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Original
2026-01-01
Modified
2026-02-28
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<p>According to the closure property, real numbers are closed under addition, subtraction,<a>multiplication</a>, and<a>division</a>(except<a>division by zero</a>, which is undefined).</p>
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<p>According to the closure property, real numbers are closed under addition, subtraction,<a>multiplication</a>, and<a>division</a>(except<a>division by zero</a>, which is undefined).</p>
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<p><strong>Closure property for<a>integers</a></strong></p>
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<p><strong>Closure property for<a>integers</a></strong></p>
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<p>The set of integers is represented as:</p>
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<p>The set of integers is represented as:</p>
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<p>Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}</p>
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<p>Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}</p>
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<p>The closure property applies to the<a>arithmetic</a>operations such as addition, subtraction, and multiplication of integers, but not to division.</p>
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<p>The closure property applies to the<a>arithmetic</a>operations such as addition, subtraction, and multiplication of integers, but not to division.</p>
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<p><strong>Closure property under addition</strong></p>
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<p><strong>Closure property under addition</strong></p>
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<p>According to this property, the sum of any two integers always results in another integer. That is, for the integers a and b, their sum (a + b) is also an integer.</p>
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<p>According to this property, the sum of any two integers always results in another integer. That is, for the integers a and b, their sum (a + b) is also an integer.</p>
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<p><strong>For instance:</strong></p>
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<p><strong>For instance:</strong></p>
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<p>\((-7) + 9 = 2.\)</p>
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<p>\((-7) + 9 = 2.\)</p>
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<p>\(5 + 12 = 17.\)</p>
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<p>\(5 + 12 = 17.\)</p>
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<p><strong>Closure property under subtraction</strong></p>
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<p><strong>Closure property under subtraction</strong></p>
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<p>The difference between two integers will always result in an integer. That is, for the integers a and b, (a - b) will also be an integer.</p>
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<p>The difference between two integers will always result in an integer. That is, for the integers a and b, (a - b) will also be an integer.</p>
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<p><strong>For instance:</strong></p>
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<p><strong>For instance:</strong></p>
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<p>\(12 - 7 = 5.\)</p>
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<p>\(12 - 7 = 5.\)</p>
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<p>\((- 8) - (- 2) = - 6\)</p>
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<p>\((- 8) - (- 2) = - 6\)</p>
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<p><strong>Property of closure under multiplication</strong></p>
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<p><strong>Property of closure under multiplication</strong></p>
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<p>When two integers are multiplied, their product will always be an integer. For the integers a and b, their product (a × b) will also be an integer.</p>
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<p>When two integers are multiplied, their product will always be an integer. For the integers a and b, their product (a × b) will also be an integer.</p>
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<p><strong>For instance:</strong></p>
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<p><strong>For instance:</strong></p>
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<p>\(4 × (-6) = -24\)</p>
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<p>\(4 × (-6) = -24\)</p>
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<p>\((-9) × (-5) = 45\)</p>
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<p>\((-9) × (-5) = 45\)</p>
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<p><strong>Property of closure under division</strong></p>
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<p><strong>Property of closure under division</strong></p>
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<p>Division does not always yield an integer. So, integers are not closed under division. </p>
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<p>Division does not always yield an integer. So, integers are not closed under division. </p>
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<p><strong>For instance:</strong></p>
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<p><strong>For instance:</strong></p>
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<p>\((-15) ÷ 3 = - 5\) (an integer) </p>
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<p>\((-15) ÷ 3 = - 5\) (an integer) </p>
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<p>\((-8) ÷ (-20) = 0.4\) (not an integer)</p>
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<p>\((-8) ÷ (-20) = 0.4\) (not an integer)</p>