{"id":30631,"date":"2023-06-16T08:19:44","date_gmt":"2023-06-16T08:19:44","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=30631"},"modified":"2024-02-05T16:10:17","modified_gmt":"2024-02-05T16:10:17","slug":"properties-of-integers-definition-examples-facts-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers","title":{"rendered":"Properties of Integers &#8211; Definition, Examples, Facts, FAQs"},"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-bc78d7ff-c170-4e07-b2f0-205f064e6b94\" 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\/properties-of-integers#0-what-are-the-properties-of-integers>What Are the Properties of Integers?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers#2-closure-property-of-integers>Closure Property of Integers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers#6-identity-property-of-integers>Identity Property of Integers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers#9-solved-examples-on-properties-of-integers>Solved Examples on Properties of Integers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers#10-practice-problems-on-properties-of-integers>Practice Problems on Properties of Integers<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers#11-frequently-asked-questions-on-properties-of-integers>Frequently Asked Questions on Properties of Integers<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-are-the-properties-of-integers\">What Are the Properties of Integers?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Properties of integers in mathematics are certain rules that can be applied to simplify calculations when we perform arithmetic operations on them.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The set of <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/integers\">integers<\/a> are represented by the symbol \u2124.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2124 $= \\left\\{\u2026,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,\u2026\\right\\}$<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"344\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/Integers-on-a-number-line.png\" alt=\"Integers on a number line\" class=\"wp-image-35887\" title=\"Integers on a number line\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/Integers-on-a-number-line.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/Integers-on-a-number-line-300x166.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The basic operation of addition, subtraction, multiplication, and division give rise to five main types of properties in integers:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. Closure Property<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2. Associative Property<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">3. Commutative Property<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">4. Distributive Property<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">5. Identity Property<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1-properties-of-integers\">Properties of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Here\u2019s a chart of properties of integers that gives an overview of each property.<\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><tbody><tr><td><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/closure-property\"><strong>Closure Property<\/strong><\/a><\/td><td>$a + b \\in$ \u2124<br>$a \\;-\\; b \\in$&nbsp;\u2124<br>$a \\times b \\in$ \u2124<br>for any two integers a and b<\/td><\/tr><tr><td><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/commutative-property\"><strong>Commutative Property<\/strong><\/a><\/td><td>$a + b = b +a$<br>$a \\times b = b \\times a$<br>for any two integers a and b<\/td><\/tr><tr><td><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/associative-property\"><strong>Associative Property<\/strong><\/a><\/td><td>$(a + b) + c = a + (b + c)$<br>$(a \\times b) \\times c = a \\times (b \\times c)$<br>for integers a, b, and c<\/td><\/tr><tr><td><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/algebra\/distributive-property\"><strong>Distributive Property<\/strong><\/a><\/td><td>$a \\times (b + c) = (a \\times b) + (a \\times c)$<br>$a \\times (b\\;-\\;c) =(a \\times b)\\;-\\;(a \\times c)$<br>for integers a, b, and c<\/td><\/tr><tr><td><strong>Identity Property<\/strong><\/td><td>$a + 0 = a<br>a \\times 1 = 1 \\times a = a$\\<br>for any integer a<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s study all properties of integers in detail.<\/p>\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\/match-the-commutative-properties-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\/match-the-commutative-properties-of-addition.jpeg\" alt=\"Match the Commutative Properties 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><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\" id=\"2-closure-property-of-integers\">Closure Property of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The closure property of integers states that the <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/addition\"><strong>addition<\/strong><\/a><strong>, <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/subtraction\/subtract\"><strong>subtraction<\/strong><\/a><strong>, or <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/multiplication\/multiplication\"><strong>multiplication<\/strong><\/a><strong> of integers always results in an integer.<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For any two integers a and b, we have<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$a + b =$ integer<\/li>\n\n\n\n<li>$a \\;-\\; b&nbsp; =$ integer<\/li>\n\n\n\n<li>$a \\times b =$ integer<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$-\\;5 + 4 = -\\;3$<\/li>\n\n\n\n<li>$8\\;-\\;5 = 3$<\/li>\n\n\n\n<li>$2 \\times 3 = 6$<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>However, the closure property of integers does not work for the division of integers because the division of two integers may not be an integer.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example: $15 \\div 2 = 7.5$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-associative-property-of-integers\">Associative Property of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>According to the associative property of integers, if we add three or more integers, the result is not affected by the way we group the integers. Similarly, the product of three or more integers does not depend on how we group them.