matha namo namaha

 
MISCELLANEOUS ITEMS
1. Straight Squaring:
We have already noticed methods useful to find out squares of numbers. But the methods are useful under some situations and conditions only. Now we go to a more general formula.
The sutra Dwandwa-yoga (Duplex combination process) is used in two different meanings. They are i) by squaring ii) by cross-multiplying.
We use both the meanings of Dwandwa-yoga in the context of finding squares of numbers as follows:
We denote the Duplex of a number by the symbol D. We define for a single digit ‘a’, D =a2. and for a two digit number of the form ‘ab’, D =2( a x b ). If it is a 3 digit number like ‘abc’, D =2( a x c ) + b2.
For a 4 digit number ‘abcd’, D = 2( a x d ) + 2( b x c ) and so on. i.e. if the digit is single central digit, D represents ‘square’: and for the case of an even number of digits equidistant from the two ends D represent the double of the cross- product.
Consider the examples:
 Number
 DuplexD
       3
 32 = 9
       6
 62 = 36
     23
 2 (2 x 3) = 12
     64
 2 (6 x 4) = 48
   128
 2 (1 x 8) + 22 = 16 + 4 = 20
   305
 2 (3 x 5) + 02 = 30 + 0 = 30
 4231
 2 (4 x 1) + 2 (2 x 3) = 8 + 12 = 20
 7346
 2 (7 x 6) + 2 (3 x 4) = 84 + 24 = 108
Further observe that for a n- digit number, the square of the number contains 2n or 2n-1 digits. Thus in this process, we take extra dots to the left one less than the number of digits in the given numbers.
Examples:1  622  Since number of digits = 2, we take one extra dot to the left. Thus

                    .62         for 2, D = 22 = 4
                   ____
                    644         for 62, D = 2 x 6 x 2 = 24
                  32           for 62, D = 2(0 x 2) + 62
                       _____                          = 36
                  3844

            622 = 3844.
Examples:2  2342  Number of digits = 3. extradots =2 Thus

                ..234    for 4, D = 42 = 16
               _____
               42546    for 34, D = 2 x 3 x 4 = 24
               1221      for 234, D = 2 x 2 x 4 + 32 = 25
               _____
               54756     for  .234, D = 2.0.4 + 2.2.3 = 12
                            for ..234, D = 2.0.4 + 2.0.3 + 22 = 4
Examples:3  14262. Number of digits = 4, extra dots = 3

                i.e
                       ...1426            6,  D = 36
                     ________
                     1808246           26,  D = 2.2.6 = 24
                       22523           426,  D = 2.4.6 + 22 = 52
                    _________
                     2033476        1426,  D = 2.1.6 + 2.4.2 = 28
                                         .1426,  D = 2.0.6 + 2.1.2 + 42 = 20
                                        ..1426,  D = 2.0.6 + 2.0.2 + 2.1.4 = 8
                                       ...1426,  D = 12 = 1

                    Thus 14262 = 2033476.
With a little bit of practice the results can be obtained mentally as a single line answer.
Algebraic Proof:
Consider the first example 622

        Now 622 = (6 x 10 + 2)2 = (10a + b)2 where a = 6, b = 2
                                             = 100a2 + 2.10a.b + b2
                                             = a2 (100) + 2ab (10) + b2
i.e. b2 in the unit place, 2ab in the 10th place and a2 in the 100th place i.e. 22 = 4 in units place, 2.6.2 = 24 in the 10th place (4 in the 10th place and with carried over to 100th place). 62=36 in the 100th place and with carried over 2 the 100th place becomes 36+2=38.
    Thus the answer 3844.
Find the squares of the numbers  54, 123, 2051, 3146.
       Applying the Vedic sutra Dwanda yoga.
2.CUBING
Take a two digit number say 14.
i) Find the ratio of the two digits i.e. 1:4

ii) Now write the cube of the first digit of the number i.e. 13

iii) Now write numbers in a row of 4 terms in such a way that the first one is the cube of the first digit and remaining three are obtained in a geometric progression with common ratio as the ratio of the original two digits (i.e. 1:4) i.e. the row is

                          1    4    16    64.

iv) Write twice the values of 2nd and 3rd terms under the terms respectively in second row.

i.e.,
        1    4    16    64
              8    32           ( 2 x 4 = 8, 2 x 16 = 32)

v) Add the numbers column wise and follow carry over process.

        1    4    16    64    Since 16 + 32 + 6 (carryover) = 54
              8    32           4 written and 5 (carryover) + 4 + 8 = 17
        ______________
        2    7     4     4      7 written and 1 (carryover) + 1 = 2.

