{"id":9134,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9134"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"properties-of-irrational-number","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/properties-of-irrational-number\/","title":{"rendered":"Properties Of Irrational Number"},"content":{"rendered":"

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

Chapter: <\/strong>Properties of Irrational Numbers<\/strong><\/h3>\n

Reference: – Introduction to Irrational Numbers, Closure Properties (Addition, Subtraction, Multiplication, Division), Commutative & Associative Properties, Distributive Property, Density Property, Comparison Properties, Properties of Square Roots, Key Differences from Rational, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

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

    \n
  • Introduction to Properties of Irrational Numbers<\/em><\/li>\n
  • Closure Properties<\/em><\/li>\n
  • Density & Comparison Properties<\/em><\/li>\n
  • Key Differences Between Rational & Irrational Numbers<\/em><\/li>\n<\/ul>\n

    Introduction to Properties of Irrational Numbers<\/strong><\/p>\n

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

    Irrational numbers are real numbers that cannot be expressed as p\/q (where p, q are integers, q ≠ 0). Their decimal expansions are non-terminating and non-repeating.<\/p>\n

    Examples: √2, √3, π, e, φ (golden ratio ≈ 1.618)<\/p>\n

    When we study properties of irrational numbers, we essentially ask:<\/p>\n

    "Do rational number properties (closure, commutativity, etc.) also hold for irrational numbers?"<\/p>\n

    The answer is: Some do, some don't.<\/p>\n

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

      \n
    • Helps understand the complete real number system<\/li>\n
    • Essential for advanced mathematics (calculus, analysis)<\/li>\n
    • Clarifies why irrationals are "between" rational<\/li>\n
    • Prevents common mistakes in algebraic manipulations<\/li>\n<\/ul>\n

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

      Group:<\/strong> { √2, √3, √5, √7 }
      \nCommon Property:<\/strong> All are irrational (square roots of non-perfect squares).
      \nSo, if "√4 = 2" was given, it would not belong (it is rational).<\/p>\n

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

      1. What Properties Do Irrationals Share with Rationals?<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
      \n

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

      		\n

      Rationals<\/p>\n<\/td>\n

      		\n

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

      \n

      Commutative (+)<\/strong><\/p>\n<\/td>\n

      		\n

      \u2705 a+b = b+a<\/p>\n<\/td>\n

      		\n

      \u2705 √2+√3 = √3+√2<\/p>\n<\/td>\n<\/tr>\n

      \n

      Commutative (×)<\/strong><\/p>\n<\/td>\n

      		\n

      \u2705 a×b = b×a<\/p>\n<\/td>\n

      		\n

      \u2705 √2×√3 = √3×√2<\/p>\n<\/td>\n<\/tr>\n

      \n

      Associative (+)<\/strong><\/p>\n<\/td>\n

      		\n

      \u2705 (a+b)+c = a+(b+c)<\/p>\n<\/td>\n

      		\n

      \u2705 (√2+√3)+√5 = √2+(√3+√5)<\/p>\n<\/td>\n<\/tr>\n

      \n

      Associative (×)<\/strong><\/p>\n<\/td>\n

      		\n

      \u2705 (a×b)×c = a×(b×c)<\/p>\n<\/td>\n

      		\n

      \u2705 (√2×√3)×√5 = √2×(√3×√5)<\/p>\n<\/td>\n<\/tr>\n

      \n

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

      		\n

      \u2705 a×(b+c)=a×b+a×c<\/p>\n<\/td>\n

      		\n

      \u2705 √2×(√3+√5)=√2×√3+√2×√5<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Key Point:<\/strong> Irrationals behave like rationals under commutative, associative, and distributive laws.<\/p>\n

      Closure Properties (The Critical Differences)<\/strong><\/p>\n

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

      Closure means: When you perform an operation on two numbers from a set, the result stays in that set.<\/p>\n

      For Irrationals: NOT closed under any operation!<\/strong><\/p>\n\n\n\n\n\n\n\n\n
      \n

