{"id":28832,"date":"2023-05-08T10:41:31","date_gmt":"2023-05-08T10:41:31","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=28832"},"modified":"2023-05-11T08:09:44","modified_gmt":"2023-05-11T08:09:44","slug":"closure-property-definition-formula-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property","title":{"rendered":"Closure Property: Definition, Formula, Examples"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-47cff434-17f0-4a4c-8160-68966de0bc2a\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property#0-what-is-closure-property>What Is Closure Property?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property#2-closure-property-of-real-numbers>Closure Property of Real Numbers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property#9-formula-for-closure-property>Formula for Closure Property<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property#12-solved-examples-on-closure-property>Solved Examples on Closure Property<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property#13-practice-problems-on-closure-property>Practice Problems on Closure Property<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property#14-frequently-asked-questions-on-closure-property>Frequently Asked Questions on Closure Property<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-closure-property\">What Is Closure Property?<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><strong>The closure property states that if a set of numbers (integers, real numbers, etc.) is closed under some operation (such as addition, subtraction, or multiplication, etc.), then performing that operation on any two numbers in the set results in the element belonging to the set.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">To understand this, let\u2019s look at some questions that we often wonder about.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Is the sum of two natural numbers a natural number? Yes!<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Is the subtraction of two integers always an integer? Yes!<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Do we always get an integer by the division of two integers? No!<\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">The closure property is directly linked with such properties of any given set with respect to an operation. Let\u2019s explore the closure property in detail for different operations and different sets.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"1-definition-of-closure-property\">Definition of Closure Property<\/h3>\n\n\n\n<p class=\" eplus-wrapper\"><strong>The closure property states that when a set of numbers is closed under an arithmetic operation, performing the operation on any two numbers in the set always results in a number belonging to the same set of numbers.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">If the operation results in even a single element outside the set, we can say that the set is not closed under the given operation.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example 1:<\/strong> The addition of two real numbers is always a real number. Thus, real numbers are closed under addition.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example 2:<\/strong> Subtraction of two natural numbers may or may not be a natural number. Thus, natural numbers are not closed under subtraction.<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-using-associative-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/add_sub_facts_together_we_flee_4_gm.png\" alt=\"Add Using Associative Property Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add Using Associative Property Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/apply-distributive-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_facts_mul_dispro_exp1_pt.png\" alt=\"Apply Distributive Property Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Apply Distributive Property Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/multiply-using-the-commutative-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_facts_mul_compro_fact2_pt.png\" alt=\"Multiply using the Commutative Property Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Multiply using the Commutative Property Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/solve-using-distributive-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_facts_mul_dispro_pProd_pt.png\" alt=\"Solve using Distributive Property Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Solve using Distributive Property Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-closure-property-of-real-numbers\">Closure Property of Real Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The set of rational numbers and irrational numbers together form the set of real numbers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Real numbers include:<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Natural numbers $\\left\\{1,\\;2,\\;3,\\; &#8230;\\right\\}$<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Whole numbers $\\left\\{0,\\;1,\\;2,\\;3,\\; &#8230;\\right\\}$<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Integers $\\left\\{&#8230;,\\;-\\;3,\\;-\\;2,\\;-\\;1,\\;0,\\;1,\\;2,\\;3,\\;&#8230;.\\right\\}$<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Rational numbers&nbsp;<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Irrational numbers&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">Real numbers are closed under the following operations: <strong>addition, subtraction, multiplication, division<\/strong>. Note that division by 0 is not defined. Division by zero is the only case where the closure property does not apply for real numbers. We can say that the real numbers are closed under non-zero division<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-htnl-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Real Numbers<\/strong><\/td><\/tr><tr><td>Real number $+$ Real number $=$ Real number<\/td><td>Closed under addition<\/td><\/tr><tr><td>Real number $\\;-\\;$ Real number $=$ Real number<\/td><td>Closed under subtraction<\/td><\/tr><tr><td>Real number Real number $=$ Real number<\/td><td>Closed under multiplication<\/td><\/tr><tr><td>Real