MATH MAGIC - PERFORMING THE
    MULTIPLICATION OF LARGE
    NUMBERS EASILY AND QUICKLY

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        Please study

        Mental Math before Math Magic

        That knowledge is a prerequisite here.

        There we applied the "the Urdhva Tiryak Sutra"
        (meaning : Vertically and cross-wise) to perform
        multiplication of three digit numbers mentally,
        easily and speedily.

        Here, we extend the method to perform
        multiplication of four digit numbers.

        There and Here we deal with the principles
        appled to Multiplication. For the application
        to other topics of math magic, go to
        Vedic Maths eBook.
        for more details, see
        near the bottom of the page.

        Multiply vertically and crosswise
        to get the digits of the product.

        Examples will clarify the method.

        Before seeing the examples, let us see
        the formula for finding the product of
        four digit numbers.

        Let us say the four digits of the first number be
        'a'(Thousands' digit) and 'b'(Hundreds' digit),
        'c'(tens' digit) and 'd'(units' digit).

        And those of the second number be
        'p'(Thousands' digit) and 'q'(Hundreds' digit),
        'r'(tens' digit) and 's'(units' digit).

        Write the digits of the two numbers
        one below the other as follows.

          a  b  c  d
          p  q  r  s

        The product of these numbers has seven parts
        which are given below seperated by '/'.

          a  b  c  d
          p  q  r  s
          -----------
          ap/(aq+pb)/(ar+pc+bq)/(as+pd+br+qc)/(bs+qd+cr)/(cs+rd)/ds
          -----------

        'ap' is the millions part which is the
        vertical product of the first column.

        (aq+pb) is the hundred thousands' part
        which is the sum of the cross-wise
        product of the first and third columns.

        (ar+pc+bq) is the ten thousands' part which is the
        sum of the cross-wise product of the first and third
        columns and the vertical product of
        the second column.

        (as+pd+br+qc) is the Thousands' part which
        is the sum of the cross-wise products of
        first and last columns and the cross-wise
        products of second and third columns.

        (bs+qd+cr) is the Hundreds' part which is
        the sum of the cross-wise products
        of second and last columns and the vertical
        product of third column.

        (cs+rd) is the Tens' part which is the sum of
        the cross-wise products of the last two columns.

        'ds' is the units' part which is
        the vertical product of the last column.

        Let us see the method by examples.





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         Example 1 of Math Magic

        To find 2011 x 1402

          2011                  
          1402
          -----
          2x1/2x4+1x0/2x0+1x1+0x4/2x2+1x1+0x0+4x1/
          0x2+4x1+1x0/1x2+0x1/1x2
          = 2/8/1/9/4/2/2
          2011 x 1402 = 2819422

        Having seen the steps for two and three
        digit numbers, following here for the
        four digit numbers should not be difficult.

        The first and last terms are vertical
        products of the first and last columns.

          2
          1
          ---
          2x1
          ---
          1
          2
          ---
          1x2
          ---

        The second term is the sum of the cross
        product of the first two columns.

          2 0
          1 4
          ----
          2x4+1x0
          ----

        Similarly the second term from last, is the sum
        of the cross product of the last two columns.

          1 1
          0 2
          ----
          1x2+0x1
          ----

        The third term is the sum of the cross product
        of the first and third columns and the
        vertical product of the second column.

          2 0 1
          1 4 0
          -----
          2x0+1x1+0x4 
          -----

        Similarly the third term from last, is the sum of
        the cross product of the second and last columns
        and the vertical product of the third column.

          0 1 1
          4 0 2
          -----
          0x2+4x1+1x0
          -----

        The fourth term is the sum of the cross product
        of the first and last columns and the cross
        product of the second and third columns.

          2 0 1 1                  
          1 4 0 2
          -------
          2x2+1x1 + 0x0+4x1
          -------

        We can do the calculations mentally and write
        the seven parts directly as shown below.