This 2744 is nothing but the cube of the number 14

Example 1:  Find 183

                   




Example 2:  Find 333





                       
Algebraic Proof:
Let a and b be two digits.
Consider the row a3     a2b     ab2     b3
     the first is a3 and the numbers are in the ratio a:b
    since a3:a2b=a2b:b3=a:b
Now twice of a2b, ab2 are 2a2b, 2ab2

            a3 +   a2b + ab2 + b3
                   2a2b + 2ab2
              ________________________________
            a3 + 3a2b + 3ab2 + b3 = (a + b)3.
Thus cubes of two digit numbers can be obtained very easily by using the vedic sutra ‘anurupyena’. Now cubing can be done by using the vedic sutra ‘Yavadunam’.
Example 3:   Consider 1063.
i) The base is 100 and excess is 6. In this context we double the excess and then add.
    i.e. 106 + 12 = 118. ( 2 X 6 =12 )
    This becomes the left - hand - most portion of the cube.
    i.e. 1063 = 118 / - - - -
ii) Multiply the new excess by the initial excess

         i.e. 18 x 6 = 108 (excess of 118 is 18)
    Now this forms the middle portion of the product of course 1 is carried over, 08 in the middle.

         i.e. 1063 = 118 / 08 / - - - - -
                             1
iii) The last portion of the product is cube of the initial excess.
    i.e. 63 = 216.
    16 in the last portion and 2 carried over.

         i.e. 1063 = 118 / 081 / 16 = 1191016
                             1      2
Example 4:  Find 10023.
i) Base = 1000. Excess = 2. Left-hand-most portion of the cube becomes 1002+(2x2)=1006.
ii) New excess x initial excess = 6 x 2 = 12.
    Thus 012 forms the middle portion of the cube.
iii) Cube of initial excess = 23 = 8.
    So the last portion is 008.
    Thus 10023 = 1006 / 012 / 008 = 1006012008.
Example 5:  Find 943.
i) Base = 100, deficit = -6. Left-hand-most portion of the cube becomes 94+(2x-6)=94-12=82.
ii) New deficit x initial deficit = -(100-82)x(-6)=-18x-6=108
    Thus middle potion of the cube = 08 and 1 is carried over.
iii) Cube of initial deficit = (-6)3 = -216
                                          __                  __
            Now 943 = 82 / 08 / 16 = 83 / 06 / 16
                                     _
                              1     2
                                            = 83 / 05 / (100 – 16)
                                            = 830584.
Find the cubes of the following numbers using Vedic sutras.

            103, 112, 91, 89, 998, 9992, 1014.
3. Equation of Straight line passing through two given points:
To find the equation of straight line passing through the points (x1, y1) and (x2, y2) , we generally consider one of the following methods.
1. General equation y = mx + c.
    It is passing through (x1, y1) then y1 = mx1 + c.
    It is passing through (x2, y2) also, then y2 = mx2 + c.
    Solving these two simultaneous equations, we get ‘m’ and ‘c’ and so the equation.
2. The formula
                                             (y2 - y1)
                                y – y1 = ________ (x – x1)   and substitution.
                                              (x2 - x1)
Some sequence of steps gives the equation. But the paravartya sutra enables us to arrive at the conclusion in a more easy way and convenient to work mentally.
Example1:  Find the equation of the line passing through the points (9,7) and (5,2).
Step1: Put the difference of the y - coordinates as the x - coefficient and vice - versa.
          i.e. x coefficient = 7 - 2 = 5
          y coefficient = 9 - 5 = 4.
          Thus L.H.S of equation is 5x - 4y.
Step 2: The constant term (R.H.S) is obtained by substituting the co-ordinates of either of the given points in
        L.H.S (obtained through step-1)
        i.e. R.H.S of the equation is    

             5(9) - 4(7) = 45 - 28 = 17

            or 5(5) - 4(2) = 25 - 8 = 17.

            Thus the equation is 5x - 4y = 17.
Example 2:  Find the equation of the line passing through (2, -3) and (4,-7).
Step 1 : x[-3-(-7)] –y[2-4] = 4x + 2y.
Step 2 : 4(2) + 2(-3) = 8 –6 = 2.
Step 3 : Equation is 4x + 2y =2 or 2x +y = 1.
Example 3 : Equation of the line passing through the points (7,9) and (3,-7).
Step 1 : x[9 - (-7)] – y(7 - 3) = 16x - 4y.
Step 2 : 16(7) - 4(9) = 112 – 36 = 76
Step 3 : 16x- 4y = 76 or 4x – y = 19
Find the equation of the line passing through the points using Vedic methods.

            1.   (1, 2), (4,-3)             2.  (5,-2), (5,-4)

            3.  (-5, -7), (13,2)         4.   (a, o) , (o,b)

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