number Real number $=$ Not always a real number<\/td><td>Closed only under non-zero division&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/add-the-numbers-using-commutative-property-of-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/add-the-numbers-using-commutative-property-of-addition.jpeg\" alt=\"Add the Numbers Using Commutative Property of Addition\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/apply-commutative-property-of-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/apply-commutative-property-of-addition.jpeg\" alt=\"Apply Commutative Property of Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/apply-commutative-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/apply-commutative-property.jpeg\" alt=\"Apply Commutative Property Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/apply-distributive-property-in-2-different-ways\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/apply-distributive-property-in-2-different-ways.jpeg\" alt=\"Apply Distributive Property in 2 Different Ways Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/apply-the-commutative-property-of-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/apply-the-commutative-property-of-addition.jpeg\" alt=\"Apply the Commutative Property of Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/break-it-down-using-distributive-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/break-it-down-using-distributive-property.jpeg\" alt=\"Break it Down using Distributive Property Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/break-multiply-add-using-distributive-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/break-multiply-add-using-distributive-property.jpeg\" alt=\"Break Multiply Add using Distributive Property Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/break-using-distributive-property\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/break-using-distributive-property.jpeg\" alt=\"Break using Distributive Property Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/commutative-property-of-addition-on-number-line\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/commutative-property-of-addition-on-number-line.jpeg\" alt=\"Commutative Property of Addition on Number Line Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/commutative-property-of-addition\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/commutative-property-of-addition.jpeg\" alt=\"Commutative Property of Addition Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-closure-property-for-integers\">Closure Property for Integers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The set of integers is given by $Z = \\left\\{&#8230;,\\; -\\;3,\\; -\\;2,\\; -\\;1,\\; 0,\\; 1,\\; 2,\\; 3,\\; &#8230; \\right\\}$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>The closure property holds true for addition, subtraction, and multiplication of integers. It does not apply for the division of two integers.<\/strong><\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\"><strong>Closure property of integers under addition:<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">The closure property of addition of integers states that the sum of any two integers will always be an integer. If a and b are any two integers, $a + b$ will be an integer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Examples:&nbsp; $(\\;-\\;5) + 8 = 3$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$10 + 6 = 16$<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\"><strong>Closure property of integers under subtraction:<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">The difference between any two integers will always be an integer. If a and b are any two integers, $a \\;-\\; b$ will also be an integer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Examples: $10 \\;-\\; 6 = 4$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$(\\;-\\;6) \\;-\\; (\\;-\\;3) = \\;-\\;3$<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\"><strong>Closure property of integers under multiplication:<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">The product of any two integers will be an integer. If a and b are any two integers, a b will also be an integer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Examples: $3 \\times (\\;-\\;9) = \\;-\\;27$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$(\\;\u2013\\;7) \\times (\\;-\\;9) = 63$<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\"><strong>Closure property of integers under division:<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\" eplus-wrapper\">If you divide an integer by another integer, you may or may not get an integer as a quotient. Also, the division by 0 is not defined.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Examples: $\\;-\\;10 \\div 5 = \\;-\\; 2$ is an integer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$(\\;-\\;5) \\div ( \\;-\\;25) = \\frac{1}{5}$ is not an integer<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-html-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Integers<\/strong><\/td><\/tr><tr><td>Integer $+$ Integer $=$ Integer<\/td><td>Closed under addition<\/td><\/tr><tr><td>Integer $-$ Integer $=$ Integer<\/td><td>Closed under subtraction<\/td><\/tr><tr><td>Integer Integer $=$ Integer<\/td><td>Closed under multiplication<\/td><\/tr><tr><td>Integer Integer $=$ Not always an integer<\/td><td>Not closed under division<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-closure-property-of-rational-numbers\">Closure Property of Rational Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Rational numbers are the numbers that can be represented in the form of $\\frac{p}{q}$, where $q \\neq 0$; p and q are integers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>The rational numbers are closed under addition, subtraction, and multiplication. The closure property isn\u2019t applicable for division since division by zero isn\u2019t defined.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">In other words, we can say that closure property is applicable for division of rational numbers except division by zero.