{"id":9456,"date":"2017-05-02T22:40:25","date_gmt":"2017-05-02T22:40:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9456"},"modified":"2020-09-11T00:39:32","modified_gmt":"2020-09-11T00:39:32","slug":"summary-properties-of-identity-inverses-and-zero-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/summary-properties-of-identity-inverses-and-zero-2\/","title":{"raw":"5.4.d - Summary: Properties of Identity, Inverses, and Zero","rendered":"5.4.d &#8211; Summary: Properties of Identity, Inverses, and Zero"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul id=\"eip-260\">\r\n \t<li><strong>Identity Properties<\/strong>\r\n<ul id=\"eip-id1170326451798\">\r\n \t<li><strong>Identity Property of Addition:<\/strong>\r\nFor any real number [latex]a[\/latex]: [latex]a+0=a(0)+a=a[\/latex].\u00a0 [latex]0[\/latex] is the <strong>additive identity<\/strong><\/li>\r\n \t<li><strong>Identity Property of Multiplication:<\/strong>\r\nFor any real number [latex]a[\/latex]:\r\n<ul>\r\n \t<li>[latex]a\\cdot 1=a[\/latex]<\/li>\r\n \t<li>[latex]1\\cdot a=a[\/latex]<\/li>\r\n \t<li>[latex]1[\/latex] is the <strong>multiplicative identity<\/strong><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Inverse Properties<\/strong>\r\n<ul id=\"eip-id1170325236694\">\r\n \t<li><strong>Inverse Property of Addition:<\/strong> For any real number [latex]a[\/latex]: [latex]a+\\left(-a\\right)=0-a[\/latex] is the <strong>additive inverse<\/strong> of [latex]a[\/latex]<\/li>\r\n \t<li><strong>Inverse Property of Multiplication:<\/strong> For any real number [latex]a[\/latex]: [latex]\\left(a\\ne 0\\right)a\\cdot {\\Large\\frac{1}{a}}=1[\/latex].\u00a0 \u00a0The <strong>multiplicative inverse<\/strong> of [latex]a[\/latex] is\u00a0[latex]{\\Large\\frac{1}{a}}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Properties of Zero<\/strong>\r\n<ul id=\"eip-id1170323884528\">\r\n \t<li><strong>Multiplication by Zero: <\/strong> For any real number [latex]a[\/latex],\r\n[latex]\\begin{array}{ccccccc}\\hfill a\\cdot 0=0\\hfill &amp; &amp; &amp; \\hfill 0\\cdot a=0\\hfill &amp; &amp; &amp; \\hfill \\text{The product of any number and 0 is 0.}\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li><strong>Division of Zero: <\/strong> For any real number [latex]a[\/latex],\r\n[latex]\\begin{array}{ccccccc}\\hfill {\\Large\\frac{0}{a}}=0\\hfill &amp; &amp; &amp; \\hfill 0+a=0\\hfill &amp; &amp; &amp; \\hfill \\text{Zero divided by any real number, except itself, is zero.}\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li><strong>Division by Zero: <\/strong>For any real number [latex]a[\/latex]<em>,<\/em> [latex]{\\Large\\frac{0}{a}}[\/latex] is undefined and [latex]a\\div 0[\/latex] is undefined. Division by zero is undefined.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1166497483045\" class=\"definition\">\r\n \t<dt><strong>Additive Identity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1166497444483050\">The additive identity is [latex]0[\/latex]. When zero is added to any number, it does not change the value.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166666497483045\" class=\"definition\">\r\n \t<dt><strong>Additive Inverse<\/strong><\/dt>\r\n \t<dd id=\"fs-id1166497483050\">The opposite of a number is its additive inverse. The additive inverse of a is [latex]-a[\/latex] .<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166777497483058\" class=\"definition\">\r\n \t<dt><strong>Multiplicative Identity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1166497483064\">The multiplicative identity is [latex]1[\/latex]. When one multiplies any number, it does not change the value.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166488897483097\" class=\"definition\">\r\n \t<dt><strong>Multiplicative Inverse<\/strong><\/dt>\r\n \t<dd id=\"fs-id1166497483103\">The reciprocal of a number is its multiplicative inverse. The multiplicative inverse of [latex]a[\/latex] is [latex]{\\Large\\frac{1}{a}}[\/latex] .