{"id":7,"date":"2017-08-23T06:40:57","date_gmt":"2017-08-23T06:40:57","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=7"},"modified":"2017-09-21T20:17:19","modified_gmt":"2017-09-21T20:17:19","slug":"real-number-properties","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/real-number-properties\/","title":{"rendered":"Real Number Properties"},"content":{"rendered":"
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Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Real Number Properties<\/h1>\n

Objective<\/h4>\n

The following lesson will examine whole numbers, rational and irrational numbers, integers, and their properties.<\/p>\n

\n

What are the different types of numbers?<\/h3>\n

The counting numbers<\/em><\/strong> or natural numbers are the\u00a0numbers 1,2,3,4,…. The whole numbers<\/em><\/strong> are the\u00a0counting numbers plus zero. The integers\u00a0<\/em><\/strong>are the whole numbers and all their opposites. Real\u00a0numbers consist of rational and irrational numbers.\u00a0Rational\u00a0numbers<\/abbr> include whole numbers, fractions, finite or repeating\u00a0decimals, and percents. Irrational\u00a0numbers <\/abbr>are numbers that cannot be written as fractions since\u00a0they are non-repeating and nonterminating decimals. A fraction\u00a0<\/em>is a ratio of two integers. The real numbers\u00a0<\/em><\/strong>are the combination of the rational and irrational numbers. The\u00a0only numbers that are not considered real numbers are the complex\u00a0<\/em><\/strong>numbers which have the form\u00a0.<\/p>\n

A prime number<\/strong><\/em> is a number divisible\u00a0only by 1 and itself. A composite number\u00a0<\/strong><\/em>is a number with factors other than 1 and itself.<\/p>\n

How do I convert between different forms of numbers?<\/p>\n

We will study three conversions:<\/p>\n

    \n
  1. between fractions and decimals,<\/li>\n
  2. between fractions and percents, and<\/li>\n
  3. between percents and decimals.<\/li>\n<\/ol>\n

    First Conversion: Between fractions and decimals<\/h3>\n

    To convert from fractions to decimals, divide the numerator by the denominator.<\/p>\n

    \"2<\/p>\n

    To convert from decimals to fractions, write the decimal as a fraction and reduce.<\/p>\n

    \"\"
    \n\"28<\/p>\n

    Second Conversion: Between fractions and percents<\/h3>\n

    Think of a percent as meaning out of 100. To convert from fractions to percents, set the fraction equal to\u00a0 and\u00a0solve for x<\/em>.<\/p>\n

    <\/p>\n

    To convert from percents to fractions, put the percent over 100 and reduce.<\/p>\n

    <\/p>\n

    Third Conversion: Between percents and decimals<\/h3>\n

    To convert from percents to decimals, move the decimal two\u00a0places to the left and drop the percent sign. If no decimal place\u00a0is shown assume the decimal place is at the right hand side of the\u00a0number.<\/p>\n

    28% = 0.28<\/p>\n

    To convert from decimals to percents, move the decimal two\u00a0places to the right and add a percent sign.<\/p>\n

    0.3 = 30%<\/p>\n

    \n

    Question<\/h4>\n
    \n

    Which of the following correctly shows 125% converted to a\u00a0fraction?<\/p>\n

      \n
    1. <\/li>\n
    2. <\/li>\n
    3. <\/li>\n
    4. <\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct answer is B. To convert the percent to a fraction,\u00a0place 125 over 100 and reduce.<\/p>\n

      <\/p>\n<\/div>\n<\/section>\n

      How are real numbers represented on a number line?<\/h3>\n

      Real numbers, decimals, fractions, and percents can be thought\u00a0of as a point on a number line, where they are placed according to\u00a0their value. The points on a number line are called coordinates,\u00a0while the zero point is called the origin. The line extends\u00a0infinitely on both sides of the origin, with positive numbers to\u00a0the right and negative numbers to the left.<\/p>\n

