(0,e.jsx)(n.a,{"aria-hidden":"true",href:"#3-\u5B8C\u5907\u6027\u516C\u7406",tabIndex:"-1",children:(0,e.jsx)(e.Fragment,{children:(0,e.jsx)(n.span,{className:"content-header-link",children:(0,e.jsxs)(n.svg,{className:"h-5 linkicon w-5",fill:"currentColor",viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",children:[(0,e.jsx)(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),(0,e.jsx)(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"3. \u5B8C\u5907\u6027\u516C\u7406"]}),(0,e.jsxs)(n.p,{children:["R\u533A\u522B\u4E8EQ\u7684",(0,e.jsx)(n.strong,{children:"\u5173\u952E\u6027\u8D28"}),"\uFF1A"]}),(0,e.jsx)(n.ul,{children:(0,e.jsxs)(n.li,{children:["\u6BCF\u4E2A\u975E\u7A7A\u4E14\u6709\u4E0A\u754C\u7684R\u5B50\u96C6\u5728R\u4E2D\u90FD\u6709",(0,e.jsx)(n.strong,{children:"\u6700\u5C0F\u4E0A\u754C\uFF08\u4E0A\u786E\u754C\uFF09"}),"\u3002 ",(0,e.jsx)(n.em,{children:"\u4F8B"}),"\uFF1A \u8BBE\u96C6\u5408S = ",(0,e.jsx)(n.code,{children:"{x in Q | x^2 < 2}"}),"\u3002\u5728Q\u4E2D\uFF0CS\u65E0\u4E0A\u786E\u754C\uFF08\u56E0sqrt(2)\u4E0D\u5C5E\u4E8EQ\uFF09\uFF0C\u4F46\u5728R\u4E2D\uFF0Csup(S) = sqrt(2)\u3002"]})}),(0,e.jsx)(n.hr,{}),(0,e.jsxs)(n.h2,{className:"content-header",id:"111-positive-negative-zero-\u6B63\u6570\u8D1F\u6570\u96F6",children:[(0,e.jsx)(n.a,{"aria-hidden":"true",href:"#111-positive-negative-zero-\u6B63\u6570\u8D1F\u6570\u96F6",tabIndex:"-1",children:(0,e.jsx)(e.Fragment,{children:(0,e.jsx)(n.span,{className:"content-header-link",children:(0,e.jsxs)(n.svg,{className:"h-5 linkicon w-5",fill:"currentColor",viewBox:"0 0 20 20",xmlns:"http://www.w3.org/2000/svg",children:[(0,e.jsx)(n.path,{d:"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"}),(0,e.jsx)(n.path,{d:"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"})]})})})}),"1.1.1 Positive, Negative, Zero \u6B63\u6570\uFF0C\u8D1F\u6570\uFF0C\u96F6"]}),(0,e.jsx)(n.p,{children:(0,e.jsx)(n.strong,{children:"Positive \u6B63(\u6570)"})}),(0,e.jsx)(n.p,{children:"adj. /\u02C8p\u0252z\u0259t\u026Av/"}),(0,e.jsx)(n.p,{children:"Definition: Greater than zero. \u6B63\u6570\u662F\u5927\u4E8E0\u7684\u6570\u3002"}),(0,e.jsx)(n.p,{children:"Notation: The set of positive numbers is denoted by R+. \u6B63\u6570\u96C6\u7528R+\u8868\u793A\u3002"}),(0,e.jsx)(n.p,{children:"Properties:"}),(0,e.jsxs)(n.ol,{children:[(0,e.jsx)(n.li,{children:(0,e.jsx)(n.p,{children:"Positive numbers are always at the right side of 0 on the number line. \u6B63\u6570\u5728\u6570\u8F74\u4E0A0\u7684\u53F3\u8FB9\u3002"})}),(0,e.jsx)(n.li,{children:(0,e.jsx)(n.p,{children:"The union of the set of positive numbers and zero is also known as the set of nonnegative numbers. \u6B63\u6570\u96C6\u548C\u96F6\u7684\u5E76\u96C6\u662F\u975E\u8D1F\u96C6\u3002"})})]}),(0,e.jsx)(n.p,{children:(0,e.jsx)(n.strong,{children:"Negative 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