{"id":41276,"date":"2026-09-14T11:52:10","date_gmt":"2026-09-14T11:52:10","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=41276"},"modified":"2026-09-14T12:00:15","modified_gmt":"2026-09-14T12:00:15","slug":"properties-of-rational-numbers","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/properties-of-rational-numbers\/","title":{"rendered":"What are the Properties of Rational Numbers?"},"content":{"rendered":"\t\t
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\r\nWhat happens when you switch two numbers around in an addition problem? Or change the way three numbers are grouped? Sometimes the answer stays exactly the same\u2014and sometimes it doesn't.\r\n<\/p>\r\n\r\n

\r\nThat is where the properties of rational numbers<\/strong> come in. They describe how rational numbers behave when we add, subtract, multiply, or divide them.\r\n<\/p>\r\n\r\n

\r\nThe six commonly taught properties are closure, commutative, associative, distributive, identity, and inverse<\/strong>.\r\n<\/p>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\nThe Six Properties of Rational Numbers at a Glance\r\n<\/h2>\r\n\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Property<\/th>\r\nWhat it means<\/th>\r\nExample<\/th>\r\n<\/tr>\r\n<\/thead>\r\n\r\n
Closure<\/strong><\/td>\r\nThe result stays rational<\/td>\r\n1\/2 + 1\/3 = 5\/6<\/td>\r\n<\/tr>\r\n\r\n
Commutative<\/strong><\/td>\r\nChanging the order does not change the answer<\/td>\r\n1\/2 + 1\/3 = 1\/3 + 1\/2<\/td>\r\n<\/tr>\r\n\r\n
Associative<\/strong><\/td>\r\nChanging the grouping does not change the answer<\/td>\r\n(1\/2 + 1\/3) + 1\/6 = 1\/2 + (1\/3 + 1\/6)<\/td>\r\n<\/tr>\r\n\r\n
Distributive<\/strong><\/td>\r\nMultiplication can be distributed over addition or subtraction<\/td>\r\n2(3 + 4) = 2\u00d73 + 2\u00d74<\/td>\r\n<\/tr>\r\n\r\n
Identity<\/strong><\/td>\r\nA special number leaves the original number unchanged<\/td>\r\n5\/7 + 0 = 5\/7<\/td>\r\n<\/tr>\r\n\r\n
Inverse<\/strong><\/td>\r\nA number can be paired with another to produce an identity<\/td>\r\n3\/5 + (-3\/5) = 0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n<\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\r\n1. Closure Property\r\n<\/h2>\r\n\r\n

\r\nThe closure property<\/strong> means that performing an operation on rational numbers gives another rational number.\r\n<\/p>\r\n\r\n

\r\nRational numbers are closed under:\r\n<\/p>\r\n\r\n

\r\n
Addition<\/div>\r\n
Subtraction<\/div>\r\n
Multiplication<\/div>\r\n
Division by a nonzero rational number<\/strong><\/div>\r\n<\/div>\r\n\r\n

\r\nFor example:
\r\n1\/2 + 2\/3 = 7\/6<\/strong>
\r\nSince 7\/6 is rational, closure holds.\r\n<\/p>\r\n\r\n

\r\n

\r\nFor division:
\r\n2\/3 \u00f7 4\/5 = 2\/3 \u00d7 5\/4 = 5\/6<\/strong>
\r\nThe answer is also rational.\r\n<\/p>\r\n<\/div>\r\n\r\n

\r\nThe important catch? You cannot divide by zero.<\/strong>\r\n<\/p>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t

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\r\n2. Commutative Property\r\n<\/h2>\r\n\r\n
\r\nThink \u201ccommutative = change the order.\u201d<\/strong>\r\n<\/div>\r\n\r\n

\r\nFor rational numbers, addition and multiplication are commutative:
\r\na + b = b + a<\/strong>
\r\na \u00d7 b = b \u00d7 a<\/strong>\r\n<\/p>\r\n\r\n

\r\nFor example:
\r\n2\/5 + 1\/3 = 1\/3 + 2\/5<\/strong>\r\n<\/p>\r\n\r\n

\r\n

\r\nBut subtraction and division are not<\/strong> commutative.
\r\n2\/3 \u2212 1\/3 \u2260 1\/3 \u2212 2\/3<\/strong>\r\n<\/p>\r\n<\/div>\r\n\r\n

\r\nSo, changing the order does not always work.\r\n<\/p>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n3. Associative Property\r\n<\/h2>\r\n\r\n
\r\nThink \u201cassociative = change the grouping.\u201d<\/strong>\r\n<\/div>\r\n\r\n

\r\nAddition and multiplication of rational numbers are associative:
\r\n(a + b) + c = a + (b + c)<\/strong>
\r\n(a \u00d7 b) \u00d7 c = a \u00d7 (b \u00d7 c)<\/strong>\r\n<\/p>\r\n\r\n

\r\nFor example:
\r\n(1\/2 + 1\/3) + 1\/6 = 1\/2 + (1\/3 + 1\/6)<\/strong>
\r\nBoth sides equal 1.\r\n<\/p>\r\n\r\n

\r\nSubtraction and division are not associative.\r\n<\/p>\r\n\r\n

\r\n

Quick memory trick<\/h3>\r\n\r\n

\r\nCommutative \u2192 change the order<\/strong>
\r\nAssociative \u2192 change the grouping<\/strong>\r\n<\/p>\r\n<\/div>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\r\n4. Distributive Property\r\n<\/h2>\r\n\r\n

\r\nThe distributive property tells us that multiplication can be distributed over addition or subtraction.\r\n<\/p>\r\n\r\n

\r\na(b + c) = ab + ac<\/strong>
\r\na(b \u2212 c) = ab \u2212 ac<\/strong>\r\n<\/div>\r\n\r\n

\r\nFor example:
\r\n2(3 + 4) = 2\u00d73 + 2\u00d74<\/strong>
\r\n14 = 6 + 8<\/strong>
\r\n14 = 14<\/strong>\r\n<\/p>\r\n\r\n

\r\n

\r\nThis property becomes especially useful when simplifying algebraic expressions.\r\n<\/p>\r\n<\/div>\r\n\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Recommended reading:<\/strong><\/p>