{"id":1799,"date":"2024-06-06T22:54:08","date_gmt":"2024-06-06T22:54:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1799"},"modified":"2024-11-21T17:45:17","modified_gmt":"2024-11-21T17:45:17","slug":"module-8-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-8-background-youll-need-2\/","title":{"raw":"Quadratic Functions: Background You'll Need 2","rendered":"Quadratic Functions: Background You’ll Need 2"},"content":{"raw":"
Identify and understand the differences between rational numbers (like fractions and whole numbers) and irrational numbers (like pi and square roots)<\/span><\/section>\r\n

Exploring Number Types: Rational, Irrational, and Real Numbers<\/h2>\r\nNumbers in mathematics are sorted into different types such as, rational, irrational, and real numbers. Rational numbers are fractions with integers on top and bottom, like [latex]\u00bd[\/latex] or [latex]-3\/4[\/latex]. Irrational numbers can't be neatly written as fractions because their decimals go on endlessly without repeating\u2014think of [latex]\u03c0[\/latex] or the square root of [latex]2[\/latex]. Both of these types are part of the real numbers, which make up the number line we use for all basic math. This page will guide you through these concepts, starting with rational numbers.\r\n\r\n
You should already know about the other number types -\u00a0 counting numbers, whole numbers and integers.\r\n
    \r\n \t
  • Counting numbers<\/strong>, also known as natural numbers, are the numbers we use to count items: [latex]1, 2, 3, [\/latex] and so on. They are a subset of whole numbers<\/strong>, which extend counting numbers to include [latex]0[\/latex], forming the set ([latex]0, 1, 2, 3,[\/latex] ...).<\/li>\r\n \t
  • Integers<\/strong> further broaden this scope by incorporating their negative counterparts, resulting in an uninterrupted sequence (... [latex]-3, -2, -1, 0, 1, 2, 3,[\/latex] ...). These foundational elements serve as the groundwork for rational numbers, since any counting number, whole number, or integer can be expressed as a fraction with one as the denominator.<\/li>\r\n<\/ul>\r\n<\/section>\r\n

    Rational Numbers<\/h3>\r\n
    \r\n
    \r\n

    rational numbers<\/h3>\r\nA rational number<\/strong> is a number that can be written in the form [latex]{\\Large\\frac{p}{q}}[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]q\\ne o[\/latex].\r\n\r\n<\/div>\r\n<\/section>Rational numbers are the counts and measures we encounter in everyday life. Whether it's in dividing a pizza into equal slices (fractions) or measuring the distance between two points (decimals), these numbers are all around us. Each can be expressed as a fraction, with both the numerator and denominator being whole numbers and the denominator never being zero. Let's put this into practice and express the following values as ratios of two integers.\r\n\r\n
    [ohm2_question hide_question_numbers=1]2861[\/ohm2_question]<\/section>\r\n

    Irrational Numbers<\/h3>\r\n
    \r\n
    \r\n

    irrational number<\/h3>\r\nAn irrational number<\/strong> is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.\r\n\r\n<\/div>\r\n<\/section>
    Let's summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number:\r\n
      \r\n \t
    • stops or repeats, the number is rational.<\/li>\r\n \t
    • does not stop and does not repeat, the number is irrational.<\/li>\r\n<\/ul>\r\n<\/section>
      [ohm2_question hide_question_numbers=1]2863[\/ohm2_question]<\/section>\r\n

