{"id":8698,"date":"2023-10-04T18:06:33","date_gmt":"2023-10-04T18:06:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=8698"},"modified":"2024-10-18T20:56:46","modified_gmt":"2024-10-18T20:56:46","slug":"real-numbers-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/real-numbers-learn-it-2\/","title":{"raw":"Real Numbers: Learn It 2","rendered":"Real Numbers: Learn It 2"},"content":{"raw":"

Recognizing Properties of Real Numbers<\/h2>\r\n

The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers<\/strong>. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated.<\/p>\r\n

The table below is a partial list of properties of real numbers.<\/p>\r\n

 <\/p>\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\nExample<\/th>\r\nIn Words<\/th>\r\n<\/tr>\r\n
Distributive Property<\/strong>
\r\n[latex]a \\times (b + c) = a \\times b + a \\times c[\/latex]<\/td>\r\n		[latex]5 \\times (3 + 4) = 5 \\times 3 + 5 \\times 4[\/latex]<\/td>\r\nMultiplication distributes across addition<\/td>\r\n<\/tr>\r\n
Commutative Property of Addition<\/strong>
\r\n[latex]a + b = b + a[\/latex]<\/td>\r\n		[latex]3 + 7 = 7 + 3[\/latex]<\/td>\r\nNumbers can be added in any order<\/td>\r\n<\/tr>\r\n
Commutative Property of Multiplication<\/strong>
\r\n[latex]a \\times b = b \\times a[\/latex]<\/td>\r\n		[latex]10 \\times 4 = 4 \\times 10[\/latex]<\/td>\r\nNumbers can be multiplied in any order<\/td>\r\n<\/tr>\r\n
Associative Property of Addition<\/strong>
\r\n[latex]a + (b + c) = (a + b) + c[\/latex]<\/td>\r\n		[latex]4 + (3 + 8) = (4 + 3) + 8[\/latex]<\/td>\r\nDoesn't matter which pair of numbers is added first<\/td>\r\n<\/tr>\r\n
Associative Property of Multiplication<\/strong>
\r\n[latex]a \\times (b \\times c) = (a \\times b) \\times c[\/latex]<\/td>\r\n		[latex]2 \\times (5 \\times 7) = (2 \\times 5) \\times 7[\/latex]<\/td>\r\nDoesn't matter which pair of numbers is multiplied first<\/td>\r\n<\/tr>\r\n
Additive Identity Property<\/strong>
\r\n[latex]a + 0 = a[\/latex]<\/td>\r\n		[latex]17 + 0 = 17[\/latex]<\/td>\r\nAny number plus [latex]0[\/latex] is the number<\/td>\r\n<\/tr>\r\n
Multiplicative Identity Property<\/strong>
\r\n[latex]a \\times 1 = a[\/latex]<\/td>\r\n		[latex]21 \\times 1 = 21[\/latex]<\/td>\r\nAny number times one is the number<\/td>\r\n<\/tr>\r\n
Additive Inverse Property<\/strong>
\r\n[latex]a + (-a) = 0[\/latex]<\/td>\r\n		[latex]14 + (-14) = 0[\/latex]<\/td>\r\nEvery number plus its negative is [latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
Multiplicative Inverse Property<\/strong>
\r\n[latex]a \\times \\frac{1}{a} = 1[\/latex], provided [latex](a \\neq 0)[\/latex]<\/td>\r\n		[latex]3 \\times \\frac{1}{3} = 1[\/latex]<\/td>\r\nEvery non-zero number times its reciprocal is [latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n

The names of the properties are suggestive. The commutative properties<\/strong>, for example, suggest commuting, or moving. Associative properties<\/strong> suggest which items are associated with others, or if order matters in the computation. The distributive property<\/strong> addresses how a number is distributed across parentheses.<\/p>\r\n<\/section>\r\n

\r\n

In each of the following, identify which property of the real numbers is being applied.<\/p>\r\n

    \r\n\t
  1. [latex]4+(8+13)=(4+8)+13[\/latex]<\/li>\r\n\t
  2. [latex]34\u00d7(\\frac{1}{34})=1[\/latex]<\/li>\r\n\t
  3. [latex]14+27=27+14[\/latex]<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]<\/p>\r\n

      \r\n\t
    1. Here, the pair of numbers that is added first is switched. This is the associative property of addition.<\/li>\r\n\t
    2. Here, a number is multiplied by its reciprocal, resulting in [latex]1[\/latex]. This is the multiplicative inverse property.<\/li>\r\n\t
    3. Here, the order in which numbers are added is switched. This is the commutative property of addition.<\/li>\r\n<\/ol>\r\n

      [\/hidden-answer]<\/p>\r\n<\/section>\r\n

      \r\n

      [ohm2_question hide_question_numbers=1]12706[\/ohm2_question]<\/p>\r\n<\/section>\r\n

      Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers that add to a multiple of [latex]10[\/latex], or multiply to a multiple of [latex]10[\/latex] or [latex]100[\/latex].<\/p>\r\n

      \r\n

      Use properties of the real numbers to calculate the following:<\/p>\r\n

        \r\n\t
      1. [latex]2\u00d713\u00d750[\/latex]<\/li>\r\n\t
      2. [latex]13+84+27[\/latex]<\/li>\r\n\t
      3. [latex]9\u00d716\u00d711[\/latex]<\/li>\r\n<\/ol>\r\n

        [reveal-answer q=\"214518\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214518\"]<\/p>\r\n

          \r\n\t
        1. Notice that [latex]2\u00d750=100[\/latex], so that becomes the multiplication to do first. Use the commutative property of multiplication to change the order of the numbers being multiplied.