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Associative property of integers is true only for the addition and multiplication but not for division and subtraction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For any three integers a, b, and c, we have<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/addition\/associative-property-of-addition\">Associative property of addition<\/a>: $a + (b + c) = (a + b) + c$<\/li>\n\n\n\n<li><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/multiplication\/associative-property-of-multiplication\">Associative property of multiplication<\/a>: $a \\times (b \\times c) = (a \\times b) \\times c$<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">However,&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Operation<\/strong><\/th><th><strong>Result<\/strong><\/th><th><strong>Examples<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\">Addition of integers<\/td><td>Associative property is satisfied.<\/td><td>$1 + (2 + 3) = 6$<br>$(1 + 2) + 3 = 6$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Multiplication of integers<\/td><td>Associative property is satisfied.<\/td><td>$4 \\times (2 \\times 3) = 24$<br>$(4 \\times 2) \\times 3 = 24$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Subtraction of integers<\/td><td>Associative property is <strong>not<\/strong> satisfied.<\/td><td>$1 \\;-\\; (2 \\;-\\; 3) = 2$<br>$(1\\;-\\;2)\\;-\\;3 = \\;-4$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Division of integers<\/td><td>Associative property is <strong>not<\/strong> satisfied.<\/td><td>$8\\div (4 \\div 2) = 4$<br>$(8 \\div 4) \\div 2 = 1$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-commutative-property-of-integers\">Commutative Property of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The commutative property states that changing the order of integers in addition and multiplication does not change the result of the operation.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Interestingly, this property is true for only the addition and multiplication and not for division and subtraction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For two integers a and b, we have<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/division\/commutative-property-of-addition\">Commutative property of addition<\/a>: $a + b = b + a$<\/li>\n\n\n\n<li><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/multiplication\/commutative-property-of-multiplication\">Commutative property of multiplication<\/a>: $a \\times b = b \\times a$<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\"><strong>Operation<\/strong><\/th><th><strong>Result<\/strong><\/th><th><strong>Example<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-left\" data-align=\"left\">Addition of integers<\/td><td>Commutative property is satisfied.<\/td><td>$1 + 2 = 3 = 2 + 1$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Multiplication of integers<\/td><td>Commutative property is satisfied.<\/td><td>$1 \\times 2 = 3 = 2 \\times 1$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Subtraction of integers<\/td><td>Commutative property is <strong>not<\/strong> satisfied.<\/td><td>$1\\;-\\;2 = 1\\;-\\;2$<\/td><\/tr><tr><td class=\"has-text-align-left\" data-align=\"left\">Division of integers<\/td><td>Commutative property is <strong>not<\/strong> satisfied.<\/td><td>$1\\;-\\;2 = 1\\;-\\;2$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-distributive-property-of-integers\">Distributive Property of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The distributive property states that the multiplication of integers can be distributed over addition and subtraction to make the calculation easier.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So, for any three integers a, b, and c, we have<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Distributive property of multiplication over addition: $a \\times (b + c) =(a \\times b) + (a \\times c)$<\/li>\n\n\n\n<li>Distributive property of multiplication over subtraction: $a \\times (b\\;-\\;c) = (a \\times b)\\;-\\;(a \\times c)$<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example:&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$5 \\times (4 + 3) = (5 \\times 4) + (5 \\times 3) = 35$<\/li>\n\n\n\n<li>$\\;-\\; 2 \\times (6 \\;-\\; 1) = (\\;-\\; 2 \\times 6) + (\\;-\\; 2 \\;-\\;1) = \\;-\\; 10$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-identity-property-of-integers\">Identity Property of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The identity property of an integer states that any integer added to zero will result in the same integer. Also, any integer multiplied by one will give the same integer.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">0 is the additive identity of integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1 is the multiplicative identity of integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So, for any integer a, we have<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$a + 0 = 0 + a = a$<\/li>\n\n\n\n<li>$a \\times 1 = 1 \\times a = a$<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example:&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$5 + 0 = 0 + 5 = 5$<\/li>\n\n\n\n<li>$5 \\times 1 = 1 \\times 5 = 5$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-facts-about-properties-of-integers\">Facts about Properties of Integers<\/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-4fc3f8e1 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>The use of the letter \u201c\u2124\u201d to denote the set of integers comes from the German word \u201cZahlen\u201d which means &#8220;numbers.&#8221;<\/li>\n\n\n\n<li>There are an infinite number of integers.<\/li>\n\n\n\n<li>0 is an integer that is neither positive nor negative.<\/li>\n\n\n\n<li>Every integer has an additive inverse. The additive inverse of an integer&nbsp;a is -a.<\/li>\n\n\n\n<li>Properties of multiplication of integers: Multiplication of integers follows closure property, commutative property, associative property, distributive property, and identity property.