{"id":13,"date":"2017-08-23T06:43:12","date_gmt":"2017-08-23T06:43:12","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=13"},"modified":"2018-01-16T18:36:15","modified_gmt":"2018-01-16T18:36:15","slug":"field-properties-of-real-numbers","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/field-properties-of-real-numbers\/","title":{"rendered":"Field Properties of Real Numbers"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/operations-with-real-numbers\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/number-sense\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Next Workshop \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Field Properties of Real Numbers<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study how to use the field properties of the real number systems.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><strong><em>Real numbers<\/em><\/strong> are the field of\u00a0all rational and irrational numbers.<\/li>\n<\/ul>\n<section>\n<h3>What are the field properties for addition of real numbers?<\/h3>\n<p>Mathematicians organize numbers into various sets, or\u00a0structures, in which all the numbers satisfy a particular group of\u00a0rules. There are several field properties for the addition of real\u00a0numbers. Mathematicians call any set of numbers that satisfies the\u00a0following properties a <strong><em>field<\/em><\/strong>: closure,\u00a0commutativity, associativity, distributivity, identity elements,\u00a0and inverses.<\/p>\n<ul>\n<li>The <strong><em><abbr title=\"if a and b are real numbers, then a+b is a real number\">closure\u00a0property for addition<\/abbr> <\/em><\/strong>states that if <em>a<\/em> and\u00a0<em>b<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image002.gif\" width=\"30\" height=\"11\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is a real number. For example, in the equation\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image004.gif\" width=\"80\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a03,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image006.gif\" width=\"16\" height=\"11\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image008.gif\" width=\"16\" height=\"11\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<li>The <abbr title=\"states that a+0=a\"><em>identity\u00a0property for addition<\/em><\/abbr> states that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image010.gif\" width=\"54\" height=\"11\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0For example in the equations\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image012.gif\" width=\"50\" height=\"11\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image014.gif\" width=\"50\" height=\"11\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a08 and <em>e<\/em> are real numbers.<\/li>\n<li>The <abbr title=\"a+(-a)=0\"><em>inverse\u00a0property of addition<\/em><\/abbr> states that for each real\u00a0number <em>a<\/em>, there is a real number\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image016.gif\" width=\"17\" height=\"7\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0such that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image018.gif\" width=\"75\" height=\"14\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0For example, in the equations\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image020.gif\" width=\"71\" height=\"14\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image022.gif\" width=\"78\" height=\"14\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a08,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image024.gif\" width=\"16\" height=\"11\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image026.gif\" width=\"11\" height=\"8\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<li>The <abbr title=\"rule that states that the order in which you add a set of numbers does not affect the sum. If a and b are real numbers, then a+b=b+a.\"><em>commutative\u00a0property of addition<\/em><\/abbr> states that if <em>a<\/em> and\u00a0<em>b<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image028.gif\" width=\"76\" height=\"11\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0For example, in the equations\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image030.gif\" width=\"111\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image032.gif\" width=\"80\" height=\"29\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a08,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image034.gif\" width=\"16\" height=\"11\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a02, and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image036.gif\" width=\"9\" height=\"29\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<li>The <abbr title=\"a rule that states that the way you group a set of numbers does not affect the sum. If a, b, and c are real numbers, then a+(b+c)=(a+b)+c.\"><em>associative\u00a0property of addition<\/em><\/abbr> states that if <em>a<\/em>, <em>b<\/em>,\u00a0and <em>c<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image038.gif\" width=\"141\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Consider the equations\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image040.gif\" width=\"137\" height=\"14\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image042.gif\" width=\"155\" height=\"32\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0In these equations, 8, 3, 5,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image044.gif\" width=\"8\" height=\"29\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image046.gif\" width=\"9\" height=\"29\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image048.gif\" width=\"9\" height=\"29\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<\/ul>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which property is illustrated by the following equation?