{"id":755,"date":"2024-09-21T10:35:45","date_gmt":"2024-09-21T02:35:45","guid":{"rendered":"https:\/\/www.math.ntnu.edu.tw\/merc\/?page_id=755"},"modified":"2025-10-16T15:34:45","modified_gmt":"2025-10-16T07:34:45","slug":"%e7%84%a1%e7%90%86%e6%95%b8%e7%9a%84%e7%99%bc%e7%8f%be","status":"publish","type":"page","link":"https:\/\/www.math.ntnu.edu.tw\/merc\/animated_math\/%e7%84%a1%e7%90%86%e6%95%b8%e7%9a%84%e7%99%bc%e7%8f%be\/","title":{"rendered":"\u7121\u7406\u6578\u7684\u767c\u73fe"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"755\" class=\"elementor elementor-755\">\n\t\t\t\t<div class=\"elementor-element elementor-element-31cb5dda e-flex e-con-boxed e-con e-parent\" data-id=\"31cb5dda\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-59178951 elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"59178951\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;bounceIn&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">\u821e\u52d5\u6578\u5b78<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-5a01465f e-flex e-con-boxed e-con e-parent\" data-id=\"5a01465f\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2310fc49 elementor-widget elementor-widget-heading\" data-id=\"2310fc49\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">\u7121\u7406\u6578\u7684\u767c\u73fe<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-2d5ef78f e-flex e-con-boxed e-con e-parent\" data-id=\"2d5ef78f\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5b4c9a65 elementor-widget elementor-widget-video\" data-id=\"5b4c9a65\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;hosted&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"e-hosted-video elementor-wrapper elementor-open-inline\">\n\t\t\t\t\t<video class=\"elementor-video\" src=\"https:\/\/cantor.math.ntnu.edu.tw\/workshop\/animated_math\/media\/\u7121\u7406\u6578\u7684\u767c\u73fe.mp4\" autoplay=\"\" loop=\"\" controls=\"\" controlsList=\"nodownload\"><\/video>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-5cb65298 e-flex e-con-boxed e-con e-parent\" data-id=\"5cb65298\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4addeb73 elementor-widget elementor-widget-text-editor\" data-id=\"4addeb73\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h3 class=\"text-center\">\u300c\u7121\u7406\u6578\u7684\u767c\u73fe\u300d<\/h3>\n<h4>\u4e00\u3001\u7121\u7406\u6578\u5230\u5e95\u6709\u591a\u7121\u7406<\/h4>\n<div>\n<ol>\n \t<li>\u6709\u7406\u6578\u7684\u82f1\u6587 rational number \u770b\uff0crational \u4e00\u822c\u7684\u7406\u89e3\u90fd\u662f\u300c\u6709\u9053\u7406 \u7684\u300d\uff0c\u90a3\u9ebc\uff0c\u61c9\u8a72\u662f\u4ec0\u9ebc\u9053\u7406\uff1f\n\u5f9e ration \u7684\u62c9\u4e01\u6587\u8a9e\u6e90\u4f86\u770b\uff0c\u5b83\u6e90\u81ea\u65bc\u62c9\u4e01\u8a9e\u7684 logos\uff08\u82f1 \u6587\u662f ratio\uff0c\u610f\u601d\u70ba\u6bd4\uff09\uff0c\u62c9\u4e01\u6587\u7684\u539f\u610f\u662f\u300c\u53ef\u8868\u9054\u7684\u300d\uff0c\n\u5c0d\u7562\u6c0f\u5b78\u6d3e\u800c\u8a00\uff0c\u6709\u7406\u6578\u5c31\u662f\u5206 \u6578\uff0c\u610f\u601d\u662f\u53ef\u8868\u9054\u7684\u6578\u3002\u6709\u7406\u6578\u662f<b>\u300c\u53ef\u8868\u9054\u7684\u6578\u300d<\/b>\uff0c\u5373\u53ef\u5beb\u6210\u5169\u500b\u6574\u6578\u4e4b\u6bd4\u3002<\/li>\n \t<li>\u7121\u7406\u6578irrational number\uff0c\u8868\u793a\u300c\u4e0d\u53ef\u8868\u9054\u7684\u6578\u300d\uff0c \u4e5f\u5c31\u662f\u4e0d\u80fd\u5beb\u6210\u5169\u500b\u6574\u6578\u4e4b\u6bd4\u7684\u6578\u3002\n\u6240\u4ee5\u7121\u7406\u6578\u662f\u6307<b>\u300c\u4e0d\u53ef\u8868\u9054\u7684\u6578\u300d<\/b>\u3002<\/li>\n<\/ol>\n<\/div>\n<h4>\u4e8c\u3001\u300c\u53ef\u516c\u5ea6\u91cf\u7684\u300d(commensurable)<\/h4>\n<div>\n\n\u6709\u7406\u6578\u662f<b>\u300c\u53ef\u8868\u9054\u7684\u6578\u300d<\/b>\uff0c\u5373\u53ef\u5beb\u6210\u5169\u500b\u6574\u6578\u4e4b\u6bd4\u3002\u5373\u53ef\u4ee5\u5316\u6210\u6700\u7c21\u5206\u6578\u7684\u5f62\u5f0f $\\frac{q}{p}$ \uff0c\u5176\u4e2d $p,q$ \u4e92\u8cea\u3002\u4f8b\u5982\uff0c\n$\\frac{12}{18}=\\frac{6\\times 2}{6\\times 3}=\\frac{2}{3}$\u3002\u5982\u679c\u7528\u5169\u500b\u7dda\u6bb5\u9577\u4f86\u8aaa\uff0c\n\u8868\u793a $12$ \u548c $18$ \u53ef\u4ee5\u627e\u5230\u5171\u540c\u55ae\u4f4d\u9577 $6$ \u91cf\u76e1\u5b83\u5011\uff0c\u6211\u5011\u7a31\u4e4b\u70ba<b>\u300c\u53ef\u516c\u5ea6\u91cf\u7684\u300d<\/b>(commensurable)\u3002\u53cd\u4e4b\uff0c\u5982\u679c\u7121\u6cd5\u8868\u793a\u6210\u5206\u6578\u5f62\u5f0f\uff0c\u4ee3\u8868\u5169\u500b\u7dda\u6bb5\u9577\n\u7121\u6cd5\u627e\u5230\u5171\u540c\u55ae\u4f4d\u9577\u540c\u6642\u91cf\u76e1\uff0c\u5c31\u7a31\u70ba<b>\u300c\u4e0d\u53ef\u516c\u5ea6\u91cf\u7684\u300d<\/b>(incommensurable)\u3002\n\u4f8b\u5982\uff1a\u53ef\u516c\u5ea6\u91cf\u7684\u8aaa\u660e\uff1a\n\u7d66\u5b9a\u9577 $18$ cm\uff0c\u5bec $10$ cm\u7684\u9577\u65b9\u5f62\u7d19\u7247\u4e00\u5f35\uff0c\u4eca\u526a\u53bb\u4e00\u500b\u6700\u5927\u6b63\u65b9\u5f62\u5f8c\uff0c\u5c07\u5269\u4e0b\u7684\u9577\u65b9\u5f62\u7d19\u7247\u518d\u526a\u53bb\u4e00\u500b\u6700\u5927\u6b63\u65b9\u5f62\uff0c\u5982\u6b64\u7e7c\u7e8c\u505a\uff0c\n\u76f4\u5230\u5269\u4e0b\u7684\u7684\u7d19\u7247\u662f\u6b63\u65b9\u5f62\u624d\u505c\u6b62\uff0c\u5247\u6700\u5f8c\u5269\u4e0b\u7684\u6b63\u65b9\u5f62\u908a\u9577\u70ba $2$ cm\uff0c\u5982\u4e0b\u5716\u3002\u6240\u4ee5\u5c31\u53ef\u5c07 $18$\uff0c$10$ \u7528 $2$ \u7576\u57fa\u672c\u55ae\u4f4d\u4f86\u5ea6\u91cf\uff0c\n\u5373 $18 = 9 \\times 2$ \uff1b $10 = 5 \\times 2$\uff1a\n<div><img decoding=\"async\" style=\"max-width: 100%;\" src=\"https:\/\/cantor.math.ntnu.edu.tw\/workshop\/animated_math\/images\/\u7121\u7406\u6578\u7684\u767c\u73fe_1.png\" \/><\/div>\n<p style=\"margin-top: 10px;\">\u7528\u73fe\u5728\u7684\u8a71\u8a9e\u4f86\u8aaa\uff0c\u4e5f\u5c31\u662f18\u548c10\u9019\u5169\u500b\u6578\uff0c\u53ef\u4ee52\u7576\u57fa\u672c\u55ae\u4f4d\u9577\u4f86\u5ea6\u91cf\uff0c\u6240\u4ee5 $\\frac{18}{10}$ \u662f\u6709\u7406\u6578\u3002<\/p>\n\n<\/div>\n<h4>\u4e09\u3001\u70ba\u4f55 $\\sqrt{2}$ \u662f\u7121\u7406\u6578<\/h4>\n<div>\n\n\u6b50\u5e7e\u91cc\u5f97\u300a\u5e7e\u4f55\u539f\u672c\u300b\u7b2c\u5341\u5377\u547d\u984c2\uff1a\n\u5982\u679c\u5f9e\u5169\u4e0d\u7b49\u91cf\u7684\u5927\u91cf\u4e2d\u9023\u7e8c\u6e1b\u53bb\u5c0f\u91cf\uff0c\u76f4\u5230\u9918\u91cf\u5c0f\u65bc\u5c0f\u91cf\uff0c\u518d\u5f9e\u5c0f\u91cf\u4e2d\u9023\u7e8c\u6e1b\u53bb\u9918\u91cf\u76f4\u5230\u5c0f\u65bc\u9918\u91cf\uff0c\u5982\u6b64\u4e00\u76f4\u4f5c\u4e0b\u53bb\uff0c\n\u7576\u6240\u9918\u7684\u91cf\u6c38\u9060\u4e0d\u80fd\u91cf\u76e1\u5b83\u524d\u9762\u7684\u91cf\u6642\uff0c\u5247\u5169\u91cf\u4e0d\u53ef\u516c\u5ea6\u3002\n\n\u6839\u64da\u4e0a\u9762\u7684\u8aaa\u660e\uff0c\u6211\u5011\u8981\u4f86\u63a8\u5c0e $\\sqrt{2}$ \u8207 $1$ \u662f\u4e0d\u53ef\u516c\u5ea6\u91cf\u7684\uff0c\u85c9\u4ee5\u8aaa\u660e $\\sqrt{2}$ \u662f\u7121\u7406\u6578\u3002\u6211\u5011\u5c07\u5229\u7528\u4e0b\u9762\u7684\u5716\u5f62\u8aaa\u660e\uff01\n\n\u4e0b\u5716(\u4e00)\u4e2d $ABCD$ \u70ba\u4e00\u6b63\u65b9\u5f62\uff0c\u6b32\u5229\u7528\u6b64\u5716\u8aaa\u660e $\\overline{AC}$\u3001 $\\overline{AD}$ \u4e0d\u53ef\u516c\u5ea6\uff0c\u5373\u5c07 $\\overline{AC}$\u3001 $\\overline{AD}$ \u4e0d\u65b7\u7684\u4e92\u6e1b\uff0c\n\u4e26\u8aaa\u660e\u5169\u500b\u7dda\u6bb5\u9577\u7121\u6cd5\u627e\u5230\u57fa\u672c\u55ae\u4f4d\u9577\u540c\u6642\u91cf\u76e1\uff0c\u5c31\u53ef\u7a31 $\\overline{AC}$\u3001 $\\overline{AD}$ \u70ba<b>\u300c\u4e0d\u53ef\u516c\u5ea6\u91cf\u7684\u300d<\/b>\u3002\u8a66\u8457\u770b\u5b8c\u5f71\u7247\u5f8c\uff0c\u56de\u7b54\u4e0b\u5217\u554f\u984c\uff1a\n<div class=\"row\">\n<div class=\"col-md-6\">\u554f\u984c1\uff1a\u5982\u5716(\u4e00)\uff0c\u5728 $\\overline{AC}$ \u4e2d\u91cf\u6e2c\u4e00\u7dda\u6bb5\u9577 $\\overline{AE}=\\overline{AD}$ \uff0c\u904e $E$ \u505a $\\overline{AC}$ \u7684\u5782\u76f4\u7dda\u4ea4 $\\overline{CD}$ \u65bc $F$ \u9ede\uff0c\u8a66\u8b49\u660e\uff1a $\\overline{DF}=\\overline{EC}$\u3002<\/div>\n<div class=\"col-md-6\">\n<figure class=\"figure\"><img decoding=\"async\" class=\"figure-img img-fluid rounded shadow-3 mb-3\" src=\"https:\/\/cantor.math.ntnu.edu.tw\/workshop\/animated_math\/images\/\u7121\u7406\u6578\u7684\u767c\u73fe_2.png\" alt=\"\u5716(\u4e00)\" \/><figcaption class=\"figure-caption\">\u5716(\u4e00)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-51a453f0 e-flex e-con-boxed e-con e-parent\" data-id=\"51a453f0\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-64ef2dc1 elementor-widget elementor-widget-text-editor\" data-id=\"64ef2dc1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"color: #306f7f;\"><a href=\"https:\/\/cantor.math.ntnu.edu.tw\/workshop\/animated_math\/media\/\u7121\u7406\u6578\u7684\u767c\u73fe(\u5b78\u7fd2\u55ae).pdf\">\u7121\u7406\u6578\u7684\u767c\u73fe(\u5b78\u7fd2\u55ae)<\/a><\/p>\n<hr style=\"margin-bottom:20px;\" \/>\n<p style=\"color: #306f7f;\">\u88fd\u4f5c\u4eba\uff1a\u8607\u9e97\u654f<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>\u821e\u52d5\u6578\u5b78 \u7121\u7406\u6578\u7684\u767c\u73fe https:\/\/cantor.math.ntnu.edu.tw\/workshop\/an [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":193,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ocean_post_layout":"full-width","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"0","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"0","ocean_custom_header_template":"0","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"0","ocean_menu_typo_font_family":"0","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"0","footnotes":""},"class_list":["post-755","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/pages\/755","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/comments?post=755"}],"version-history":[{"count":12,"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/pages\/755\/revisions"}],"predecessor-version":[{"id":777,"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/pages\/755\/revisions\/777"}],"up":[{"embeddable":true,"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/pages\/193"}],"wp:attachment":[{"href":"https:\/\/www.math.ntnu.edu.tw\/merc\/wp-json\/wp\/v2\/media?parent=755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}