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">1)&nbsp; $\\frac{4}{7} + \\frac{2}{3} = \\frac{26}{21}$&nbsp; is a rational number<\/p>\n\n\n\n<p class=\" eplus-wrapper\">2)&nbsp; $\\frac{4}{3} \\;\u2013\\; \\frac{2}{4} = \\frac{6}{12}$&nbsp; is a rational number<\/p>\n\n\n\n<p class=\" eplus-wrapper\">3)&nbsp; $\\frac{3}{5} \\times \\frac{2}{3} = \\frac{6}{15}$&nbsp; is a rational number<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-html-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Rational Numbers<\/strong><\/td><\/tr><tr><td>Rational number $+$ Rational number $=$ Rational number<\/td><td>Closed under addition<\/td><\/tr><tr><td>Rational number $\u2013$ Rational number $=$ Rational number<\/td><td>Closed under subtraction<\/td><\/tr><tr><td>Rational number Rational number $=$ Rational number<\/td><td>Closed under multiplication<\/td><\/tr><tr><td>Rational number $\\div$ Rational number $=$ Not always a Rational number<\/td><td>Closed only under non-zero division<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-closure-property-of-whole-numbers\">Closure Property of Whole Numbers<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The set of whole numbers is given by $W = \\left\\{0,\\; 1,\\; 2,\\; 3,\\; 4,\\; \u2026\\right\\}$. Thus, whole numbers include natural numbers and 0.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>The whole numbers are closed under addition and the multiplication. If a and b are two whole numbers, <\/strong>$a + b$<strong> is a whole number and <\/strong>$a \\times b$<strong> is also a whole number.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">For example,<\/p>\n\n\n\n<p class=\" eplus-wrapper\">1) $2 + 6 = 8$ \u2026a whole number<\/p>\n\n\n\n<p class=\" eplus-wrapper\">2) $2 \\times 6 = 12$ \u2026a whole number<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Whole numbers are not closed under subtraction and division. If a and b are two whole numbers, then <\/strong>$a \\;-\\; b$<strong> and <\/strong>$a \\div b$<strong> is not always a whole number.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">For example:&nbsp;&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">1) $6 \\;-\\; 1 = 5$ is a whole number.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">2) $1\\;-\\; 6 = \\;-\\; 5$&nbsp; is not a whole number.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">3)&nbsp;$\\frac{12}{6} = 2$ is a whole number.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">4)&nbsp; $\\frac{1}{6} = 0.167$ is a decimal number, not a whole number.<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-html-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Whole Numbers<\/strong><\/td><\/tr><tr><td>Whole number $+$ Whole number $=$ Whole number<\/td><td>Closed under addition<\/td><\/tr><tr><td>Whole number Whole number $=$ Whole number<\/td><td>Closed under multiplication<\/td><\/tr><tr><td>Whole number $-$ Whole number $=$ Not always a whole number<\/td><td>Not closed under subtraction<\/td><\/tr><tr><td>Whole number $\\div$ Whole number $=$ Not always a whole number<\/td><td>Not closed under division<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-closure-property-of-addition\">Closure Property of Addition<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">If the addition of two numbers in a given set of numbers belongs to the set, then we say that the given set of numbers is closed under addition.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The set of real numbers, natural numbers, whole numbers, rational numbers, and integers are closed under addition.<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-html-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Addition<\/strong><\/td><\/tr><tr><td>Real Numbers<\/td><td>$a + b =$ Real number (a, b are real numbers.)<\/td><\/tr><tr><td>Rational Numbers<\/td><td>$a + b =$ Rational number (a, b are real numbers.)<\/td><\/tr><tr><td>Integers Numbers<\/td><td>$a + b =$ Integer (a, b are integers.)<\/td><\/tr><tr><td>Natural Numbers<\/td><td>$a + b =$ Natural number (a, b are natural numbers)<\/td><\/tr><tr><td>Whole Numbers<\/td><td>$a + b =$ Whole number (a, b are whole numbers)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-closure-property-of-multiplication\">Closure Property of Multiplication<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">If the multiplication or product of two numbers in a given set of numbers belongs to the set, then we say that the given set of numbers is closed under multiplication.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The closure property of multiplication is applicable for natural numbers, whole numbers, integers, and rational numbers.<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-html-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Multiplication<\/strong><\/td><\/tr><tr><td>Real Numbers<\/td><td>$a \\times b =$ Real number (a, b are real numbers.)<\/td><\/tr><tr><td>Rational Numbers<\/td><td>$a \\times b =$ Rational number (a, b are rational numbers.)<\/td><\/tr><tr><td>Integers Numbers<\/td><td>$a \\times b =$ Integer (a, b are integers.)<\/td><\/tr><tr><td>Natural Numbers<\/td><td>$a \\times b =$ Natural number (a, b are natural numbers)<\/td><\/tr><tr><td>Whole Numbers<\/td><td>$a \\times b =$ Whole number (a, b are whole numbers)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-closure-property-of-subtraction\">Closure Property of Subtraction<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">If the subtraction of two numbers in a given set of numbers belongs to the set, then we say that the given set of numbers is closed under subtraction.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">This property is applicable for real numbers, integers, and rational numbers.<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-html-table\"><tbody><tr class=\"wj-html-tr\"><td colspan=\"2\"><strong>Closure Property of Subtraction<\/strong><\/td><\/tr><tr><td>Real Numbers<\/td><td>$a \\;-\\; b =$ Real number (a, b are real numbers.)<\/td><\/tr><tr><td>Rational Numbers<\/td><td>$a \\;-\\; b =$ Rational number (a, b are real numbers.)