      Real Numbers<\/h3>\r\n
      \r\n
      \r\n

      real number<\/h3>\r\nReal numbers<\/strong> are numbers that are either rational or irrational.\r\n\r\n<\/div>\r\n<\/section>
      Determine whether each of the numbers in the following list is a\r\n
        \r\n \t
      1. whole number<\/li>\r\n \t
      2. integer<\/li>\r\n \t
      3. rational number<\/li>\r\n \t
      4. irrational number<\/li>\r\n \t
      5. real number<\/li>\r\n<\/ol>\r\n
        [latex]-7,\\Large\\frac{14}{5}\\normalsize ,8,\\sqrt{5},5.9,-\\sqrt{64}[\/latex]<\/center>\r\n[reveal-answer q=\"688739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688739\"]\r\n
          \r\n \t
        1. The whole numbers are [latex]0,1,2,3\\dots[\/latex] The number [latex]8[\/latex] is the only whole number given.<\/li>\r\n \t
        2. The integers are the whole numbers, their opposites, and [latex]0[\/latex]. From the given numbers, [latex]-7[\/latex] and [latex]8[\/latex] are integers. Also, notice that [latex]64[\/latex] is the square of [latex]8[\/latex] so [latex]-\\sqrt{64}=-8[\/latex]. So the integers are [latex]-7,8,-\\sqrt{64}[\/latex].<\/li>\r\n \t
        3. Since all integers are rational, the numbers [latex]-7,8,\\text{and}-\\sqrt{64}[\/latex] are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so [latex]\\Large\\frac{14}{5}\\normalsize\\text{ and }5.9[\/latex] are rational.<\/li>\r\n \t
        4. The number [latex]5[\/latex] is not a perfect square, so [latex]\\sqrt{5}[\/latex] is irrational.<\/li>\r\n \t
        5. All of the numbers listed are real.<\/li>\r\n<\/ol>\r\nWe'll summarize the results in a table.\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
          Number<\/th>\r\nWhole<\/th>\r\nInteger<\/th>\r\nRational<\/th>\r\nIrrational<\/th>\r\nReal<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
          [latex]-7[\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]\\Large\\frac{14}{5}[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]8[\/latex]<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]\\sqrt{5}[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]5.9[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]-\\sqrt{64}[\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/td>\r\n[latex]\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>
          [ohm2_question hide_question_numbers=1]2865[\/ohm2_question]<\/section>","rendered":"
          Identify and understand the differences between rational numbers (like fractions and whole numbers) and irrational numbers (like pi and square roots)<\/span><\/section>\n

          Exploring Number Types: Rational, Irrational, and Real Numbers<\/h2>\n

          Numbers in mathematics are sorted into different types such as, rational, irrational, and real numbers. Rational numbers are fractions with integers on top and bottom, like [latex]\u00bd[\/latex] or [latex]-3\/4[\/latex]. Irrational numbers can’t be neatly written as fractions because their decimals go on endlessly without repeating\u2014think of [latex]\u03c0[\/latex] or the square root of [latex]2[\/latex]. Both of these types are part of the real numbers, which make up the number line we use for all basic math. This page will guide you through these concepts, starting with rational numbers.<\/p>\n

          You should already know about the other number types –\u00a0 counting numbers, whole numbers and integers.<\/p>\n
            \n
          • Counting numbers<\/strong>, also known as natural numbers, are the numbers we use to count items: [latex]1, 2, 3,[\/latex] and so on. They are a subset of whole numbers<\/strong>, which extend counting numbers to include [latex]0[\/latex], forming the set ([latex]0, 1, 2, 3,[\/latex] …).<\/li>\n
          • Integers<\/strong> further broaden this scope by incorporating their negative counterparts, resulting in an uninterrupted sequence (… [latex]-3, -2, -1, 0, 1, 2, 3,[\/latex] …). These foundational elements serve as the groundwork for rational numbers, since any counting number, whole number, or integer can be expressed as a fraction with one as the denominator.<\/li>\n<\/ul>\n<\/section>\n

            Rational Numbers<\/h3>\n
            \n
            \n

            rational numbers<\/h3>\n

            A rational number<\/strong> is a number that can be written in the form [latex]{\\Large\\frac{p}{q}}[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]q\\ne o[\/latex].<\/p>\n<\/div>\n<\/section>\n

            Rational numbers are the counts and measures we encounter in everyday life. Whether it’s in dividing a pizza into equal slices (fractions) or measuring the distance between two points (decimals), these numbers are all around us. Each can be expressed as a fraction, with both the numerator and denominator being whole numbers and the denominator never being zero. Let’s put this into practice and express the following values as ratios of two integers.<\/p>\n