{"id":10878,"date":"2026-11-02T02:03:32","date_gmt":"2026-11-02T02:03:32","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10878"},"modified":"2026-11-02T02:03:32","modified_gmt":"2026-11-02T02:03:32","slug":"introduction-types-of-integers","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/introduction-types-of-integers\/","title":{"rendered":"Introduction & Types Of Integers"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Number System<\/strong><\/h2>\n

Chapter: <\/strong>Introduction & Types of Integers<\/strong><\/h3>\n

Reference: – Introduction to Real Numbers, Classification of Numbers, Rational & Irrational Numbers, Properties of Real Numbers, Operations on Real Numbers, The Number Line, Surds (Radicals), Laws of Exponents for Real Numbers, Density Property, Decimal Representation, Comparison & Ordering, Real-Life Applications<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Introduction to Real Numbers<\/li>\n
  • Rational and Irrational Numbers<\/li>\n
  • Properties & Operation on Real Numbers<\/li>\n
  • Decimal Representation & The Number Line<\/li>\n<\/ul>\n

    Introduction to Real Numbers<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Real Numbers are the set of all numbers that can be represented on a number line. They include rational numbers (such as integers, fractions, terminating and repeating decimals) and irrational numbers (such as \u221a2, \u03c0, e).<\/p>\n

    The set of real numbers is denoted by the symbol\u00a0\u211d<\/strong>.<\/p>\n

    When we classify real numbers, we essentially ask:<\/p>\n

    “What type of number is this \u2014 rational or irrational? Can it be expressed as a fraction?”<\/p>\n

    Once we identify the type, we can determine its properties, perform operations, and locate it on the number line.<\/p>\n

    Importance of Real Numbers<\/u><\/strong><\/p>\n

      \n
    • Forms the foundation of algebra, calculus, and higher mathematics.<\/li>\n
    • Enables precise measurement of continuous quantities (length, time, temperature).<\/li>\n
    • Used in science, engineering, economics, and everyday calculations.<\/li>\n
    • Bridges the gap between discrete counting numbers and continuous quantities.<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      Group:<\/strong>\u00a0{ -3, 0, \u00bd, \u221a2, \u03c0, 4.75 }
      \nCommon Property:<\/strong>\u00a0All can be placed on a number line.
      \nSo, if “\u221a-1” (imaginary number) was given, we could say it does not belong (since it is not a real number).
      \n\u00a0<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Concept of Real Numbers<\/strong><\/p>\n

      Real numbers include every number you normally use in daily life \u2014 temperatures, bank balances, measurements, and more.<\/p>\n

      Key Points:<\/strong><\/p>\n

        \n
      • Real numbers can be\u00a0positive, negative, or zero.<\/li>\n
      • They can be\u00a0rational\u00a0(fractions) or\u00a0irrational\u00a0(non-repeating, non-terminating decimals).<\/li>\n
      • Every point on the number line corresponds to exactly one real number, and vice versa \u2014 this is called the\u00a0completeness property.<\/li>\n<\/ul>\n

        2. Finding the Group Basis (Property)<\/strong><\/p>\n

        The group basis for real numbers is usually whether a number is\u00a0rational<\/strong>\u00a0or\u00a0irrational<\/strong>, or which subset (natural, whole, integer, rational, irrational) it belongs to.<\/p>\n

        Steps to Identify Real Number Subsets:<\/strong><\/p>\n

          \n
        1. Observe\u00a0the number carefully (look for decimal form, square roots, \u03c0, etc.).<\/li>\n
        2. Check\u00a0if it can be written as p\/q where p, q are integers and q \u2260 0.<\/li>\n
        3. Identify\u00a0the smallest set it belongs to (Natural \u2192 Whole \u2192 Integer \u2192 Rational \u2192 Real).<\/li>\n
        4. Apply\u00a0properties like density, closure, commutativity.<\/li>\n<\/ol>\n

          Example 1 \u2013 Classifying numbers:<\/strong>
          \nNumbers: { -2, 0, 3, \u00bd, \u221a4 }<\/p>\n

            \n
          • \u221a4 = 2, so all are rational.<\/li>\n
          • Common Property: All are rational numbers.<\/li>\n<\/ul>\n