<\/td><\/tr><tr><td>Integers<\/td><td>$a \\;-\\; b =$ Rational number (a, b are real numbers<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-formula-for-closure-property\">Formula for Closure Property<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">For two real numbers a and b, the closure property formula can be given as<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$a + b = R$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$a \\;-\\; b = R$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$a \\times b = R$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$a \\div b \\neq R$<strong> \u2026since the division by 0 is not defined<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Similarly, you can write the formulas for different sets with respect to each operation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-facts-about-closure-property\">Facts about Closure Property<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Closure Property is a mathematical operation that is used in addition, subtraction and multiplication of numbers.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">If a set satisfies the closure property under a particular operation then it is said to be \u201cclosed under the operation.\u201d<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The closure property of division tells us that the result of the division of two whole numbers is not always a whole number.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Division of integers doesn\u2019t follow the closure property as the quotient of any two integers a and b, may or may not be an integer.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The result of the subtraction and division is not always a whole number.&nbsp;<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"11-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">In this article, we learned that the closure property is one of the most significant properties in mathematics. The closure property denotes that the set is complete. This indicates that any operation performed on elements within a set produces a result that belongs to the same set. Now let&#8217;s apply this knowledge to solve some closure property examples.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"12-solved-examples-on-closure-property\">Solved Examples on Closure Property<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><strong>1. Write whether the following statements are true or false.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">a. Rational numbers are closed under division.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">b. Natural numbers are closed under division.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">c. Whole numbers are not closed under subtraction and division.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>a. Rational numbers are closed under division.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">False&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Rational numbers are closed under addition, subtraction, and multiplication but not under division.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>b.<\/strong> <strong>Natural numbers are closed under division.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">False<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Consider two natural numbers 1 and 2.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$1\\div2 = 0.5$, which is not a natural number.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>c. Whole numbers are not closed under subtraction and division.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">True<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Whole numbers are only closed under addition and multiplication.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>2. Are integers closed under division? Explain why or why not.&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Integers are not closed under division.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Consider a counter example.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The numbers 6 and 7 are integers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">However, $6\\div 7 = 0.85$ is not an integer.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>3. Give a counterexample to support the statement: Whole numbers are not closed under subtraction.&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Whole numbers include natural numbers and 0.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$W = \\left\\{0,\\; 1,\\; 2,\\; 3,\\; \u2026\\right\\}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">For any two whole numbers a and b, the difference $a \\;-\\; b$ may or may not be a whole number.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Consider a counterexample.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">0 and 1 are whole numbers.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">However, $0 \\;-\\; 1 = \\;-\\;1$ is not a whole number.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"13-practice-problems-on-closure-property\">Practice Problems on Closure Property<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Closure Property: Definition, Formula, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Closure property holds true for whole numbers under  ______ and ______ .<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">addition and subtraction<\/div><div class=\"spq_answer_block\" data-value=\"1\">addition and division<\/div><div class=\"spq_answer_block\" data-value=\"2\">addition and multiplication<\/div><div class=\"spq_answer_block\" data-value=\"3\">multiplication and division.<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: addition and multiplication<br\/>Whole numbers are closed under addition and multiplication.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">If $m$ and $n$ are two rational numbers, then the closure property of addition states that______ .<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$m + n = n + m$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$m + n$ is a rational number.<\/div><div class=\"spq_answer_block\" data-value=\"2\">$m \\times n$ is a rational number.<\/div><div class=\"spq_answer_block\" data-value=\"3\">$m = n$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $m + n$ is a rational number.