<\/li>\n\n\n\n<li>Rules for multiplying integers with different signs:<br>Positive integer&nbsp;$\\times$ Positive integer $=$ Positive integer<br>Positive integer&nbsp;$\\times$ Negative integer $=$ Negative integer<br>Negative integer&nbsp;$\\times$ Negative integer $=$ Positive integer<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In this article, we have learned the various properties of integers. These properties of integers can be used to speed up and simplify a number of operations on integers. Now, let&#8217;s solve some examples and practice problems to understand the concept better.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-solved-examples-on-properties-of-integers\">Solved Examples on Properties of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Identify the correct properties of integers used in the following equations:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>a) <\/strong>$a + (b + c) = (a + b) + c$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>b) <\/strong>$a + 0 = 0 + a = a$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) $a + (b + c) = (a + b) + c$&nbsp; shows the associative property of integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It shows that the addition of three or more integers does not depend on how we group them.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">b)&nbsp; $a + 0 = 0 + a = a$&nbsp; shows the identity property of integers<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Addition of any integer and 0 is the integer itself.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. Evaluate the expression: <\/strong>$(10 \\times 5) + (10 \\times 5)$<strong> using the properties of integers.<\/strong><br><br><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The given expression is $(10 \\times 5) + (10 \\times 4)$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It can be solved by using the distributive property of integers, which states that,&nbsp;&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For any three integers a, b, and c:&nbsp;&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$(a \\times b) + (a \\times c) = a \\times (b + c)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$(10 \\times 5) + (10 \\times 4) = 10 \\times (5 + 4)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;&nbsp;&nbsp; $= 10 \\times 9$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;&nbsp;  $= 90$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, $(10 \\times 5) + (10 \\times 4) = 90$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. If <\/strong>$a = \\;-\\;35$<strong>, <\/strong>$b = 10$<strong> units and <\/strong>$c = \\;-\\;5$<strong>, verify that:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>(i)&nbsp; <\/strong>$a + (b + c) = (a + b) + c$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>(ii) <\/strong>$a \\times (b + c) = (a \\times b) + (a \\times c)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Given that $a = \\;-\\;35\\;, b = 10\\;, c = \\;-\\;5$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>(i) <\/strong>$a + (b + c) = (a + b) + c$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L.H.S. $= a + (b + c) = (\\;-\\;35) + \\left[10 + (\\;-\\;5)\\right] = (\\;-\\;35) + 5 = \\;-\\;30$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">R.H.S. $= (a + b) + c = \\left[(\\;-\\;35) + 10\\right] + (\\;-\\;5) = (\\;-\\;25) + (\\;-\\;5) = \\;-\\;(25 + 5) = \\;-\\;30$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, L.H.S. = R.H.S.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, verified.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>(ii) <\/strong>$a \\times (b + c) = (a \\times b) + (a \\times c)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L.H.S. $= a(b + c)= (\\;-\\;35) \\left[10 + (\\;-\\;5)\\right] = (\\;-\\;35) 5 = \\;-\\;175$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">R.H.S. $= (a \\times b) + (a \\times c) = (\\;-\\;35) \\times 10 + (\\;-\\;35) \\times (\\;-\\;5) = \\;-\\;350 + 175 = \\;-\\;175$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, L.H.S. $=$ R.H.S.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4. State whether the following statements are true or false.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>a) Closure property holds true for addition, subtraction and multiplication of integers.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>b) The product of two negative integers is a negative integer.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>c) According to commutative property of multiplication: <\/strong>$a \\times (b \\times c) = (a \\times b) \\times c$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>a)<\/strong> <strong>Closure property holds true for addition, subtraction and multiplication of integers.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">True.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Closure property holds true for addition, subtraction and multiplication of integers. It fails for division of integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>b)<\/strong> <strong>The product of two negative integers is a negative integer.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">False.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The product of two negative integers is a positive integer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>c) According to commutative property of multiplication: <\/strong>$a \\times (b \\times c) = (a \\times b) \\times c$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">False.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The commutative property of multiplication states that the order of integers in multiplication operation does not change the result of the operation.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Commutative property of multiplication: $a \\times b = b \\times a$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-practice-problems-on-properties-of-integers\">Practice Problems on Properties of Integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Properties of Integers \u2013 Definition, Examples, Facts, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">$8 + (3 + 4) = (8 + 3) + 4$ is an example of __________.