            Example 2 \u2013 Rational vs Irrational:<\/strong>
            \nGroup: { \u221a4, \u2153, 0.75, 2 }
            \nCommon Property: All are rational (\u221a4 = 2).
            \nOdd one out in { \u221a2, \u221a3, \u221a4, \u221a5 } \u2192\u00a0\u221a4<\/strong>\u00a0(it is rational, others irrational).<\/p>\n

            Example 3 \u2013 Number line representation:<\/strong>
            \nGroup: { -1.5, 0, 2.3, \u221a2 }
            \nCommon Property: All can be plotted on a number line.<\/p>\n

            Rational Numbers<\/strong><\/p>\n

            Definition<\/u><\/strong><\/p>\n

            A rational number is any number that can be expressed in the form\u00a0p\/q, where p and q are integers and\u00a0q \u2260 0.<\/p>\n

            The set of rational numbers is denoted by\u00a0\u211a<\/strong>\u00a0(from “quotient”).<\/p>\n

            Importance of Rational Numbers<\/strong><\/p>\n

              \n
            • Used in fractions, ratios, proportions, and percentages.<\/li>\n
            • All terminating and repeating decimals are rational.<\/li>\n
            • Essential for measurements, cooking, construction, and finance.<\/li>\n<\/ul>\n

              Examples<\/strong><\/p>\n

                \n
              • Group 1:<\/strong>\u00a0{ \u00bd, \u2154, \u00be, \u215d } \u2192 All are positive fractions.<\/li>\n
              • Group 2:<\/strong>\u00a0{ -3, 0, 5, \u00bd } \u2192 All are rational numbers.<\/li>\n<\/ul>\n

                Subtopics<\/u><\/strong><\/p>\n

                1. Integers as Rational Numbers<\/strong><\/p>\n

                Every integer is rational because it can be written with denominator 1.
                \nExamples: 5 = 5\/1, -3 = -3\/1, 0 = 0\/1.<\/p>\n

                Quick Tip:<\/strong>
                \nNatural numbers (1,2,3\u2026), Whole numbers (0,1,2\u2026), and Integers (\u2026-2,-1,0,1,2\u2026) are all subsets of rational numbers.<\/p>\n

                2. Fractions (Proper and Improper)<\/strong><\/p>\n

                  \n
                • Proper fraction:<\/strong>\u00a0Numerator < denominator (e.g., \u2154, \u215e)<\/li>\n
                • Improper fraction:<\/strong>\u00a0Numerator \u2265 denominator (e.g., 5\/3, 7\/4)<\/li>\n
                • Mixed number:<\/strong>\u00a0Whole number + proper fraction (e.g., 2\u00bd)<\/li>\n<\/ul>\n

                  3. Terminating and Repeating Decimals<\/strong><\/p>\n

                    \n
                  • Terminating decimals:<\/strong>\u00a0Decimal ends after finite digits.
                    \n\tExample: 0.75 = \u00be, 0.125 = \u215b
                    \n\tReason:<\/em>\u00a0Denominator has only prime factors 2 and\/or 5.<\/li>\n
                  • Repeating (recurring) decimals:<\/strong>\u00a0Decimal repeats a pattern infinitely.
                    \n\tExample: 0.333\u2026 = \u2153, 0.142857142857\u2026 = 1\/7<\/li>\n<\/ul>\n

                    Special Note:<\/strong>
                    \nAll terminating and repeating decimals are rational numbers.<\/p>\n

                    \u00a0<\/p>\n

                    Irrational Numbers<\/strong><\/p>\n

                    Definition<\/u><\/strong><\/p>\n

                    An irrational number is a real number that\u00a0cannot\u00a0be expressed as p\/q, where p and q are integers and q \u2260 0. Its decimal expansion is\u00a0non-terminating and non-repeating.<\/p>\n

                    The set of irrational numbers has no standard symbol but is often written as\u00a0\u211a<\/strong>‘<\/strong>\u00a0or\u00a0\u211d<\/strong> <\/strong>\u211a<\/strong>.<\/p>\n

                    Importance of Irrational Numbers<\/strong><\/p>\n

                      \n
                    • Essential for geometry (diagonals, circles, spirals).<\/li>\n
                    • Appear in physics (\u03c0 in waves, e in growth\/decay).<\/li>\n
                    • Show that the number line has “gaps” that fractions cannot fill.<\/li>\n<\/ul>\n

