{"id":6980,"date":"2021-07-23T23:20:33","date_gmt":"2021-07-23T14:20:33","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6980"},"modified":"2026-09-27T16:53:53","modified_gmt":"2026-09-27T07:53:53","slug":"real-numbers","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/real-numbers\/","title":{"rendered":"\uc2e4\uc218\uacc4"},"content":{"rendered":"<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\uc2e4\uc218\uacc4\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\ub2e4. \uc5ec\uae30\uc11c\ub294 \uc2e4\uc218\uacc4\uac00 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub300\uc218\uc801 \uc131\uc9c8, \uc21c\uc11c\uc5d0 \uad00\ud55c \uc131\uc9c8, \uadf8\ub9ac\uace0 \uc644\ube44\uc131\uc5d0 \uad00\ud55c \uc131\uc9c8\uc744 \uacf5\ub9ac\ub85c \ubc1b\uc544\ub4e4\uc784\uc73c\ub85c\uc368 \uc2e4\uc218\uacc4\ub97c \ub2e4\ub8e8\uae30\ub85c \ud55c\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc2e4\uc218\uacc4\uc758 \uacf5\ub9ac<\/h2>\n<p><span class=\"defined\">\uc2e4\uc218\uacc4<\/span>(real number system)\ub780 \uc9d1\ud569 \\(\\mathbb{R}\\)\uc5d0 <span class=\"defined\">\ub367\uc148<\/span>\uc774\ub77c\uace0 \ubd88\ub9ac\ub294 \uc774\ud56d\uc5f0\uc0b0 \u2018\\(+\\)\u2019\uc640 <span class=\"defined\">\uacf1\uc148<\/span>\uc774\ub77c\uace0 \ubd88\ub9ac\ub294 \uc774\ud56d\uc5f0\uc0b0 \u2018\\(\\cdot\\)\u2019, \uadf8\ub9ac\uace0 \uc21c\uc11c\uad00\uacc4 \u2018\\(\\le\\)\u2019\uac00 \uc8fc\uc5b4\uc838 \uc788\uace0 \ub2e4\uc74c\uc758 \ub300\uc218 \uacf5\ub9ac\uc640 \uc21c\uc11c \uacf5\ub9ac, \uadf8\ub9ac\uace0 \ub4a4\uc5d0\uc11c \uc124\uba85\ud560 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218 \uccb4\uacc4\uc774\ub2e4.<\/p>\n<p><strong>\ub300\uc218 \uacf5\ub9ac.<\/strong> \uc784\uc758\uc758 \\(a,b,c\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ub367\uc148\uacfc \uacf1\uc148\uc740 \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<br \/>\n\\[<br \/>\na+b=b+a,<br \/>\n\\qquad<br \/>\na\\cdot b=b\\cdot a.<br \/>\n\\]\n<\/li>\n<li>\ub367\uc148\uacfc \uacf1\uc148\uc740 \uacb0\ud569\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<br \/>\n\\[<br \/>\n(a+b)+c=a+(b+c),<br \/>\n\\qquad<br \/>\n(a\\cdot b)\\cdot c=a\\cdot(b\\cdot c).<br \/>\n\\]\n<\/li>\n<li>\\(\\mathbb{R}\\)\uc5d0 \uc6d0\uc18c \\(0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na+0=a<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774 \uc6d0\uc18c\ub97c <span class=\"defined\">\ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uac01 \\(a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na+b=0<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(b\\in\\mathbb{R}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774 \uc6d0\uc18c\ub97c \\(a\\)\uc758 <span class=\"defined\">\ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\\(\\mathbb{R}\\)\uc5d0 \\(0\\)\uacfc \ub2e4\ub978 \uc6d0\uc18c \\(1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na\\cdot1=a<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774 \uc6d0\uc18c\ub97c <span class=\"defined\">\uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uac01 \\(0\\ne a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na\\cdot b=1<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(b\\in\\mathbb{R}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774 \uc6d0\uc18c\ub97c \\(a\\)\uc758 <span class=\"defined\">\uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uacf1\uc148\uc740 \ub367\uc148\uc5d0 \ub300\ud558\uc5ec \ubd84\ubc30\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<br \/>\n\\[<br \/>\na\\cdot(b+c)=a\\cdot b+a\\cdot c.