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ALTERNATIVE ALGORITHM No.14

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  • Diosdado |Fragata
    HELLO!   Here is your alternative algorithm for today:   ALTERNATIVE ALGORITHM # 14 RAISING A NUMBER TO AN INTEGER EXPONENT   The usual method of raising a
    Message 1 of 2 , Nov 1, 2011
      HELLO!
       
      Here is your alternative algorithm for today:
       
      ALTERNATIVE ALGORITHM # 14
      RAISING A NUMBER TO AN INTEGER EXPONENT
       
      The usual method of raising a number to an integer exponent is done by multiplying the number by itself successively (n-1) times as in 25 = 2x2x2x2x2 = 32. This method is manageable if the base number is not more than two digits. But if the base number is more than two digits, the task becomes exasperating. However, there are manual methods for approximating the values of exponentials.

      a. THE DIFFERENTIAL EXCESS METHOD
      If a number is expressed as a binomial which is raised to exponent n, the binomial expansion is:
       (a + b)n = an + nban-1 + n(n-1)b2an-2  +…bn
      If b is very small compared to a, the approximate value of the binomial is equal to the sum of the values of the first two terms in the expansion thus:
                        (a + b)n = an + n ban-1       (5)
      The value obtained by using Eq.(5) is considered fairly accurate for as long as a is at least a two digit integer and b is the decimal complement of the given base number.
      Example:
                Find the value of 15.1255
      Solution:

      For the first term;
      15.1255 = (15 + 0.125)5 = 155 + 5 x 0.125 x 154
      15 2= 225
      154 = 2252 = (22x23) affix 25
      =23 x (20 +2) = 460+46 affix 25 = 50625
      155 = 50625 x (10 +5) = 506250 + 253125
             = 75375
      For the second term;
      50625 x 5 = 253125
      253125 x 0.125 = 253125x (0.1+ 0.2+0.005)
                                 = 25312.5
                                 5062.5 
                                 1265.625
                               31640.625
      Thus;            
                         15.1255 = 739375
                                           31640 
                                         791015
       The six digit value of 15.1255 obtained with a scientific calculator is 791547. The error in the approximation is only 0.067% of the true value.
      For base numbers with one digit or more than two digit integer part, the number should be converted to a Special Exponential Notation form containing a two digit integer part of the significant figure before transforming it to the equivalent binomial form.
      Examples:
       
      1.  2.3745 =(23.74 x 10-1)5  =23.745 x 104
                                       =(23 + 0.74)5 x 104
       
      2.  3764 = (37.6 x 10)4 = 37.64 x 104
           = (37+ 0.6)4 x 104
      In Example 1 the binomial (23 + 0.74) may be changed to (24 - 0.26) to improve the accuracy of the approximation since the ratio 0.26/24 is much smaller than 0.74/23.
      b. INTERPOLATION
       Interpolation may be used for evaluating exponentials. As in the previous method, the significant figure of the Exponential should have a two- digit integer part to maintain an acceptable accuracy.
      Example: 
                                 
           Evaluate 26.253 by interpolation
      Solution:                                                                       
               X                       X3  
                       26                   17576 
                       26.25           18102.75 
                       27                   19683       
                        
       The value of 26.253 obtained with a scientific calculator is 18087.89. There is an error of 14.86 in the approximation which is only 0.082% of the true value.
      Your next algorithm is how to raise a number containing a decimal part to an exponent that also has a decimal part.
    • FEROJ AL MAMUN
      Dear Admin, Please remove my name from the group list. Best wishes for all the members of the group. may we found great mathematicians from this group.   Best
      Message 2 of 2 , Nov 2, 2011
        Dear Admin,

        Please remove my name from the group list.


        Best wishes for all the members of the group. may we found great mathematicians from this group.



         
        Best regards

        Feroz

        From: Diosdado |Fragata <dadofragata@...>
        To: "math_club@yahoogroups.com" <math_club@yahoogroups.com>
        Sent: Wednesday, November 2, 2011 9:28 AM
        Subject: [math_club] ALTERNATIVE ALGORITHM No.14

         
        HELLO!
         
        Here is your alternative algorithm for today:
         
        ALTERNATIVE ALGORITHM # 14
        RAISING A NUMBER TO AN INTEGER EXPONENT
         
        The usual method of raising a number to an integer exponent is done by multiplying the number by itself successively (n-1) times as in 25 = 2x2x2x2x2 = 32. This method is manageable if the base number is not more than two digits. But if the base number is more than two digits, the task becomes exasperating. However, there are manual methods for approximating the values of exponentials.

        a. THE DIFFERENTIAL EXCESS METHOD
        If a number is expressed as a binomial which is raised to exponent n, the binomial expansion is:
         (a + b)n = an + nban-1 + n(n-1)b2an-2  +…bn
        If b is very small compared to a, the approximate value of the binomial is equal to the sum of the values of the first two terms in the expansion thus:
                          (a + b)n = an + n ban-1       (5)
        The value obtained by using Eq.(5) is considered fairly accurate for as long as a is at least a two digit integer and b is the decimal complement of the given base number.
        Example:
                  Find the value of 15.1255
        Solution:

        For the first term;
        15.1255 = (15 + 0.125)5 = 155 + 5 x 0.125 x 154
        15 2= 225
        154 = 2252 = (22x23) affix 25
        =23 x (20 +2) = 460+46 affix 25 = 50625
        155 = 50625 x (10 +5) = 506250 + 253125
               = 75375
        For the second term;
        50625 x 5 = 253125
        253125 x 0.125 = 253125x (0.1+ 0.2+0.005)
                                   = 25312.5
                                   5062.5 
                                   1265.625
                                 31640.625
        Thus;            
                           15.1255 = 739375
                                             31640 
                                           791015
         The six digit value of 15.1255 obtained with a scientific calculator is 791547. The error in the approximation is only 0.067% of the true value.
        For base numbers with one digit or more than two digit integer part, the number should be converted to a Special Exponential Notation form containing a two digit integer part of the significant figure before transforming it to the equivalent binomial form.
        Examples:
         
        1.  2.3745 =(23.74 x 10-1)5  =23.745 x 104
                                         =(23 + 0.74)5 x 104
         
        2.  3764 = (37.6 x 10)4 = 37.64 x 104
             = (37+ 0.6)4 x 104
        In Example 1 the binomial (23 + 0.74) may be changed to (24 - 0.26) to improve the accuracy of the approximation since the ratio 0.26/24 is much smaller than 0.74/23.
        b. INTERPOLATION
         Interpolation may be used for evaluating exponentials. As in the previous method, the significant figure of the Exponential should have a two- digit integer part to maintain an acceptable accuracy.
        Example: 
                                   
             Evaluate 26.253 by interpolation
        Solution:                                                                       
                 X                       X3  
                         26                   17576 
                         26.25           18102.75 
                         27                   19683       
                          
         The value of 26.253 obtained with a scientific calculator is 18087.89. There is an error of 14.86 in the approximation which is only 0.082% of the true value.
        Your next algorithm is how to raise a number containing a decimal part to an exponent that also has a decimal part.


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