                      Examples<\/strong><\/p>\n

                        \n
                      • Group 1:<\/strong>\u00a0{ \u221a2, \u221a3, \u221a5, \u221a7 } \u2192 All are square roots of non-perfect squares.<\/li>\n
                      • Group 2:<\/strong>\u00a0{ \u03c0, e, \u03c6 (golden ratio = (1+\u221a5)\/2) } \u2192 Famous mathematical constants.<\/li>\n<\/ul>\n

                        Subtopics<\/u><\/strong><\/p>\n

                        1. Square Roots of Non-Perfect Squares<\/strong><\/p>\n

                        Numbers like \u221a2, \u221a3, \u221a5, \u221a6, \u221a7, \u221a8, \u221a10, etc., are irrational.<\/p>\n

                        Quick Check:<\/strong>
                        \nIf a positive integer is\u00a0not\u00a0a perfect square (1,4,9,16,25\u2026), its square root is irrational.<\/p>\n

                        2. Cube Roots and Higher Roots<\/strong><\/p>\n

                        Similarly, \u221b2, \u221b3, \u221b5, etc. (where the radicand is not a perfect cube) are irrational.<\/p>\n

                        3. Famous Irrational Constants<\/strong><\/p>\n

                          \n
                        • \u03c0 (pi)<\/strong>\u00a0\u2248 3.1415926535\u2026 (ratio of circumference to diameter of a circle)<\/li>\n
                        • e (Euler’s number)<\/strong>\u00a0\u2248 2.7182818284\u2026 (base of natural logarithms)<\/li>\n
                        • \u03c6 (golden ratio)<\/strong>\u00a0\u2248 1.6180339887\u2026<\/li>\n<\/ul>\n

                          4. Sums and Products Involving Irrationals<\/strong><\/p>\n

                            \n
                          • \u221a2 + \u221a3 is irrational.<\/li>\n
                          • \u221a2 \u00d7 \u221a3 = \u221a6 is irrational.<\/li>\n
                          • But \u221a2 \u00d7 \u221a2 = 2 (rational) \u2014 irrational \u00d7 irrational can be rational.<\/li>\n<\/ul>\n

                            Special Note:<\/strong>
                            \n\u03c0 and e are\u00a0transcendental numbers<\/strong>\u00a0(a stronger type of irrational \u2014 not roots of any polynomial with integer coefficients). \u221a2 is algebraic irrational.<\/p>\n

                            \u00a0<\/p>\n

                            Decimal Representation of Real Numbers<\/strong><\/p>\n

                            Definition<\/u><\/strong><\/p>\n

                            Every real number has a unique decimal representation (except that terminating decimals can also be written as repeating 9’s).<\/p>\n

                            \n\n\n\n\n\n\n\n
                            \n

                            Type of Number<\/p>\n<\/td>\n

                            		\n

                            Decimal Form<\/p>\n<\/td>\n

                            		\n

                            Example<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                            \n

                            Rational (terminating)<\/p>\n<\/td>\n

                            		\n

                            Ends after finite digits<\/p>\n<\/td>\n

                            		\n

                            0.75, 2.5, 3.0<\/p>\n<\/td>\n<\/tr>\n

                            \n

                            Rational (repeating)<\/p>\n<\/td>\n

                            		\n

                            Infinite repeating pattern<\/p>\n<\/td>\n

                            		\n

                            overline{3}, overline{6}<\/p>\n<\/td>\n<\/tr>\n

                            \n

                            Irrational<\/p>\n<\/td>\n

                            		\n

                            Infinite, no repeating pattern<\/p>\n<\/td>\n

                            		\n

                            1.41421356\u2026 (\u221a2)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                            Source: Kapdec.com<\/p>\n<\/div>\n

                            Quick Rule:<\/strong><\/p>\n

                              \n
                            • If decimal\u00a0terminates\u00a0or\u00a0repeats\u00a0\u2192\u00a0Rational<\/li>\n
                            • If decimal\u00a0never terminates and never repeats\u00a0\u2192\u00a0Irrational<\/li>\n<\/ul>\n

                              \u00a0<\/p>\n

                              Properties of Real Numbers<\/strong><\/p>\n

                              Real numbers follow several important properties under addition and multiplication.<\/p>\n