      Operation<\/p>\n<\/td>\n

      		\n

      Closed?<\/p>\n<\/td>\n

      		\n

      Why?<\/p>\n<\/td>\n

      		\n

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

      \n

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

      		\n

      \u274c No<\/p>\n<\/td>\n

      		\n

      Sum can be rational<\/p>\n<\/td>\n

      		\n

      √2 + (-√2) = 0 (rational)<\/p>\n<\/td>\n<\/tr>\n

      \n

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

      		\n

      \u274c No<\/p>\n<\/td>\n

      		\n

      Difference can be rational<\/p>\n<\/td>\n

      		\n

      √5 – √5 = 0 (rational)<\/p>\n<\/td>\n<\/tr>\n

      \n

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

      		\n

      \u274c No<\/p>\n<\/td>\n

      		\n

      Product can be rational<\/p>\n<\/td>\n

      		\n

      √2 × √2 = 2 (rational)<\/p>\n<\/td>\n<\/tr>\n

      \n

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

      		\n

      \u274c No<\/p>\n<\/td>\n

      		\n

      Quotient can be rational<\/p>\n<\/td>\n

      		\n

      √8 ÷ √2 = √4 = 2 (rational)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Special Cases to Remember<\/strong><\/p>\n

      Case 1: Sum of Two Irrationals<\/strong><\/p>\n\n\n\n\n\n\n\n\n
      \n

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

      		\n

      Result<\/p>\n<\/td>\n

      		\n

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

      \n

      √2 + √3<\/p>\n<\/td>\n

      		\n

      ≈ 3.146<\/p>\n<\/td>\n

      		\n

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

      \n

      √2 + (-√2)<\/p>\n<\/td>\n

      		\n

      0<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n

      \n

      (1+√2) + (1-√2)<\/p>\n<\/td>\n

      		\n

      2<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n

      \n

      √2 + √8 = √2 + 2√2<\/p>\n<\/td>\n

      		\n

      3√2<\/p>\n<\/td>\n

      		\n

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

      Case 2: Product of Two Irrationals<\/strong><\/p>\n\n\n\n\n\n\n\n\n
      \n

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

      		\n

      Result<\/p>\n<\/td>\n

      		\n

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

      \n

      √2 × √3<\/p>\n<\/td>\n

      		\n

      √6<\/p>\n<\/td>\n

      		\n

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

      \n

      √2 × √2<\/p>\n<\/td>\n

      		\n

      2<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n

      \n

      √2 × √8 = √16<\/p>\n<\/td>\n

      		\n

      4<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n

      \n

      (√5 + 1) × (√5 – 1)<\/p>\n<\/td>\n

      		\n

      5 – 1 = 4<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Case 3: Rational × Irrational<\/strong><\/p>\n\n\n\n\n\n\n\n
      \n

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

      		\n

      Result<\/p>\n<\/td>\n

      		\n

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

      \n

      2 × √2<\/p>\n<\/td>\n

      		\n

      2√2<\/p>\n<\/td>\n

      		\n

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

      \n

      0 × √2<\/p>\n<\/td>\n

      		\n

      0<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n

      \n

      5 × π<\/p>\n<\/td>\n

      		\n

      5π<\/p>\n<\/td>\n

      		\n

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

      Case 4: Irrational ÷ Irrational<\/strong><\/p>\n\n\n\n\n\n\n\n
      \n

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

      		\n

      Result<\/p>\n<\/td>\n

      		\n

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

      \n

      √6 ÷ √2<\/p>\n<\/td>\n

      		\n

      √3<\/p>\n<\/td>\n

      		\n

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

      \n

      √8 ÷ √2<\/p>\n<\/td>\n

      		\n

      √4 = 2<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n

      \n

      π ÷ π<\/p>\n<\/td>\n

      		\n

      1<\/p>\n<\/td>\n

      		\n

      Rational<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

       <\/p>\n

      Density Property<\/strong><\/p>\n

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

      Between any two distinct irrational numbers, there exists:<\/p>\n

        \n
      1. Infinitely many irrational numbers<\/strong><\/li>\n
      2. Infinitely many rational numbers<\/strong><\/li>\n<\/ol>\n

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

        Between √2 (≈1.4142) and √3 (≈1.7320):<\/p>\n

          \n
        • Irrational between: √2.5 ≈ 1.581 (since 2.5 is not perfect square)<\/li>\n
        • Rational between: 1.5 = 3\/2, 1.6 = 8\/5<\/li>\n<\/ul>\n

          Key Point:<\/strong> Both rationals and irrationals are dense on the number line. Neither has "gaps" — they are interwoven.<\/p>\n

          Comparison Properties<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
          \n

