Absolute Value



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Chapter 3 : Absolute Value



Absolute Value arrow_upward


  • The absolute value of a real number is its numerical value without regard to its sign.
  • It is denoted as |number|.
  • It is always non – negative.
  • is the absolute value of both  and
  • The absolute value is related to the distance from the origin.
  • The absolute value of a real or complex number is the distance from that number to the origin, along the real number line.

  • Formal Definition arrow_upward



    Graph of the Modulus Function arrow_upward



    Properties of the Absolute Value Function arrow_upward



    The square root notation without sign represents the positive square root

    Non-negativity

    Positive-definiteness

    Multiplicativeness

    Subadditivity

    Symmetry

    Identity of indiscernibles (equivalent to positive-definiteness)

    Triangle inequality (equivalent to subadditivity)

    Preservation of division (equivalent to multiplicativeness)

    (equivalent to subadditivity)



    Absolute Value of Complex Numbers arrow_upward


  • The absolute value of a complex number is the distance from  to the origin
  • For any complex number

  • Where  and  are real numbers, the absolute value or modulus of is
  • denoted  and is defined as
  • The absolute value of a real number  is equal to its absolute value when considered as a complex number

    Operations on the Absolute Value arrow_upward



    Case 1

  • and are both positive:
    • 0 so
    • | xy |= 0 = ( 0 )( 0 )= | x || y |




    Thank You from Kimavi arrow_upward


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