                              \n\n\n\n\n\n\n\n\n\n\n
                              \n

                              Property<\/p>\n<\/td>\n

                              		\n

                              Addition ( + )<\/p>\n<\/td>\n

                              		\n

                              Multiplication ( \u00d7 )<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                              \n

                              Closure<\/strong><\/p>\n<\/td>\n

                              		\n

                              a + b is real<\/p>\n<\/td>\n

                              		\n

                              a \u00d7 b is real<\/p>\n<\/td>\n<\/tr>\n

                              \n

                              Commutative<\/strong><\/p>\n<\/td>\n

                              		\n

                              a + b = b + a<\/p>\n<\/td>\n

                              		\n

                              a \u00d7 b = b \u00d7 a<\/p>\n<\/td>\n<\/tr>\n

                              \n

                              Associative<\/strong><\/p>\n<\/td>\n

                              		\n

                              (a+b)+c = a+(b+c)<\/p>\n<\/td>\n

                              		\n

                              (a\u00d7b)\u00d7c = a\u00d7(b\u00d7c)<\/p>\n<\/td>\n<\/tr>\n

                              \n

                              Identity<\/strong><\/p>\n<\/td>\n

                              		\n

                              a + 0 = a<\/p>\n<\/td>\n

                              		\n

                              a \u00d7 1 = a<\/p>\n<\/td>\n<\/tr>\n

                              \n

                              Inverse<\/strong><\/p>\n<\/td>\n

                              		\n

                              a + (-a) = 0<\/p>\n<\/td>\n

                              		\n

                              a \u00d7 (1\/a) = 1 (a \u2260 0)<\/p>\n<\/td>\n<\/tr>\n

                              \n

                              Distributive<\/strong><\/p>\n<\/td>\n

                              		\n

                              a \u00d7 (b + c) = a\u00d7b + a\u00d7c<\/p>\n<\/td>\n

                              		\n

                              1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                              Source: Kapdec.com<\/p>\n<\/div>\n

                              Key Point:<\/strong>
                              \nReal numbers are\u00a0closed\u00a0under addition, subtraction, multiplication, and division (except division by zero).<\/p>\n

                              Operations on Real Numbers<\/strong><\/p>\n

                              1. Addition and Subtraction<\/strong>
                              \nCombine like terms. For irrationals, only combine if the irrational part is identical.<\/p>\n

                              Example: 3\u221a2 + 5\u221a2 = 8\u221a2
                              \nBut 3\u221a2 + 4\u221a3 cannot be simplified further.<\/p>\n

                              2. Multiplication<\/strong>
                              \nMultiply coefficients and multiply radicands separately.<\/p>\n

                              Example: (2\u221a3)(5\u221a6) = 10\u221a18 = 10 \u00d7 3\u221a2 = 30\u221a2<\/p>\n

                              3. Division<\/strong>
                              \nRationalize the denominator when needed.<\/p>\n

                              Example: 1\/\u221a2 = \u221a2\/2<\/p>\n

                              4. Rationalisation<\/strong>
                              \nProcess of removing a radical from the denominator using conjugate.<\/p>\n

                              Example: 1\/(\u221a3+\u221a2) = (\u221a3\u2212\u221a2)\/((\u221a3+\u221a2)(\u221a3\u2212\u221a2)) = (\u221a3\u2212\u221a2)\/(3\u22122) = \u221a3\u2212\u221a2<\/p>\n

                              \u00a0<\/p>\n

                              The Number Line<\/strong><\/p>\n

                              Every real number corresponds to exactly one point on the number line.<\/p>\n

                              Density Property:<\/strong>
                              \nBetween any two distinct real numbers, there exists infinitely many rational numbers\u00a0and<\/strong>\u00a0infinitely many irrational numbers.<\/p>\n

                              \u00a0<\/p>\n

                              Surds (Radicals)<\/strong><\/p>\n

                              Definition<\/u><\/strong><\/p>\n

                              A surd is an irrational root of a rational number.
                              \nExample: \u221a2, \u221b5, \u221a(10) are surds.
                              \n\u221a4 = 2 is\u00a0not<\/strong>\u00a0a surd (it’s rational).<\/p>\n

                              Types of Surds:<\/strong><\/p>\n

                                \n
                              • Pure surd:<\/strong>\u00a0\u221aa where a has no factor that is a perfect power. Example: \u221a3<\/li>\n
                              • Mixed surd:<\/strong>\u00a0k\u221aa where k is rational. Example: 2\u221a3<\/li>\n<\/ul>\n