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

          		\n

          Statement<\/p>\n<\/td>\n

          		\n

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

          \n

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

          		\n

          Exactly one of: a < b, a = b, a > b<\/p>\n<\/td>\n

          		\n

          √2 < √3 (true)<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          If a < b and b < c, then a < c<\/p>\n<\/td>\n

          		\n

          √2 < √5 and √5 < π ⇒ √2 < π<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          If a < b, then a + c < b + c<\/p>\n<\/td>\n

          		\n

          √2 < √3 ⇒ √2+1 < √3+1<\/p>\n<\/td>\n<\/tr>\n

          \n

          Multiplication Property<\/strong> (c > 0)<\/p>\n<\/td>\n

          		\n

          If a < b, then ac < bc<\/p>\n<\/td>\n

          		\n

          √2 < √3 ⇒ 2√2 < 2√3<\/p>\n<\/td>\n<\/tr>\n

          \n

          Multiplication Property<\/strong> (c < 0)<\/p>\n<\/td>\n

          		\n

          If a < b, then ac > bc (reverses)<\/p>\n<\/td>\n

          		\n

          √2 < √3 ⇒ -2√2 > -2√3<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

           <\/p>\n

          Properties of Square Roots (Common Irrationals)<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n
          \n

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

          		\n

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

          \n

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

          		\n

          √6 = √2 × √3<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          √(2\/3) = √2\/√3<\/p>\n<\/td>\n<\/tr>\n

          \n

          (√a)² = a<\/p>\n<\/td>\n

          		\n

          (√2)² = 2<\/p>\n<\/td>\n<\/tr>\n

          \n

          √a² = a (for a > 0)<\/p>\n<\/td>\n

          		\n

          √(3²) = 3<\/p>\n<\/td>\n<\/tr>\n

          \n

          √a + √b ≠ √(a+b)<\/p>\n<\/td>\n

          		\n

          √2 + √3 ≠ √5<\/p>\n<\/td>\n<\/tr>\n

          \n

          √a – √b ≠ √(a-b)<\/p>\n<\/td>\n

          		\n

          √5 – √3 ≠ √2<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Common Mistake Alert:<\/strong> √(a+b) = √a + √b is FALSE for irrationals (and most rationals too).<\/p>\n

           <\/p>\n

          Key Differences: Rationals vs Irrationals<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n
          \n

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

          		\n

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

          		\n

          Irrational Numbers<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

          \n

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

          		\n

          \u2705 Yes<\/p>\n<\/td>\n

          		\n

          \u274c No<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          \u2705 Yes<\/p>\n<\/td>\n

          		\n

          \u274c No<\/p>\n<\/td>\n<\/tr>\n

          \n

          Decimal form<\/strong><\/p>\n<\/td>\n

          		\n

          Terminating or repeating<\/p>\n<\/td>\n

          		\n

          Non-terminating, non-repeating<\/p>\n<\/td>\n<\/tr>\n

          \n

          Can be written as p\/q<\/strong><\/p>\n<\/td>\n

          		\n

          \u2705 Yes<\/p>\n<\/td>\n

          		\n

          \u274c No<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          Dense (between any two)<\/p>\n<\/td>\n

          		\n

          Dense (between any two)<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          Countable (ℵ\u2080)<\/p>\n<\/td>\n

          		\n

          Uncountable<\/p>\n<\/td>\n<\/tr>\n

          \n

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

          		\n

          1\/a exists (a≠0)<\/p>\n<\/td>\n

          		\n

          1\/a exists and is irrational (except if a=√2\/√2 type)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

           <\/p>\n

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

          Example 1:<\/strong> Is the sum of √2 and 1\/√2 rational or irrational?<\/p>\n

          Solution:<\/strong> √2 + 1\/√2 = √2 + √2\/2 = (2√2\/2 + √2\/2) = 3√2\/2 → Irrational<\/p>\n

          Answer:<\/strong> Irrational<\/p>\n

           <\/p>\n

          Example 2:<\/strong> Give an example to show that irrational numbers are NOT closed under multiplication.<\/p>\n

          Solution:<\/strong> √3 × √3 = 3 (rational)<\/p>\n

          Answer:<\/strong> √3 × √3 = 3 (product is rational, not irrational)<\/p>\n

           <\/p>\n

          Example 3:<\/strong> Find a rational number between √5 and √6.<\/p>\n

          Solution:<\/strong> √5 ≈ 2.236, √6 ≈ 2.449 → Choose 2.4 = 12\/5 = 2.4<\/p>\n

          Answer:<\/strong> 12\/5<\/p>\n

          Example 4:<\/strong> Is π ÷ e rational or irrational? (π and e are irrational constants)<\/p>\n