<br\/>For any rational numbers a and b, a + b is also a rational number.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">If a and b are integers, then ______ is also an integer.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$a + b$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$ab$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$a \\;-\\; b$<\/div><div class=\"spq_answer_block\" data-value=\"3\">All of the above<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: All of the above<br\/>Integers are closed under addition, multiplication, and subtraction.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">Closure property does not hold true for  ______ of integers.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">addition<\/div><div class=\"spq_answer_block\" data-value=\"1\">subtraction<\/div><div class=\"spq_answer_block\" data-value=\"2\">multiplication<\/div><div class=\"spq_answer_block\" data-value=\"3\">division<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: division<br\/>Division of integers doesn\u2019t follow the closure property.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">Closure property does not hold true for  ______ of rational numbers.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">addition<\/div><div class=\"spq_answer_block\" data-value=\"1\">subtraction<\/div><div class=\"spq_answer_block\" data-value=\"2\">multiplication<\/div><div class=\"spq_answer_block\" data-value=\"3\">division<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: division<br\/>Rational numbers are not closed under division since the division by 0 is not defined.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Closure Property: Definition, Formula, Examples\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Closure Property\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Closure property holds true for whole numbers under  ______ and ______ .\",\n                    \"text\": \"Closure property holds true for whole numbers under  ______ and ______ .\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Whole numbers are closed under addition and multiplication.\"\n                    },\n    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\"@type\": \"Comment\",\n                      \"text\": \"For any rational numbers a and b, a + b is also a rational number.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$m + n = n + m$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For any rational numbers a and b, a + b is also a rational number.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$m \\\\times n$$ is a rational number.\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For any rational numbers a and b, a + b is also a rational number.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$m = n$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For any rational numbers a and b, a + b is also a rational number.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$m + n$$ is a rational number.\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"For any rational numbers a and b, a + b is also a rational number.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"For any rational numbers a and b, a + b is also a rational number.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If a and b are integers, then ______ is also an integer.\",\n                    \"text\": \"If a and b are integers, then ______ is also an integer.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Integers are closed under addition, multiplication, and subtraction.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$a + b$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Integers are closed under addition, multiplication, and subtraction.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$ab$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Integers are closed under addition, multiplication, and subtraction.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$a \\\\;-\\\\; b$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Integers are closed under addition, multiplication, and subtraction.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"All of the above\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Integers are closed under addition, multiplication, and subtraction.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Integers are closed under addition, multiplication, and subtraction.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Closure property does not hold true for  ______ of integers.\",\n                    \"text\": \"Closure property does not hold true for  ______ of integers.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Division of integers doesn\u2019t follow the closure property.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"addition\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Division of integers doesn\u2019t follow the closure property.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"subtraction\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Division of integers doesn\u2019t follow the closure property.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"multiplication\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Division of integers doesn\u2019t follow the closure property.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"division\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Division of integers doesn\u2019t follow the closure property.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Division of integers doesn\u2019t follow the closure property.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Closure property does not hold true for  ______ of rational numbers.\",\n                    \"text\": \"Closure property does not hold true for  ______ of rational numbers.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Rational numbers are not closed under division since the division by 0 is not defined.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"addition\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Rational numbers are not closed under division since the division by 0 is not defined.