                                Operations with Surds:<\/strong><\/p>\n

                                  \n
                                • Addition:<\/strong>\u00a03\u221a5 + 2\u221a5 = 5\u221a5<\/li>\n
                                • Multiplication:<\/strong>\u00a0\u221aa \u00d7 \u221ab = \u221a(ab)<\/li>\n
                                • Division:<\/strong>\u00a0\u221aa \/ \u221ab = \u221a(a\/b)<\/li>\n<\/ul>\n

                                  Simplification:<\/strong>\u00a0\u221a72 = \u221a(36\u00d72) = 6\u221a2<\/p>\n

                                  \u00a0<\/p>\n

                                  Laws of Exponents for Real Numbers<\/strong><\/p>\n

                                  For a, b > 0 (real numbers) and rational exponents p, q:<\/p>\n

                                  \n\n\n\n\n\n\n\n\n\n\n
                                  \n

                                  Law<\/p>\n<\/td>\n

                                  		\n

                                  Example<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                                  \n

                                  a\u1d56 \u00d7 a\u1d60 = a\u1d56\u207a\u1d60<\/p>\n<\/td>\n

                                  		\n

                                  2\u00b3 \u00d7 2\u00b2 = 2\u2075 = 32<\/p>\n<\/td>\n<\/tr>\n

                                  \n

                                  (a\u1d56)\u1d60 = a\u1d56\u1d60<\/p>\n<\/td>\n

                                  		\n

                                  (2\u00b3)\u00b2 = 2\u2076 = 64<\/p>\n<\/td>\n<\/tr>\n

                                  \n

                                  (ab)\u1d56 = a\u1d56 b\u1d56<\/p>\n<\/td>\n

                                  		\n

                                  (4\u00d79)\u00bd = 4\u00bd \u00d7 9\u00bd = 2\u00d73 = 6<\/p>\n<\/td>\n<\/tr>\n

                                  \n

                                  a\u1d56 \/ a\u1d60 = a\u1d56\u207b\u1d60<\/p>\n<\/td>\n

                                  		\n

                                  3\u2075 \/ 3\u00b2 = 3\u00b3 = 27<\/p>\n<\/td>\n<\/tr>\n

                                  \n

                                  a\u207b\u1d56 = 1\/a\u1d56<\/p>\n<\/td>\n

                                  		\n

                                  2\u207b\u00b2 = \u00bc<\/p>\n<\/td>\n<\/tr>\n

                                  \n

                                  a^(p\/q) = \u1d60\u221a(a\u1d56)<\/p>\n<\/td>\n

                                  		\n

                                  8^(\u2154) = (8\u00b2)\u2153 = 64\u2153 = 4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \u00a0<\/p>\n

                                  Comparison & Ordering of Real Numbers<\/strong><\/p>\n

                                  To compare two real numbers:<\/strong><\/p>\n

                                    \n
                                  1. If both rational \u2192 convert to decimals or common denominator.<\/li>\n
                                  2. If one irrational \u2192 approximate decimal value.<\/li>\n
                                  3. Use square comparison: For a,b > 0, if a\u00b2 > b\u00b2 then a > b.<\/li>\n<\/ol>\n

                                    Example:<\/strong>\u00a0Compare \u221a7 and 2.8
                                    \n\u221a7 \u2248 2.64575 < 2.8<\/p>\n

                                    Example:<\/strong>\u00a0Compare \u221a5 + \u221a3 and \u221a6 + \u221a2
                                    \nSquare both sides \u2192 (\u221a5+\u221a3)\u00b2 = 5+3+2\u221a15 = 8+2\u221a15 \u2248 8+7.746=15.746
                                    \n(\u221a6+\u221a2)\u00b2 = 6+2+2\u221a12 = 8+2\u221a12 \u2248 8+6.928=14.928
                                    \nSince 15.746 > 14.928, \u221a5+\u221a3 > \u221a6+\u221a2.<\/p>\n