          Solution:<\/strong> π\/e is irrational (though not proven simply; known result)<\/p>\n

          Answer:<\/strong> Irrational<\/p>\n

           <\/p>\n

          Example 5 – Odd One Out:<\/strong><\/p>\n

          Examine the five expressions. Exactly one yields a RATIONAL result. Identify it.<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
          \n

          Item<\/p>\n<\/td>\n

          		\n

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

          \n

          1<\/p>\n<\/td>\n

          		\n

          √2 + √3<\/p>\n<\/td>\n<\/tr>\n

          \n

          2<\/p>\n<\/td>\n

          		\n

          √5 + (-√5)<\/p>\n<\/td>\n<\/tr>\n

          \n

          3<\/p>\n<\/td>\n

          		\n

          √2 × √8<\/p>\n<\/td>\n<\/tr>\n

          \n

          4<\/p>\n<\/td>\n

          		\n

          √6 ÷ √3<\/p>\n<\/td>\n<\/tr>\n

          \n

          5<\/p>\n<\/td>\n

          		\n

          (√3 + 1) × (√3 – 1)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

          Item<\/p>\n<\/td>\n

          		\n

          Calculation<\/p>\n<\/td>\n

          		\n

          Result<\/p>\n<\/td>\n

          		\n

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

          \n

          1<\/p>\n<\/td>\n

          		\n

          √2 + √3<\/p>\n<\/td>\n

          		\n

          ≈ 3.146<\/p>\n<\/td>\n

          		\n

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

          \n

          2<\/p>\n<\/td>\n

          		\n

          √5 – √5<\/p>\n<\/td>\n

          		\n

          0<\/p>\n<\/td>\n

          		\n

          Rational<\/strong> \u2713<\/p>\n<\/td>\n<\/tr>\n

          \n

          3<\/p>\n<\/td>\n

          		\n

          √2 × √8 = √16<\/p>\n<\/td>\n

          		\n

          4<\/p>\n<\/td>\n

          		\n

          Rational<\/strong> \u2713<\/p>\n<\/td>\n<\/tr>\n

          \n

          4<\/p>\n<\/td>\n

          		\n

          √6 ÷ √3 = √2<\/p>\n<\/td>\n

          		\n

          ≈ 1.414<\/p>\n<\/td>\n

          		\n

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

          \n

          5<\/p>\n<\/td>\n

          		\n

          (√3)² – 1² = 3 – 1<\/p>\n<\/td>\n

          		\n

          2<\/p>\n<\/td>\n

          		\n

          Rational<\/strong> \u2713<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Wait — items 2, 3, and 5 all yield rational results! That's three rational results, not one. Let me recheck:<\/p>\n

            \n
          • Item 2: √5 + (-√5) = √5 – √5 = 0 → Rational<\/li>\n
          • Item 3: √2 × √8 = √16 = 4 → Rational<\/li>\n
          • Item 5: (√3+1)(√3-1) = 3 – 1 = 2 → Rational<\/li>\n<\/ul>\n

            Items 1 and 4 are irrational. So "exactly one yields rational" would be incorrect. Perhaps the intended odd one out is different.<\/p>\n

            Alternative – Which yields IRRATIONAL?<\/strong> Then 1 and 4 are irrational — still two.<\/p>\n

            Let me reconsider: If the question says "exactly one yields RATIONAL", then the set is flawed. But if the question is "exactly one does NOT yield a rational result" — then items 2,3,5 give rational; item 4 gives irrational? No, item 4 = √2 (irrational), item 1 = irrational. That's two.<\/p>\n

            Given this, perhaps the intended single odd one out is Item 1<\/strong> if we look for a different pattern:<\/p>\n

            Three reasons why Item 1 might be odd:<\/strong><\/p>\n

            (A) Operation type:<\/strong> Item 1 is a sum of two unlike surds; others are sums with cancellation (Item 2), products (Item 3,5), or division (Item 4).<\/p>\n

            (B) Simplifiability:<\/strong> Items 2,3,4,5 all simplify to a rational or a single surd; Item 1 remains a sum of two distinct surds (cannot combine).<\/p>\n