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"subtraction\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Rational numbers are not closed under division since the division by 0 is not defined.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"multiplication\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Rational numbers are not closed under division since the division by 0 is not defined.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"division\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Rational numbers are not closed under division since the division by 0 is not defined.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Rational numbers are not closed under division since the division by 0 is not defined.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"14-frequently-asked-questions-on-closure-property\">Frequently Asked Questions on Closure Property<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-bde05706-4a89-48c7-97af-0367d9807ec8\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-bde05706-4a89-48c7-97af-0367d9807ec8\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-bde05706-4a89-48c7-97af-0367d9807ec8\"><strong>What is the commutative property of addition?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-bde05706-4a89-48c7-97af-0367d9807ec8\">\n\n<p class=\" eplus-wrapper\">Commutative property of addition states that the order of the numbers being added does not matter. In other words,$a + b = b + a$.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-bde05706-4a89-48c7-97af-0367d9807ec8\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-bde05706-4a89-48c7-97af-0367d9807ec8\"><strong>What does the term &#8220;closure property&#8221; indicate?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-bde05706-4a89-48c7-97af-0367d9807ec8\">\n\n<p class=\" eplus-wrapper\">The closure property or the property of closure denotes that the set is closed. Closure in mathematics means that any operation carried out on elements within a set results in a number that belongs to the same set.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">There\u2019s no chance of getting a number outside the given set. The term &#8220;operation closed&#8221; or &#8220;operation closure property satisfied&#8221; is used to describe any operation that meets this criterion.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-bde05706-4a89-48c7-97af-0367d9807ec8\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-bde05706-4a89-48c7-97af-0367d9807ec8\"><strong>Why are real numbers not closed under division?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-bde05706-4a89-48c7-97af-0367d9807ec8\">\n\n<p class=\" eplus-wrapper\">Division of real numbers will always give us a real number EXCEPT in the case of division by 0.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The division by zero is undefined. That\u2019s the only scenario where the closure property of division fails for the set of real numbers.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">However, we can say that the real numbers are closed under non-zero division (division except division by 0). The same is the case when it comes to the set of rational numbers.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-bde05706-4a89-48c7-97af-0367d9807ec8\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-bde05706-4a89-48c7-97af-0367d9807ec8\"><strong>Is closure property applicable for irrational numbers ?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-bde05706-4a89-48c7-97af-0367d9807ec8\">\n\n<p class=\" eplus-wrapper\">No. The set of irrational numbers is not closed under any of the operations.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Take a look at the counterexamples.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Addition: &nbsp; $\\sqrt{2} + (\\;-\\;\\sqrt{2}) = 0$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Subtraction:&nbsp; $\\sqrt{3} \\;-\\; \\sqrt{3}&nbsp; = 0$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Multiplication: &nbsp; $\\sqrt{5}\\times \\sqrt{5}&nbsp; = 5$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Division: &nbsp; $\\frac{\\sqrt{125}}{\\sqrt{5}} = 25$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">In each example, the answer is not an irrational number.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-bde05706-4a89-48c7-97af-0367d9807ec8\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-bde05706-4a89-48c7-97af-0367d9807ec8\"><strong>Is the set of real numbers closed under addition ?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-bde05706-4a89-48c7-97af-0367d9807ec8\">\n\n<p class=\" eplus-wrapper\">Yes. The set of real numbers is closed under addition. If you add two real numbers, you will get another real number. There is no possibility of ever getting anything other than a real number.For example: $5 + 10 = 15,\\; 2.5 + 2.5 = 5,\\;&nbsp; 2\\frac{1}{2} + 5 = 7\\frac{1}{2},\\; \\sqrt{3} + 2\\sqrt{3} = 3\\sqrt{3}$ , etc.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is Closure Property? The closure property states that if a set of numbers (integers, real numbers, etc.) is closed under some operation (such as addition, subtraction, or multiplication, etc.), then performing that operation on any two numbers in the set results in the element belonging to the set. To understand this, let\u2019s look at &#8230; <a title=\"Closure Property: Definition, Formula, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property\" aria-label=\"More on Closure Property: Definition, Formula, Examples\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-28832","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/28832","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=28832"}],"version-history":[{"count":15,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/28832\/revisions"}],"predecessor-version":[{"id":29226,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/28832\/revisions\/29226"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=28832"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=28832"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=28832"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}