<\/dd>\r\n<\/dl>","rendered":"<h2>Key Concepts<\/h2>\n<ul id=\"eip-260\">\n<li><strong>Identity Properties<\/strong>\n<ul id=\"eip-id1170326451798\">\n<li><strong>Identity Property of Addition:<\/strong><br \/>\nFor any real number [latex]a[\/latex]: [latex]a+0=a(0)+a=a[\/latex].\u00a0 [latex]0[\/latex] is the <strong>additive identity<\/strong><\/li>\n<li><strong>Identity Property of Multiplication:<\/strong><br \/>\nFor any real number [latex]a[\/latex]:<\/p>\n<ul>\n<li>[latex]a\\cdot 1=a[\/latex]<\/li>\n<li>[latex]1\\cdot a=a[\/latex]<\/li>\n<li>[latex]1[\/latex] is the <strong>multiplicative identity<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>Inverse Properties<\/strong>\n<ul id=\"eip-id1170325236694\">\n<li><strong>Inverse Property of Addition:<\/strong> For any real number [latex]a[\/latex]: [latex]a+\\left(-a\\right)=0-a[\/latex] is the <strong>additive inverse<\/strong> of [latex]a[\/latex]<\/li>\n<li><strong>Inverse Property of Multiplication:<\/strong> For any real number [latex]a[\/latex]: [latex]\\left(a\\ne 0\\right)a\\cdot {\\Large\\frac{1}{a}}=1[\/latex].\u00a0 \u00a0The <strong>multiplicative inverse<\/strong> of [latex]a[\/latex] is\u00a0[latex]{\\Large\\frac{1}{a}}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Properties of Zero<\/strong>\n<ul id=\"eip-id1170323884528\">\n<li><strong>Multiplication by Zero: <\/strong> For any real number [latex]a[\/latex],<br \/>\n[latex]\\begin{array}{ccccccc}\\hfill a\\cdot 0=0\\hfill & & & \\hfill 0\\cdot a=0\\hfill & & & \\hfill \\text{The product of any number and 0 is 0.}\\hfill \\end{array}[\/latex]<\/li>\n<li><strong>Division of Zero: <\/strong> For any real number [latex]a[\/latex],<br \/>\n[latex]\\begin{array}{ccccccc}\\hfill {\\Large\\frac{0}{a}}=0\\hfill & & & \\hfill 0+a=0\\hfill & & & \\hfill \\text{Zero divided by any real number, except itself, is zero.}\\hfill \\end{array}[\/latex]<\/li>\n<li><strong>Division by Zero: <\/strong>For any real number [latex]a[\/latex]<em>,<\/em> [latex]{\\Large\\frac{0}{a}}[\/latex] is undefined and [latex]a\\div 0[\/latex] is undefined. Division by zero is undefined.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1166497483045\" class=\"definition\">\n<dt><strong>Additive Identity<\/strong><\/dt>\n<dd id=\"fs-id1166497444483050\">The additive identity is [latex]0[\/latex]. When zero is added to any number, it does not change the value.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166666497483045\" class=\"definition\">\n<dt><strong>Additive Inverse<\/strong><\/dt>\n<dd id=\"fs-id1166497483050\">The opposite of a number is its additive inverse. The additive inverse of a is [latex]-a[\/latex] .<\/dd>\n<\/dl>\n<dl id=\"fs-id1166777497483058\" class=\"definition\">\n<dt><strong>Multiplicative Identity<\/strong><\/dt>\n<dd id=\"fs-id1166497483064\">The multiplicative identity is [latex]1[\/latex]. When one multiplies any number, it does not change the value.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166488897483097\" class=\"definition\">\n<dt><strong>Multiplicative Inverse<\/strong><\/dt>\n<dd id=\"fs-id1166497483103\">The reciprocal of a number is its multiplicative inverse. The multiplicative inverse of [latex]a[\/latex] is [latex]{\\Large\\frac{1}{a}}[\/latex] .<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9456\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"0c45ee8136a34daaa48082e45b140edb","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9456","chapter","type-chapter","status-publish","hentry"],"part":7349,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9456\/revisions"}],"predecessor-version":[{"id":20271,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9456\/revisions\/20271"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/7349"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9456\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=9456"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=9456"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=9456"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=9456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}