          2011                  
          1402
          -----
          2/8/1/9/4/2/2
          -----

          2011 x 1402 = 2819422

        Thus 2011 x 1402 = 2,819,422

         Example 2 of Math Magic

        To find 4201 x 5130

          4201
          5130
          ----
          4x5/4x1+5x2/4x3+5x0+2x1/4x0+5x1+2x3+1x0/
          2x0+1x1+0x3/0x0+3x1/1x0
          =20/14/14/11/1/3/0
          =20/14/14+1/1/1/3/0
          =20/14+1/5/1/1/3/0
          =20+1/5/5/1/1/3/0
          =21/5/5/1/1/3/0

          4201 x 5130 = 21551130

        Here, the seven parts are obtained as in previous
        example. But some parts have more than one digit. In
        such cases, units's digit is retained and the other
        digits are carried over to the immediate left place.

        We can do the first step mentally and write
        the second step directly. Also, we can do the
        carry over from right mentally and write it
        as the next step as shown below.


          4201
          5130
          ----
          20/14/14/11/1/3/0
          = 21/5/5/1/1/3/0  (worked out from right)
          4201x5130 = 21551130 

        Thus, 4201 x 5130 = 21,551,130

         Example 3 of Math Magic

        To find 1234 x 5678

          1234
          5678
          ----
          1x5/1x6+5x2/1x7+5x3+2x6/1x8+5x4+2x7+6x3/
          2x8+6x4+3x7/3x8+7x4/4x8
          =5/16/34/60/61/52/32
          =5/16/34/60/61/52+3/2
          =5/16/34/60/61+5/5/2
          =5/16/34/60+6/6/5/2
          =5/16/34+6/6/6/5/2
          =5/16+4/0/6/6/5/2
          =5+2/0/0/6/6/5/2
          =7/0/0/6/6/5/2

          1234 x 5678 = 7006652

        We can do the process mentally and write
        the answer in two steps as shown below.


          1234
          5678
          ----
          5/16/34/60/61/52/32
          =7/0/0/6/6/5/2 (worked out from right)
          1234x5678 = 7006652

        Thus 1234 x 5678 = 7,006,652

         Example 4 of Math Magic

        To find 3456 x 789


          3456
          0789
          ----
          0/21/52/94/118/93/54
          =2/7/2/6/7/8/4 (worked out from right)
          3456x789 = 2726784

        Thus, 3456 x 789 = 2,726,784

        See the math magic of arriving
        at the answer in two steps.

         Example 5 of Math Magic

        To find 9876 x 543


          9876
          0543
          ----
          0/45/76/94/82/45/18
          =5/3/6/2/6/6/8 (worked out from right)
          9876x543 = 5362668

        Thus, 9876 x 543 = 5,362,668

        See the math magic of arriving
        at the answer in two steps.



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Proof of the method adopted : Math Magic

We know
(ay3 + by2 + cy + d)(py3 + qy2 + ry + s)
= y6(ap) + y5(aq + pb) + y4(ar + pc + bq)
+ y3(as + pd + br + qc) + y2(bs + qd + cr) + y(cs + rd) + ds

y6 termy5 termy4 termy3 termy2 termy termconstant term
ap(aq+pb)(ar+pc+bq)(as+pd+br+qc)(bs+qd+cr)(cs+rd)ds

Multiplying four digit numbers is similar with y = 10

106term and units term are vertical products
and 105 term, 104 term,
103 term, 102 term,
10 term are cross-wise products' sums.

[a(10)3 + b(10)2 + c(10) + d][p(10)3 + q(10)2 + 10y + s]
= 106(ap) + 105(aq + pb) + 104(ar + pc + bq)
+ 103(as + pd + br + qc) + 102(bs + qd + cr) + 10(cs + rd) + ds

Millions'Hundred
Thousands'
Ten
Thousands'
Thousands'Hundreds'Tens'Units'
ap(aq+pb)(ar+pc+bq)(as+pd+br+qc)(bs+qd+cr)(cs+rd)ds

The digits of the two numbers :

          a  b  c  d
          p  q  r  s

         The seven parts of the product :

ap/(aq+pb)/(ar+pc+bq)/(as+pd+br+qc)/(bs+qd+cr)/(cs+rd)/ds (proved.)

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