<br \/>\n\\]\n<\/li>\n<\/ul>\n<p>\uc704\uc758 \uacf5\ub9ac\ub85c\ubd80\ud130 \\(0\\)\uacfc \\(1\\), \uadf8\ub9ac\uace0 \uac01 \uc6d0\uc18c\uc758 \uc5ed\uc6d0\uc740 \uc720\uc77c\ud568\uc744 \uc54c \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(0\\)\uacfc \\(0&#8217;\\)\uc774 \ubaa8\ub450 \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uc774\uba74<br \/>\n\\[<br \/>\n0=0+0&#8217;=0&#8242;<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub610\ud55c \\(b\\)\uc640 \\(c\\)\uac00 \ubaa8\ub450 \\(a\\)\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc774\uba74<br \/>\n\\[<br \/>\nb=b+0=b+(a+c)=(b+a)+c=0+c=c.<br \/>\n\\]<br \/>\n\uacf1\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uacfc \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc758 \uc720\uc77c\uc131\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ud655\uc778\ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(a\\)\uc758 \ub367\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc744 \\(-a\\), \\(0\\ne a\\)\uc758 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc744 \\(a^{-1}\\) \ub610\ub294 \\(1\/a\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p><strong>\uc21c\uc11c \uacf5\ub9ac.<\/strong> \uc784\uc758\uc758 \\(a,b,c\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(a\\le a\\)\uc774\ub2e4.<\/li>\n<li>\\(a\\le b\\)\uc774\uace0 \\(b\\le a\\)\uc774\uba74 \\(a=b\\)\uc774\ub2e4.<\/li>\n<li>\\(a\\le b\\)\uc774\uace0 \\(b\\le c\\)\uc774\uba74 \\(a\\le c\\)\uc774\ub2e4.<\/li>\n<li>\\(a\\le b\\) \ub610\ub294 \\(b\\le a\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/li>\n<li>\\(a\\le b\\)\uc774\uba74 \ubaa8\ub4e0 \\(c\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na+c\\le b+c<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li>\\(0\\le a\\)\uc774\uace0 \\(0\\le b\\)\uc774\uba74<br \/>\n\\[<br \/>\n0\\le a\\cdot b<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\\(a\\le b\\)\uc774\uace0 \\(a\\ne b\\)\uc77c \ub54c \\(a< b\\)\ub77c\uace0 \uc4f4\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(a>b\\)\ub294 \\(b< a\\)\ub77c\ub294 \ub73b\uc774\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ub300\uc218 \uacf5\ub9ac\uc640 \uc21c\uc11c \uacf5\ub9ac\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218 \uccb4\uacc4\ub97c <span class=\"defined\">\uc21c\uc11c\uccb4<\/span>(ordered field)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2e4\uc218\uacc4\ub294 \uc21c\uc11c\uccb4\uc77c \ubfd0 \uc544\ub2c8\ub77c \ub2e4\uc74c \uc808\uc5d0\uc11c \uc124\uba85\ud558\ub294 \uc644\ube44\uc131\ub3c4 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8<\/h2>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nx\\le b<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(b\\)\uac00 \uc874\uc7ac\ud558\uba74 \\(E\\)\ub97c <span class=\"defined\">\uc704\ub85c \uc720\uacc4<\/span>(bounded above)\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\ub7ec\ud55c \\(b\\)\ub97c \\(E\\)\uc758 <span class=\"defined\">\uc0c1\uacc4<\/span>(upper bound)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(E\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \\(b\\)\uac00 \\(E\\)\uc758 \uc0c1\uacc4\uc774\uace0, \\(E\\)\uc758 \uc784\uc758\uc758 \uc0c1\uacc4 \\(c\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nb\\le c<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74 \\(b\\)\ub97c \\(E\\)\uc758 <span class=\"defined\">\ucd5c\uc18c\uc0c1\uacc4<\/span>(least upper bound) \ub610\ub294 <span class=\"defined\">\uc0c1\ud55c<\/span>(supremum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ucd5c\uc18c\uc0c1\uacc4\uac00 \uc874\uc7ac\ud558\uba74 \uadf8\uac83\uc740 \uc720\uc77c\ud558\uba70<br \/>\n\\[<br \/>\nb=\\sup E<br \/>\n\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8.