                                    \u00a0<\/p>\n

                                    Real-Life Applications of Real Numbers<\/strong><\/p>\n

                                      \n
                                    • Measurements:<\/strong>\u00a0Length, weight, volume, temperature (all continuous quantities)<\/li>\n
                                    • Time:<\/strong>\u00a0Hours, minutes, seconds (and fractional seconds)<\/li>\n
                                    • Finance:<\/strong>\u00a0Interest rates, stock prices, currency exchange<\/li>\n
                                    • Science:<\/strong>\u00a0Velocity, acceleration, force, energy<\/li>\n
                                    • Engineering:<\/strong>\u00a0Dimensions, tolerances, stress\/strain calculations<\/li>\n
                                    • Medicine:<\/strong>\u00a0Dosages, vital signs, lab results<\/li>\n
                                    • Everyday life:<\/strong>\u00a0Speed, distance, fuel efficiency, cooking measurements<\/li>\n<\/ul>\n

                                      \u00a0<\/p>\n

                                      Example Problem Set \u2013 Odd One Out (Classification Style)<\/strong><\/p>\n

                                      Examine the six numbers below. Exactly one does NOT belong with the rest. Identify it and give three independent reasons (A) rational\/irrational classification, (B) decimal expansion property, (C) algebraic \/ surd simplification property.<\/strong><\/p>\n

                                      Items:<\/strong><\/p>\n

                                        \n
                                      1. \u221a16<\/li>\n
                                      2. \u221a2<\/li>\n
                                      3. 0.333…<\/li>\n
                                      4. 22\/7<\/li>\n
                                      5. 1.41421356…<\/li>\n
                                      6. (\u221a3)\u00b2<\/li>\n<\/ol>\n

                                        Solution:<\/strong><\/p>\n

                                        (A) Rational \/ Irrational Classification<\/strong><\/p>\n

                                          \n
                                        • \u221a16 = 4 \u2192 Rational<\/li>\n
                                        • \u221a2 \u2192 Irrational<\/li>\n
                                        • 0.333\u2026 = \u2153 \u2192 Rational<\/li>\n
                                        • 22\/7 \u2192 Rational<\/li>\n
                                        • 1.41421356\u2026 (non-terminating, no pattern) \u2192 Irrational<\/li>\n
                                        • (\u221a3)\u00b2 = 3 \u2192 Rational<\/li>\n<\/ul>\n

                                          So \u221a2 and 1.41421356\u2026 are irrational; others rational.\u00a0This alone doesn’t single out one.<\/em><\/p>\n

                                          (B) Decimal Expansion Property<\/strong><\/p>\n

                                            \n
                                          • \u221a16 = 4.000\u2026 (terminating)<\/li>\n
                                          • \u221a2 = 1.41421356\u2026 (non-terminating, non-repeating)<\/li>\n
                                          • 0.333\u2026 = 0.{3} (repeating)<\/li>\n
                                          • 22\/7 = 3.142857142857\u2026 (repeating pattern of length 6)<\/li>\n
                                          • 1.41421356\u2026 (non-terminating, non-repeating)<\/li>\n
                                          • (\u221a3)\u00b2 = 3.000\u2026 (terminating)<\/li>\n<\/ul>\n

                                            Here,\u00a0two numbers<\/em>\u00a0(\u221a2 and 1.41421356\u2026) share non-terminating non-repeating property. Still not unique.<\/p>\n

                                            (C) Surd Simplification \/ Exact form<\/strong><\/p>\n

                                              \n
                                            • \u221a16 = 4 (exact integer)<\/li>\n
                                            • \u221a2 = surd (cannot simplify)<\/li>\n
                                            • 0.333\u2026 = \u2153 (exact fraction)<\/li>\n
                                            • 22\/7 = exact fraction<\/li>\n
                                            • 1.41421356\u2026 =\u00a0approximation<\/strong>\u00a0of \u221a2, not exact representation<\/li>\n
                                            • (\u221a3)\u00b2 = 3 (exact integer)<\/li>\n<\/ul>\n

                                              Conclusion:<\/strong>
                                              \nThe number\u00a01.41421356…\u00a0is the odd one out because it is presented as a\u00a0decimal approximation\u00a0of \u221a2 rather than in its exact surd form \u221a2. All others are given in exact form (integer, fraction, repeating decimal with bar notation, or surd symbol). This is a semantic\/representational oddity.<\/p>\n

                                              Thus the odd item is\u00a0Item 5 (1.41421356…).<\/p>\n

                                              \u00a0<\/p>\n

                                              <\/p>\n