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">associative addition<\/div><div class=\"spq_answer_block\" data-value=\"1\">associative multiplication<\/div><div class=\"spq_answer_block\" data-value=\"2\">commutative addition<\/div><div class=\"spq_answer_block\" data-value=\"3\">commutative multiplication<\/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: associative addition<br\/>According to the associative property, for any three integers a, b, and c, we have $a + (b + c) = (a + b) + c$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">$3(a + 4) = 3a + 12$ is an example of __________.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">inverse property<\/div><div class=\"spq_answer_block\" data-value=\"1\">associative addition<\/div><div class=\"spq_answer_block\" data-value=\"2\">commutative addition<\/div><div class=\"spq_answer_block\" data-value=\"3\">distributive property<\/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: distributive property<br\/>According to distributive property of multiplication over addition, for any three integers a, b, and c, we have $a \\times (b + c) = (a \\times b) + (a \\times c)$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">The _______________ property of addition states that any integer plus zero equals the same number.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">identity<\/div><div class=\"spq_answer_block\" data-value=\"1\">commutative<\/div><div class=\"spq_answer_block\" data-value=\"2\">associative<\/div><div class=\"spq_answer_block\" data-value=\"3\">closure<\/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: identity<br\/>The identity property of integers states that any integer added to zero will result in the same integer. So, for any integer a, $a + 0 = 0 + a = a$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">The ______________ property states that changing the position of integers during addition and multiplication does not change the result of the operation.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">commutative<\/div><div class=\"spq_answer_block\" data-value=\"1\">associative<\/div><div class=\"spq_answer_block\" data-value=\"2\">distributive<\/div><div class=\"spq_answer_block\" data-value=\"3\">identity<\/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: commutative<br\/>The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$a + b = b + a$  and  $a \\times b = b \\times a$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">The __________ of integers doesn't follow the closure property.<\/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\">division<\/div><div class=\"spq_answer_block\" data-value=\"3\">multiplication<\/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\/>The closure property states that the addition, subtraction, or multiplication of integers results in an integer. Division of integers doesn\u2019t follow the closure property, i.e., the quotient of any two integers a and b, may or may not be an integer.<\/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\": \"Properties of Integers \u2013 Definition, Examples, Facts, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Properties of Integers\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    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                              }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"distributive property\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"According to distributive property of multiplication over addition, for any three integers a, b, and c, we have $$a \\\\times (b + c) = (a \\\\times b) + (a \\\\times c)$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"According to distributive property of multiplication over addition, for any three integers a, b, and c, we have $$a \\\\times (b + c) = (a \\\\times b) + (a \\\\times c)$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The _______________ property of addition states that any integer plus zero equals the same number.\",\n                    \"text\": \"The _______________ property of addition states that any integer plus zero equals the same number.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The identity property of integers states that any integer added to zero will result in the same integer. 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So, for any integer a, $$a + 0 = 0 + a = a$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"associative\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The identity property of integers states that any integer added to zero will result in the same integer. So, for any integer a, $$a + 0 = 0 + a = a$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"closure\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The identity property of integers states that any integer added to zero will result in the same integer. 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So, for any integer a, $$a + 0 = 0 + a = a$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The ______________ property states that changing the position of integers during addition and multiplication does not change the result of the operation.\",\n                    \"text\": \"The ______________ property states that changing the position of integers during addition and multiplication does not change the result of the operation.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$$a + b = b + a$$  and  $$a \\\\times b = b \\\\times a$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"associative\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$$a + b = b + a$$  and  $$a \\\\times b = b \\\\times a$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"distributive\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$$a + b = b + a$$  and  $$a \\\\times b = b \\\\times a$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"identity\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$$a + b = b + a$$  and  $$a \\\\times b = b \\\\times a$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"commutative\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$$a + b = b + a$$  and  $$a \\\\times b = b \\\\times a$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The commutative property states that changing the position of integers during addition and multiplication does not change the result of the operation.<br>\r\nSo, for two integers a and b:<br>\r\n$$a + b = b + a$$  and  $$a \\\\times b = b \\\\times a$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The __________ of integers doesn't follow the closure property.