<\/span><\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uc704\ub85c \uc720\uacc4\uc774\uba74, \\(E\\)\uc758 \ucd5c\uc18c\uc0c1\uacc4\uac00 \uc874\uc7ac\ud558\uba70<br \/>\n\\[<br \/>\n\\sup E\\in\\mathbb{R}<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc131\uc9c8\uc744 <span class=\"defined\">\uc644\ube44\uc131 \uacf5\ub9ac<\/span>(completeness axiom) \ub610\ub294 <span class=\"defined\">\uc0c1\ud55c \uacf5\ub9ac<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \ub300\uc218 \uacf5\ub9ac\uc640 \uc21c\uc11c \uacf5\ub9ac\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\ub9cc\uc73c\ub85c\ub294 \uc2e4\uc218\uacc4\uc758 \uc911\uc694\ud55c \uc131\uc9c8\uc744 \ubaa8\ub450 \uc5bb\uc744 \uc218 \uc5c6\uc73c\uba70, \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc774 \uc2e4\uc218\uc640 \uc720\ub9ac\uc218\ub97c \uad6c\ubcc4\ud558\ub294 \ud575\uc2ec\uc801\uc778 \uc870\uac74\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30.<\/span> \uc720\ub9ac\uc218 \uc9d1\ud569 \\(\\mathbb{Q}\\)\ub294 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc744 \ub9cc\uc871\uc2dc\ud0a4\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\ub2e4\uc74c \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[<br \/>\nE=\\left\\{q\\in\\mathbb{Q}\\,\\vert\\,q>0\\text{ and }q^2<2\\right\\}.\n\\]\n\\(1\\in E\\)\uc774\ubbc0\ub85c \\(E\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0, \\(2\\)\ub294 \\(E\\)\uc758 \uc0c1\uacc4\uc774\ubbc0\ub85c \\(E\\)\ub294 \\(\\mathbb{Q}\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\\(E\\)\uac00 \\(\\mathbb{Q}\\)\uc5d0\uc11c \ucd5c\uc18c\uc0c1\uacc4 \\(b\\)\ub97c \uac00\uc9c4\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(1\\in E\\)\uc774\uace0 \\(2\\)\uac00 \uc0c1\uacc4\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n1\\le b\\le2.<br \/>\n\\]<br \/>\n\uc774\uc81c \\(b^2\\)\uc758 \uac12\uc5d0 \ub530\ub77c \ub098\ub204\uc5b4 \uc0dd\uac01\ud558\uc790.<\/p>\n<p>\uba3c\uc800 \\(b^2<2\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \ub193\ub294\ub2e4.\n\\[\n\\delta=\\frac{2-b^2}{2b+1}.\n\\]\n\uadf8\ub7ec\uba74 \\(\\delta\\)\ub294 \uc591\uc758 \uc720\ub9ac\uc218\uc774\uace0 \\(0 < \\delta < 1\\)\uc774\ub2e4. \ub530\ub77c\uc11c\n\\[\n\\begin{aligned}\n(b+\\delta)^2\n&#038;=b^2+2b\\delta+\\delta^2\\\\\n&#038;< b^2+(2b+1)\\delta\\\\\n&#038;=2.\n\\end{aligned}\n\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(b+\\delta\\in E\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(b+\\delta>b\\)\uc774\ubbc0\ub85c \\(b\\)\uac00 \\(E\\)\uc758 \uc0c1\uacc4\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\uc774\ubc88\uc5d0\ub294 \\(b^2>2\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \ub193\ub294\ub2e4.<br \/>\n\\[<br \/>\n\\delta=\\frac{b^2-2}{2b+1}.<br \/>\n\\]<br \/>\n\uadf8\ub7ec\uba74 \\(0< \\delta< b\\)\uc774\uace0\n\\[\n\\begin{aligned}\n(b-\\delta)^2\n&#038;=b^2-2b\\delta+\\delta^2\\\\\n&#038;>b^2-2b\\delta\\\\<br \/>\n&#038;>2.