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image050.gif\" width=\"138\" height=\"14\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<ol>\n<li>Associative property of addition<\/li>\n<li>Commutative property of addition<\/li>\n<li>Inverse property of addition<\/li>\n<li>Identity property of addition<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is B. The left side of the equation shows\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image052.gif\" width=\"27\" height=\"11\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and the right side of the equation shows<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image054.gif\" width=\"27\" height=\"11\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This arrangement shows the commutative property of addition.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which property is illustrated by the following equation?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p2_clip_image056.gif\" width=\"72\" height=\"14\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<ol>\n<li>Associative property of addition<\/li>\n<li>Commutative property of addition<\/li>\n<li>Inverse property of addition<\/li>\n<li>Identity property of addition<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C. The statement shows two real numbers\u00a0that, when added together, result in 0. This is the inverse property of addition.<\/p>\n<\/div>\n<\/section>\n<h3>What are the field properties for multiplication of real numbers?<\/h3>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>There are similar field\u00a0properties for the multiplication of real numbers. The <strong><em> <abbr title=\"if a and b are real numbers, then ab is a real number\">closure\u00a0property for multiplication<\/abbr> <\/em><\/strong>states that if <em>a<\/em> and <em>b<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image002.gif\" width=\"15\" height=\"11\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is a real number. For example, in the equation\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image004.gif\" width=\"74\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a03,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image006.gif\" width=\"16\" height=\"11\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image008.gif\" width=\"24\" height=\"11\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<li>The <strong><em><abbr title=\"states that a(1)=a\"><em>identity\u00a0property for multiplication<\/em><\/abbr> <\/em><\/strong>states that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image010.gif\" width=\"48\" height=\"14\" name=\"graphics7\" align=\"ABSBOTTOM\" border=\"0\" \/>.\u00a0For example, in\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image012.gif\" width=\"45\" height=\"14\" name=\"graphics8\" align=\"ABSBOTTOM\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image014.gif\" width=\"45\" height=\"14\" name=\"graphics9\" align=\"ABSBOTTOM\" border=\"0\" \/>,\u00a08 and <em>e<\/em> are real numbers.<\/li>\n<li>The <strong><em><abbr title=\"For each real number a (except 0), there is a real number 1\/a such that a(1\/a)=1\"><em>inverse\u00a0property of multiplication<\/em><\/abbr> <\/em><\/strong>states that for each\u00a0real number <em>a <\/em>(except 0) , there is a real number\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image016.gif\" width=\"10\" height=\"29\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0such that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image018.gif\" width=\"54\" height=\"32\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0For example, in the equations\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image020.gif\" width=\"50\" height=\"32\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image022.gif\" width=\"57\" height=\"32\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a08,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image024.gif\" width=\"16\" height=\"11\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image026.gif\" width=\"11\" height=\"8\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<li>The <strong><abbr title=\" rule that states that the way you group a set of numbers does not affect the product. If a and b are real numbers, then ab=ba\"><em>commutative\u00a0property of multiplication<\/em><\/abbr><\/strong> states that if <em>a\u00a0<\/em>and <em>b<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image028.gif\" width=\"47\" height=\"11\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0For example, in the equation\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image030.gif\" width=\"83\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image032.gif\" width=\"78\" height=\"32\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a08,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image034.gif\" width=\"16\" height=\"11\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a02, and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image036.gif\" width=\"9\" height=\"29\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<li>The <strong><abbr title=\"a rule that states that the way you group a set of numbers does not affect the product. If a, b, and c are real numbers, then a(bc)=(ab)c\"><em>associative\u00a0property of multiplication<\/em><\/abbr><\/strong> states that if <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image038.gif\" width=\"85\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Consider the equations,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image040.gif\" width=\"95\" height=\"14\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image042.gif\" width=\"118\" height=\"32\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0In these equations, 8, 3, 5,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image044.gif\" width=\"8\" height=\"29\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image046.gif\" width=\"9\" height=\"29\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image048.gif\" width=\"9\" height=\"29\" name=\"graphics26\" align=\"ABSBOTTOM\" border=\"0\" \/>\u00a0are all real numbers.