          <\/p>

          [latex]2\u00d713\u00d750=2\u00d750\u00d713=100\u00d713=1,300[\/latex]<\/center>

          <\/p><\/li>\r\n\t

        2. Notice that [latex]13+27=40[\/latex], so that becomes the addition to do first. Use the commutative property of addition to change the order in which the numbers are added.

          <\/p>

          [latex]13+84+27=13+27+84=40+84=124[\/latex]<\/center>

          <\/p><\/li>\r\n\t

        3. Notice that [latex]9\u00d711=99[\/latex]. Using that, the problem can be changed to [latex]99\u00d716[\/latex]. That, however, doesn't look easy at all. But [latex]99=(100\u22121)[\/latex]. Using the distributive property, we rewrite and expand this as [latex]99\u00d716=(100\u22121)\u00d716=100\u00d716\u22121\u00d716=1,600\u221216[\/latex].

          <\/p>The last step is subtraction, so the final answer is [latex]1,584[\/latex]. So, multiplying by [latex]99[\/latex] is the same as multiplying by [latex]100[\/latex], and then subtracting the other number once.<\/li>\r\n<\/ol>\r\n

          [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"

          Recognizing Properties of Real Numbers<\/h2>\n

          The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers<\/strong>. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated.<\/p>\n

          The table below is a partial list of properties of real numbers.<\/p>\n

           <\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
          Property<\/th>\nExample<\/th>\nIn Words<\/th>\n<\/tr>\n
          Distributive Property<\/strong>
          \n[latex]a \\times (b + c) = a \\times b + a \\times c[\/latex]<\/td>\n          		[latex]5 \\times (3 + 4) = 5 \\times 3 + 5 \\times 4[\/latex]<\/td>\nMultiplication distributes across addition<\/td>\n<\/tr>\n
          Commutative Property of Addition<\/strong>
          \n[latex]a + b = b + a[\/latex]<\/td>\n          		[latex]3 + 7 = 7 + 3[\/latex]<\/td>\nNumbers can be added in any order<\/td>\n<\/tr>\n
          Commutative Property of Multiplication<\/strong>
          \n[latex]a \\times b = b \\times a[\/latex]<\/td>\n          		[latex]10 \\times 4 = 4 \\times 10[\/latex]<\/td>\nNumbers can be multiplied in any order<\/td>\n<\/tr>\n
          Associative Property of Addition<\/strong>
          \n[latex]a + (b + c) = (a + b) + c[\/latex]<\/td>\n          		[latex]4 + (3 + 8) = (4 + 3) + 8[\/latex]<\/td>\nDoesn’t matter which pair of numbers is added first<\/td>\n<\/tr>\n
          Associative Property of Multiplication<\/strong>
          \n[latex]a \\times (b \\times c) = (a \\times b) \\times c[\/latex]<\/td>\n          		[latex]2 \\times (5 \\times 7) = (2 \\times 5) \\times 7[\/latex]<\/td>\nDoesn’t matter which pair of numbers is multiplied first<\/td>\n<\/tr>\n
          Additive Identity Property<\/strong>
          \n[latex]a + 0 = a[\/latex]<\/td>\n          		[latex]17 + 0 = 17[\/latex]<\/td>\nAny number plus [latex]0[\/latex] is the number<\/td>\n<\/tr>\n
          Multiplicative Identity Property<\/strong>
          \n[latex]a \\times 1 = a[\/latex]<\/td>\n          		[latex]21 \\times 1 = 21[\/latex]<\/td>\nAny number times one is the number<\/td>\n<\/tr>\n
          Additive Inverse Property<\/strong>
          \n[latex]a + (-a) = 0[\/latex]<\/td>\n          		[latex]14 + (-14) = 0[\/latex]<\/td>\nEvery number plus its negative is [latex]0[\/latex]<\/td>\n<\/tr>\n
          Multiplicative Inverse Property<\/strong>
          \n[latex]a \\times \\frac{1}{a} = 1[\/latex], provided [latex](a \\neq 0)[\/latex]<\/td>\n          		[latex]3 \\times \\frac{1}{3} = 1[\/latex]<\/td>\nEvery non-zero number times its reciprocal is [latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
          \n

          The names of the properties are suggestive. The commutative properties<\/strong>, for example, suggest commuting, or moving. Associative properties<\/strong> suggest which items are associated with others, or if order matters in the computation. The distributive property<\/strong> addresses how a number is distributed across parentheses.<\/p>\n<\/section>\n

          \n

          In each of the following, identify which property of the real numbers is being applied.<\/p>\n

            \n
          1. [latex]4+(8+13)=(4+8)+13[\/latex]<\/li>\n
          2. [latex]34\u00d7(\\frac{1}{34})=1[\/latex]<\/li>\n
          3. [latex]14+27=27+14[\/latex]<\/li>\n<\/ol>\n