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(q\\in E\\)\uc774\uba74 \\(q>b-\\delta\\)\uc77c \uc218 \uc5c6\ub2e4. \uc2e4\uc81c\ub85c \\(q>b-\\delta>0\\)\uc774\uba74<br \/>\n\\[<br \/>\nq^2>(b-\\delta)^2>2<br \/>\n\\]<br \/>\n\uac00 \ub418\uc5b4 \\(q\\in E\\)\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(b-\\delta\\)\ub3c4 \\(E\\)\uc758 \uc0c1\uacc4\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(b-\\delta< b\\)\uc774\ubbc0\ub85c \\(b\\)\uac00 \ucd5c\uc18c\uc0c1\uacc4\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(b^2=2\\)\uc778 \uc720\ub9ac\uc218 \\(b\\)\ub3c4 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc2e4\uc81c\ub85c \\(b=m\/n\\)\uc744 \uae30\uc57d\ubd84\uc218\ub85c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[<br \/>\nm^2=2n^2<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(m\\)\uc740 \uc9dd\uc218\uc774\uace0 \\(m=2k\\)\ub85c \uc4f8 \uc218 \uc788\ub2e4. \uc774\ub97c \ub300\uc785\ud558\uba74<br \/>\n\\[<br \/>\nn^2=2k^2<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(n\\)\ub3c4 \uc9dd\uc218\uac00 \ub418\uc5b4 \\(m\/n\\)\uc774 \uae30\uc57d\ubd84\uc218\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\ub530\ub77c\uc11c \uc138 \uacbd\uc6b0\uac00 \ubaa8\ub450 \ubd88\uac00\ub2a5\ud558\ubbc0\ub85c \\(E\\)\ub294 \\(\\mathbb{Q}\\)\uc5d0\uc11c \ucd5c\uc18c\uc0c1\uacc4\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n<p>\uc2e4\uc218\uacc4\uc758 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc740 \ub4a4\uc5d0\uc11c \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uc804\uac1c\ud560 \ub54c \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uc720\uacc4\uc778 \ub2e8\uc870\uc2e4\uc218\uc5f4\uc774 \ubc18\ub4dc\uc2dc \uc218\ub834\ud55c\ub2e4\ub294 \ub2e8\uc870\uc218\ub834\uc815\ub9ac\ub294 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4\ub2e4. \uc774\ub7ec\ud55c \uc758\ubbf8\uc5d0\uc11c \uc2e4\uc218\uacc4\ub294 <span class=\"defined\">\uc644\ube44<\/span>(complete)\ud558\ub2e4\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uad6c\uac04<\/h2>\n<p>\\(I\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(x,y\\in I\\)\uc774\uace0<br \/>\n\\[<br \/>\nx\\le z\\le y<br \/>\n\\]<br \/>\n\uc778 \uc2e4\uc218 \\(z\\)\uac00 \ud56d\uc0c1 \\(I\\)\uc5d0 \uc18d\ud558\uba74 \\(I\\)\ub97c <span class=\"defined\">\uad6c\uac04<\/span>(interval)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(a,b\\in\\mathbb{R}\\)\uc774\uace0 \\(a\\le b\\)\ub77c\uace0 \ud558\uc790. \ub450 \uc720\ud55c\ud55c \ub05d\uc810 \\(a,b\\)\ub97c \uac16\ub294 \uad6c\uac04\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n[a,\\,b]<br \/>\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,a\\le x\\le b\\right\\},\\\\[6pt]<br \/>\n(a,\\,b)<br \/>\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,a< x< b\\right\\},\\\\[6pt]\n[a,\\,b)\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,a\\le x< b\\right\\},\\\\[6pt]\n(a,\\,b]\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,a< x\\le b\\right\\}.\n\\end{align}\n\\]\n\uccab \ubc88\uc9f8 \ud615\ud0dc\uc758 \uad6c\uac04\uc744 <span class=\"defined\">\ub2eb\ud78c\uad6c\uac04<\/span>(closed interval), \ub450 \ubc88\uc9f8 \ud615\ud0dc\uc758 \uad6c\uac04\uc744 <span class=\"defined\">\uc5f4\ub9b0\uad6c\uac04<\/span>(open interval)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub098\uba38\uc9c0 \ub450 \ud615\ud0dc\ub97c <span class=\"defined\">\ubc18\uc5f4\ub9b0\uad6c\uac04<\/span>(half-open interval) \ub610\ub294 <span class=\"defined\">\ubc18\ub2eb\ud78c\uad6c\uac04<\/span>(half-closed interval)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(a=b\\)\uc774\uba74<br \/>\n\\[<br \/>\n[a,a]=\\{a\\},<br \/>\n\\qquad<br \/>\n(a,a)=[a,a)=(a,a]=\\varnothing.