\",\n                    \"text\": \"The __________ of integers doesn't follow the closure property.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The closure property states that the addition, subtraction, or multiplication of integers results in an integer. Division of integers doesn\u2019t follow the closure property, i.e., the quotient of any two integers a and b, may or may not be an integer.\"\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\": \"The closure property states that the addition, subtraction, or multiplication of integers results in an integer. 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Division of integers doesn\u2019t follow the closure property, i.e., the quotient of any two integers a and b, may or may not be an integer.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"11-frequently-asked-questions-on-properties-of-integers\">Frequently Asked Questions on Properties of Integers<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-8260242b-736a-44b4-ae8a-f8bf6455d896\" 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-0-8260242b-736a-44b4-ae8a-f8bf6455d896\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8260242b-736a-44b4-ae8a-f8bf6455d896\"><strong>Is 0 an integer?<\/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-0-8260242b-736a-44b4-ae8a-f8bf6455d896\">\n\n<p class=\"wp-block-paragraph\">Yes, 0 is an integer. According to the definition, integers are the numbers that include whole numbers and negative natural numbers. It is a number without any fractional part, and zero has no fractional part. However, 0 is neither positive nor negative.<\/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-1-8260242b-736a-44b4-ae8a-f8bf6455d896\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8260242b-736a-44b4-ae8a-f8bf6455d896\"><strong>Are integers real 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-1-8260242b-736a-44b4-ae8a-f8bf6455d896\">\n\n<p class=\"wp-block-paragraph\">Yes. Integers are real numbers. The set of integers form a subset of the set of integers.<\/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-2-8260242b-736a-44b4-ae8a-f8bf6455d896\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8260242b-736a-44b4-ae8a-f8bf6455d896\"><strong>What are perfect 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-2-8260242b-736a-44b4-ae8a-f8bf6455d896\">\n\n<p class=\"wp-block-paragraph\">Perfect number is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6. 6 has divisors 1, 2 and 3 (excluding itself), and $1 + 2 + 3 = 6$, so 6 is a perfect number. Other perfect numbers are 28, 496, and 8,128.<\/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-3-8260242b-736a-44b4-ae8a-f8bf6455d896\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8260242b-736a-44b4-ae8a-f8bf6455d896\"><strong>Is an integer always a whole number?<\/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-3-8260242b-736a-44b4-ae8a-f8bf6455d896\">\n\n<p class=\"wp-block-paragraph\">No, each integer is not a whole number. However, each whole number is an integer.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set of whole numbers $= \\left\\{ 0, 1, 2, 3, 4, 5,\u2026\\right)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set of Integers $= \\left\\{\u2026\u2026\u2026, \\;-\\;4, \\;-\\;3, \\;-\\;2, \\;-\\;1, 0, 1, 2, 3, 4, \u2026\\right\\}$So, all whole numbers are integers, but not all integers are whole numbers. For example, -6 is an integer but not a whole 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-4-8260242b-736a-44b4-ae8a-f8bf6455d896\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8260242b-736a-44b4-ae8a-f8bf6455d896\"><strong>Are all integers natural 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-4-8260242b-736a-44b4-ae8a-f8bf6455d896\">\n\n<p class=\"wp-block-paragraph\">No, all integers are not natural numbers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Natural numbers $= \\left\\{1, 2, 3,\u2026\\right\\}$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Integers $= \\left\\{\u2026\u2212 3,\u2212 2,\u2212 1,0,1,2,3,\u2026\\right\\}$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are many integers that are not natural numbers (negatives and 0).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example, \u22121 is an integer, but not a natural number. So, not all integers are natural 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-5-8260242b-736a-44b4-ae8a-f8bf6455d896\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-8260242b-736a-44b4-ae8a-f8bf6455d896\"><strong>What are properties of addition and subtraction of integers?<\/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-5-8260242b-736a-44b4-ae8a-f8bf6455d896\">\n\n<p class=\"wp-block-paragraph\">The addition of integers follows the closure, commutative, associative, distributive, and identity property. The subtraction of integers follows the closure property. Also, integers follow distributive property over subtraction.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Are the Properties of Integers? Properties of integers in mathematics are certain rules that can be applied to simplify calculations when we perform arithmetic operations on them. The set of integers are represented by the symbol \u2124. \u2124 $= \\left\\{\u2026,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,\u2026\\right\\}$ The basic operation of &#8230; <a title=\"Properties of Integers &#8211; Definition, Examples, Facts, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/properties-of-integers\" aria-label=\"More on Properties of Integers &#8211; Definition, Examples, Facts, FAQs\">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-30631","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\/30631","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=30631"}],"version-history":[{"count":29,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/30631\/revisions"}],"predecessor-version":[{"id":39958,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/30631\/revisions\/39958"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=30631"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=30631"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=30631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}