            (C) Conjugate pattern:<\/strong> Items 5 uses conjugate (a+b)(a-b); Items 2 uses additive inverse; Items 3 and 4 use multiplicative relationships; Item 1 uses simple addition with no special structure.<\/p>\n

            Conclusion:<\/strong> If forced to pick one odd item, Item 1 is structurally different.<\/p>\n

             <\/p>\n

            Example 6 – Quick Odd One Out:<\/strong><\/p>\n

            Which one is rational?<\/strong><\/p>\n

            A) √3 + √2
            \nB) √3 – √3
            \nC) (√5)²
            \nD) √16<\/p>\n

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

              \n
            • A: Irrational<\/li>\n
            • B: 0 (Rational)<\/li>\n
            • C: 5 (Rational)<\/li>\n
            • D: 4 (Rational)<\/li>\n<\/ul>\n

              Odd one out = A<\/strong> (only irrational)<\/p>\n

              Common Mistakes to Avoid<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n
              \n

              Mistake<\/p>\n<\/td>\n

              		\n

              Why It's Wrong<\/p>\n<\/td>\n

              		\n

              Correct Understanding<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

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

              		\n

              Test: √4+√9=2+3=5, √13≈3.6 \u274c<\/p>\n<\/td>\n

              		\n

              No such property exists<\/p>\n<\/td>\n<\/tr>\n

              \n

              All surd products are irrational<\/p>\n<\/td>\n

              		\n

              √2 × √2 = 2 (rational)<\/p>\n<\/td>\n

              		\n

              Products can be rational<\/p>\n<\/td>\n<\/tr>\n

              \n

              Irrationals are closed under addition<\/p>\n<\/td>\n

              		\n

              √2 + (-√2) = 0 (rational)<\/p>\n<\/td>\n

              		\n

              Not closed<\/p>\n<\/td>\n<\/tr>\n

              \n

              π and e are the only irrationals<\/p>\n<\/td>\n

              		\n

              √2, √3, φ, etc. are also irrational<\/p>\n<\/td>\n

              		\n

              Infinitely many irrationals<\/p>\n<\/td>\n<\/tr>\n

              \n

              Irrationals can't be compared<\/p>\n<\/td>\n

              		\n

              √2 < √3 is true<\/p>\n<\/td>\n

              		\n

              Irrationals can be ordered<\/p>\n<\/td>\n<\/tr>\n

              \n

              Between two irrationals there are no rationals<\/p>\n<\/td>\n

              		\n

              Between √2 and √3 lies 1.5<\/p>\n<\/td>\n

              		\n

              Rationals exist between irrationals<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              \n

              Quick Reference Card – Irrational Properties<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
              \n

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

              		\n

              Holds for Irrationals?<\/p>\n<\/td>\n

              		\n

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

              \n

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

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              √2+√3 = √3+√2<\/p>\n<\/td>\n<\/tr>\n

              \n

              Commutative (×)<\/p>\n<\/td>\n

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              √2×√3 = √3×√2<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              (√2+√3)+√5 = √2+(√3+√5)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Associative (×)<\/p>\n<\/td>\n

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              (√2×√3)×√5 = √2×(√3×√5)<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              √2×(√3+√5)=√2√3+√2√5<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              \u274c No<\/p>\n<\/td>\n

              		\n

              √2 + (-√2) = 0 (rational)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Closure (×)<\/p>\n<\/td>\n

              		\n

              \u274c No<\/p>\n<\/td>\n

              		\n

              √2 × √2 = 2 (rational)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Density<\/p>\n<\/td>\n

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              Between any two irrationals, infinitely many irrationals<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              -√2 exists and is irrational<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              \u2705 Yes<\/p>\n<\/td>\n

              		\n

              1\/√2 = √2\/2 is irrational<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

               <\/p>\n","protected":false},"excerpt":{"rendered":"

              Unit: Number System Chapter: Properties of Irrational Numbers Reference: – Introduction to Irrational Numbers, Closure Properties (Addition, Subtraction, Multiplication, Division), Commutative & Associative Properties, Distributive Property, Density Property, Comparison Properties, Properties of Square Roots, Key Differences from Rational, Solved Examples, Odd-One-Out Problems, Common Mistakes After studying this chapter, you should be able to understand: Introduction […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[593],"tags":[],"class_list":["post-9134","post","type-post","status-publish","format-standard","hentry","category-grade-8"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9134","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9134"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9134\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}