<br \/>\n\\]\n<\/p>\n<p>\ud55c\ucabd \ub610\ub294 \uc591\ucabd\uc73c\ub85c \ub05d\uc774 \uc5c6\ub294 \uad6c\uac04\ub3c4 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n(-\\infty,\\,b)<br \/>\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,x< b\\right\\},\\\\[6pt]\n(-\\infty,\\,b]\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,x\\le b\\right\\},\\\\[6pt]\n(a,\\,\\infty)\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,a< x\\right\\},\\\\[6pt]\n[a,\\,\\infty)\n&#038;=\\left\\{x\\in\\mathbb{R}\\,\\vert\\,a\\le x\\right\\},\\\\[6pt]\n(-\\infty,\\,\\infty)\n&#038;=\\mathbb{R}.\n\\end{align}\n\\]\n\uc5ec\uae30\uc11c \\(\\infty\\)\uc640 \\(-\\infty\\)\ub294 \uc2e4\uc218\uac00 \uc544\ub2c8\ub77c \uad6c\uac04\uc744 \ub098\ud0c0\ub0b4\uae30 \uc704\ud558\uc5ec \uc0ac\uc6a9\ud558\ub294 \uae30\ud638\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\infty\\)\uac00 \ubd99\uc740 \ucabd\uc5d0\ub294 \ub300\uad04\ud638\ub97c \uc0ac\uc6a9\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\\(a\\le b\\)\uc77c \ub54c \uc704\uc758 \uc720\uacc4\uad6c\uac04<br \/>\n\\[<br \/>\n[a,b],\\qquad(a,b),\\qquad[a,b),\\qquad(a,b]<br \/>\n\\]<br \/>\n\uc758 <span class=\"defined\">\uae38\uc774<\/span>(length)\ub97c \\(b-a\\)\ub85c \uc815\uc758\ud55c\ub2e4. \\(a=b\\)\uc774\uba74 \uae38\uc774\ub294 \\(0\\)\uc774\uace0, \\(a< b\\)\uc774\uba74 \uae38\uc774\ub294 \uc591\uc218\uc774\ub2e4. \\(a< b\\)\uc778 \uad6c\uac04\uc744 <span class=\"defined\">\ud1f4\ud654\ub418\uc9c0 \uc54a\uc740 \uad6c\uac04<\/span>(nondegenerate interval)\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/complex-numbers\">\ud589\ub82c\uacfc \ubcf5\uc18c\uc218<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc2e4\uc218\uacc4\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\ub2e4. \uc5ec\uae30\uc11c\ub294 \uc2e4\uc218\uacc4\uac00 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub300\uc218\uc801 \uc131\uc9c8, \uc21c\uc11c\uc5d0 \uad00\ud55c \uc131\uc9c8, \uadf8\ub9ac\uace0 \uc644\ube44\uc131\uc5d0 \uad00\ud55c \uc131\uc9c8\uc744 \uacf5\ub9ac\ub85c \ubc1b\uc544\ub4e4\uc784\uc73c\ub85c\uc368 \uc2e4\uc218\uacc4\ub97c \ub2e4\ub8e8\uae30\ub85c \ud55c\ub2e4. \uc2e4\uc218\uacc4\uc758 \uacf5\ub9ac \uc2e4\uc218\uacc4(real number system)\ub780 \uc9d1\ud569 \\(\\mathbb{R}\\)\uc5d0 \ub367\uc148\uc774\ub77c\uace0 \ubd88\ub9ac\ub294 \uc774\ud56d\uc5f0\uc0b0 \u2018\\(+\\)\u2019\uc640 \uacf1\uc148\uc774\ub77c\uace0 \ubd88\ub9ac\ub294 \uc774\ud56d\uc5f0\uc0b0 \u2018\\(\\cdot\\)\u2019, \uadf8\ub9ac\uace0 \uc21c\uc11c\uad00\uacc4 \u2018\\(\\le\\)\u2019\uac00 \uc8fc\uc5b4\uc838 \uc788\uace0 \ub2e4\uc74c\uc758 \ub300\uc218 \uacf5\ub9ac\uc640 \uc21c\uc11c \uacf5\ub9ac, \uadf8\ub9ac\uace0 \ub4a4\uc5d0\uc11c \uc124\uba85\ud560 \ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8\uc744&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":14,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-6980","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6980","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6980"}],"version-history":[{"count":12,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6980\/revisions"}],"predecessor-version":[{"id":10126,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6980\/revisions\/10126"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6980"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}