      \"number<\/p>\n

      \n

      Question<\/h4>\n

      Which number line shows the correct placement of 1.25,\u00a0,\u00a0250%?<\/p>\n

        \n
      1. <\/li>\n
      2. <\/li>\n
      3. <\/li>\n
      4. <\/li>\n<\/ol>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is D. First, convert all the numbers to the\u00a0same form. 1.25 is already in decimal form, so convert\u00a0\u00a0to .75 by dividing the numerator by the denominator. Then convert\u00a0250 % to 2.50, by moving the decimal two places left and removing\u00a0the percent sign. The points that should be graphed are .75,\u00a01.25, and 2.50. Choice D shows the correct placement of the\u00a0numbers on the number line.<\/p>\n<\/div>\n<\/section>\n

        How many numbers are between two real numbers?<\/h3>\n

        Between every two real numbers, there are an infinite number of\u00a0rational and irrational numbers.<\/p>\n

        Important Tidbit<\/h4>\n

        Change numbers to the same form to easily see where they fit on the number\u00a0line.<\/p>\n

        \n

        Question<\/h4>\n

        Which number line shows the correct number placement?<\/p>\n

          \n
        1. \"Number<\/li>\n
        2. \"1,<\/li>\n
        3. <\/li>\n
        4. \"1,<\/li>\n<\/ol>\n

          Reveal Answer <\/a><\/p>\n

          \n

          The correct answer is C. Each number is changed to a decimal\u00a0or a decimal approximation; the numbers become 1, 1.3, 1.4, 1.5,\u00a01.85, and 2. Place these numbers on the number line.<\/p>\n<\/div>\n<\/section>\n

          What is the associative property?<\/h3>\n

          The associative property of addition<\/abbr> states that the way you group a set of numbers does not affect the sum. For example, .<\/p>\n

          The associative property of multiplication<\/abbr> states that the way you group a set of numbers does not affect the product. For example,.<\/p>\n

          What is the distributive property?<\/h3>\n

          The distributive property<\/abbr> states that the for any numbers a, b, and c, . For example, .<\/p>\n

          What is the commutative property?<\/h3>\n

          The commutative property of addition<\/abbr> states that the order in which you add a set of numbers does not affect the sum. For example, .<\/p>\n

          The commutative property of multiplication<\/abbr> states that the way you group a set of numbers does not affect the product. For example, .<\/p>\n

          \n

          Be Aware!<\/h4>\n

          While the commutative and associative properties work for addition and multiplication, they do not work for subtraction and division. When in doubt, use a simple example to decide.<\/p>\n<\/div>\n

          What are irrational numbers?<\/h3>\n

          Irrational numbers are numbers that cannot be written as a\u00a0fraction since they are non-repeating and non-terminating numbers.\u00a0Examples of irrational numbers are\u00a0\u00a0and\u00a0.\u00a0It is helpful to use approximations of irrational numbers in\u00a0problem situations.<\/p>\n

          \n

          Question<\/h4>\n

          A support beam is needed for a 10 ft square wall. The beam will\u00a0form the diagonal of the square. About how long will the beam be?<\/p>\n

            \n
          1. 10 ft.<\/li>\n
          2. 14 ft.<\/li>\n
          3. 20 ft.<\/li>\n
          4. 28 ft.<\/li>\n<\/ol>\n

            Reveal Answer <\/a><\/p>\n

            \n

            The correct answer is B. To find the length of the diagonal of\u00a0a square, use the Pythagorean theorem. Since the legs of the\u00a0triangle have the same length, both a<\/em> and b<\/em> are 10.\u00a0Substitute into the Pythagorean Theorem and solve for c<\/em>.<\/p>\n

            <\/p>\n

            Use the approximation,\u00a0but remember that\u00a0\u00a0is not really\u00a0.<\/p>\n

            <\/p>\n<\/div>\n<\/section>\n

            <\/p>\n

            Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

            Back to Top<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

            Workshop Index\u00a0Next Lesson \u27a1 Real Number Properties Objective The following lesson will examine whole numbers, rational and irrational numbers, integers, and their properties. What are the different types of numbers? The counting numbers or natural numbers are the\u00a0numbers 1,2,3,4,…. The whole numbers are the\u00a0counting numbers plus zero. The integers\u00a0are the whole numbers and all their […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/7","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=7"}],"version-history":[{"count":16,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/7\/revisions"}],"predecessor-version":[{"id":793,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/7\/revisions\/793"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=7"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}