11X1 T02 02 rational & irrational (2011)

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  • Rational Numbers

  • Rational NumbersRational numbers can be expressed in the form where a and b are integers. b

    a

  • Rational NumbersRational numbers can be expressed in the form where a and b are integers. b

    a

    Irrational Numbers

  • Rational NumbersRational numbers can be expressed in the form where a and b are integers. b

    a

    Irrational NumbersIrrational numbers are numbers which are not rational.

    All irrational numbers can be expressed as a unique infinite decimal.

  • irrational is 2 Prove ..ge

  • irrational is 2 Prove ..geProof by contradiction

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factors

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k where k is an integer

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k where k is an integer2 2b k

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k

    factorcommon a havemust and 2by divisibleboth are and So 22 ba

    where k is an integer2 2b k

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k

    factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

    where k is an integer2 2b k

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k

    factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

    rationalnot is 2so

    where k is an integer2 2b k

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k

    factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

    rationalnot is 2soirrational is2

    where k is an integer2 2b k

  • irrational is 2 Prove ..geProof by contradiction

    rationalis 2 Assume

    ba 2 where a and b are integers with no common factorsab 2

    222 ab 2by divisible bemust Thus 2a

    4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

    2 2 4b k

    factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

    rationalnot is 2soirrational is2 Exercise 2B; 2a, 3, 4 (root 3), 8, 14*

    where k is an integer2 2b k

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