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which property is illustrated by the following equation?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image050.gif\" width=\"51\" height=\"32\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<ol>\n<li>Inverse property of multiplication<\/li>\n<li>Identity property of multiplication<\/li>\n<li>Commutative property of multiplication<\/li>\n<li>Associative property of multiplication<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A. The statement shows two real numbers\u00a0that, when multiplied, result in 1.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which property is illustrated by the following equation?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/052.gif\" width=\"108\" height=\"21\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<ol>\n<li>Inverse property of multiplication<\/li>\n<li>Identity property of multiplication<\/li>\n<li>Commutative property of multiplication<\/li>\n<li>Associative property of multiplication<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C. The statement shows two real numbers\u00a0multiplied to result in a real number.<\/p>\n<\/div>\n<\/section>\n<table border=\"2\" width=\"75%\" cellpadding=\"10\" align=\"center\">\n<tbody>\n<tr>\n<td>\n<p class=\"notebox_text\" align=\"CENTER\"><strong>Be Aware! <\/strong><\/p>\n<p class=\"notebox_text\" align=\"center\">While the commutative and associative properties work for addition<br \/>\nand multiplication, they do not work for subtraction and division. When in doubt, use a simple example<br \/>\nto decide.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s7_p3_html_2fe15a2e.gif\" width=\"84\" height=\"19\" name=\"graphics38\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p class=\"notebox_text\" align=\"center\">The above example shows that the commutative property does not hold<br \/>\nfor subtraction since one side of the equation gives a positive result and the other a negative<br \/>\nresult.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>What is the distributive property?<\/strong><\/p>\n<p>The <strong><em>distributive property<\/em><\/strong> states that\u00a0for any numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image054.gif\" width=\"110\" height=\"14\" name=\"graphics29\" align=\"ABSBOTTOM\" border=\"0\" \/>.\u00a0For example,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image056.gif\" width=\"175\" height=\"14\" name=\"graphics30\" align=\"ABSBOTTOM\" border=\"0\" \/>.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which equation best illustrates the distributive property?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image058.gif\" width=\"117\" height=\"14\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image060.gif\" width=\"122\" height=\"14\" name=\"graphics32\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image062.gif\" width=\"46\" height=\"14\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image064.gif\" width=\"51\" height=\"32\" name=\"graphics34\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is B. The distributive property says that\u00a0the for any numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s5_p3_clip_image065.gif\" width=\"110\" height=\"14\" name=\"graphics35\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Choice C illustrates the identity property. Choice D illustrates\u00a0the inverse property.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/white_shim.jpg\" width=\"10\" height=\"2\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<ul>\n<li>An <strong><em>algorithm <\/em><\/strong>is\u00a0a finite set of instructions for accomplishing some task when\u00a0given a problem in its initial state.<\/li>\n<li>To <strong><em>compare\u00a0<\/em><\/strong>numbers that are in different forms, change the numbers to the\u00a0same form.<\/li>\n<li>To <strong><em>order numbers\u00a0<\/em><\/strong>that are in different forms, write all numbers in the same form,\u00a0then place them in the correct order.<\/li>\n<li><strong><em>Add<\/em><\/strong> whole numbers\u00a0from right to left, carrying as needed.<\/li>\n<li><strong><em>Subtract<\/em><\/strong> whole\u00a0numbers from right to left, borrowing as needed.<\/li>\n<li><strong><em>Multiply <\/em><\/strong>whole\u00a0numbers from right to left, carrying as needed. Insert a place\u00a0holder for the second row of multiplication. Add the numbers to\u00a0arrive at the answer.<\/li>\n<li>To <strong><em>divide<\/em><\/strong> whole\u00a0numbers, determine the number of times the divisor goes into the\u00a0dividend. Subtract the result. Carry down the next digit.\u00a0Determine the number of times the divisor goes into the dividend.\u00a0Subtract the result. Repeat until there is nothing left to bring\u00a0down. If there is a remainder place it over the divisor.<\/li>\n<li>To <strong><em>add decimals<\/em><\/strong>,\u00a0line up the decimals. Fill in holes, using zeros as place\u00a0holders. Add as usual, carrying as needed. Bring the decimal\u00a0straight down.<\/li>\n<li>To <strong><em>subtract decimals<\/em><\/strong>,\u00a0line up the decimals. Fill in holes, using zeros as place\u00a0holders. Subtract as usual, borrowing as needed. Bring the\u00a0decimal straight down.<\/li>\n<li>To <strong><em>multiply decimals<\/em><\/strong>,\u00a0do not line up the decimals. Multiply as usual, carrying as\u00a0needed and ignoring the decimal. Count the number of digits after\u00a0the decimal in the multiplicand. Move the decimal left that many\u00a0places in the answer.<\/li>\n<li>To <strong><em>divide decimals<\/em><\/strong>,\u00a0move the decimal to the right of the divisor to create a whole\u00a0number. Move the decimal the same number of places in the\u00a0dividend. Divide as usual. Move the decimal straight up, filling\u00a0in with zeros as needed.<\/li>\n<li>The <strong><em>closure property\u00a0for addition <\/em><\/strong>states that if <em>a<\/em> and <em>b\u00a0<\/em>are real numbers, then\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image002.gif\" width=\"30\" height=\"11\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is a real number.<\/li>\n<li>The <strong><em>identity\u00a0property for addition<\/em><\/strong> states that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image004.gif\" width=\"54\" height=\"11\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>inverse property\u00a0of addition<\/em><\/strong> states that for each real number <em>a<\/em>,\u00a0there is a real number\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image006.gif\" width=\"17\" height=\"7\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0such that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image008.gif\" width=\"75\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>commutative\u00a0property of addition<\/em><\/strong> states that if <em>a<\/em> and <em>b\u00a0<\/em>are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image010.gif\" width=\"76\" height=\"11\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>associative\u00a0property of addition<\/em><\/strong> states that if <em>a<\/em>, <em>b<\/em>,\u00a0and <em>c<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image012.gif\" width=\"141\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>closure property\u00a0for multiplication<\/em><\/strong> states that if <em>a<\/em> and <em>b\u00a0<\/em>are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image014.gif\" width=\"15\" height=\"11\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is a real number.<\/li>\n<li>The <strong><em>identity\u00a0property for multiplication<\/em><\/strong> states that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image016.gif\" width=\"48\" height=\"14\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>inverse property\u00a0of multiplication<\/em><\/strong> states that for each real number\u00a0<em>a<\/em>, there is a real number\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image018.gif\" width=\"10\" height=\"29\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0such that\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image020.gif\" width=\"54\" height=\"32\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>commutative\u00a0property of multiplication<\/em><\/strong> states that if <em>a\u00a0<\/em>and <em>b<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image022.gif\" width=\"47\" height=\"11\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>associative property of multiplication<\/em><\/strong> states that if <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em> are real numbers,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image024.gif\" width=\"85\" height=\"14\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>distributive property<\/em><\/strong> states\u00a0that for any numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/1\/images2\/s6_p1_clip_image026.gif\" width=\"110\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<\/ul>\n<h3>Further Reading in Number Sense<\/h3>\n<ol class=\"biblio\">\n<li style=\"list-style-type: none;\">\n<ol class=\"biblio\">\n<li><em>CRC Concise Encyclopedia of Mathematics<\/em>. (Eric W. Weisstein):\u00a0Chapman &amp; Hall, 2002.<\/li>\n<li><em>Fundamental College Math<\/em>. (Sandra P. Clarkson): Houghton Mifflin College Div., 1994.<\/li>\n<li><em>How to Read and Do Proofs: An Introduction to Mathematical Thought Processes<\/em>. (Daniel Solow) John Wiley and\u00a0Sons, 2001.<\/li>\n<li><em>Don&#8217;t forget to test your knowledge with the <a href=\" http:\/\/www.abcte.org\/drupal\/courses\/mrc\/quizzes\/numsense\" target=\"popsome\"> Number Sense Chapter Quiz; click on the\u00a0Quizzes link in the left-hand navigation bar to take the quiz.<\/a>\u00a0<\/em><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/operations-with-real-numbers\">\u2b05 Previous Lesson<\/a>\u00a0\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/number-sense\">Workshop Index<\/a>\u00a0\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Next Workshop \u27a1<\/a><\/div>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Workshop \u27a1 Field Properties of Real Numbers Objective In this lesson, you will study how to use the field properties of the real number systems. Previously Covered: Real numbers are the field of\u00a0all rational and irrational numbers. What are the field properties for addition of real numbers? Mathematicians organize numbers into [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-13","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/13","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=13"}],"version-history":[{"count":29,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/13\/revisions"}],"predecessor-version":[{"id":845,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/13\/revisions\/845"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}