Vedic Math - Square roots

Earlier we discussed "Squaring numbers near base" and "General Squaring through Duplex Process" and now we will find out how to calculate the square root of numbers. To understand this, let us first learn basic rules for finding the square root.

(1). The given number is first arranged in two-digit groups from right to left. If on left hand side, a single digit is left, that will also be counted as a group.
(2). The number of digits in the square root will be the same as the number of groups derived from the number. Examples are:

• 25 will be having one group as '25', hence square root should be of one digit.
• 144 will be having two groups as '44' and '1', hence the square root should be of two digits.
• 1024 will be having two groups as '24' and '10', hence the square root should be of two digits.

(3). If the given number has 'n' digits then the square root will have n/2 or (n+1)/2 digits
(4). The squares of the first nine natural numbers are 1,4,9,16,25,36,49,64, and 81. All of these squares end with 1, 4, 5, 6, 9, 0. This means

• An exact square never ends in 2, 3, 7 or 8
• If a number ends in 2, 3, 7 or 8, its square root will always be an irrational number
• If an exact square ends in 1, its square root ends in 1 or 9
• If an exact square ends in 4, its square root ends in 2 or 8
• If an exact square ends in 5, its square root ends in 5
• If an exact square ends in 6, its square root ends in 4 or 6
• If an exact square ends in 9, its square root ends in 3 or 7

(5). If a perfect square is an odd number, the square root is also an odd number
(6). If a perfect square is an even number, the square root is also an even number
(7). A whole number, which ends with an odd numbers of 0's, can never be the square of a whole number
(8). An exact square never ends in a 6 if the penultimate digit(digit that is next to the last digit) is even (eg. exact squares can not end in 26, 46, 86, etc.)
(9).An exact square never has an odd penultimate digit unless the final digit is a 6 (thus, exact squares can not end in 39,71, etc.)
(10).An exact square never ends with an even number when the last two digits taken together are not divisible by 4 (thus, no exact square can end in 22, 34 and other non-multiples of 4 if the last digit is even)

Firstly, we use "The First by the First and the Last by the Last" technique to solve the square root.

(1). √6889
     There are two groups of figures, '68' and '89'. So we expect 2-digit answer.
     Now see since 68 is greater than 64(8²) and less than 81(9²), the first figure must be 8.

     So, 6889 is between 6400 and 8100, that means, between 80² and 90².
     Now look at the last figure of 6889, which is 9.
     Squaring of numbers 3 and 7 ends with 9.
     So, either the answer is 83 or 87.
     There are two easy ways of deciding. One is to use the digit sums.
     If 87² = 6889
     Then converting to digit sums
     (L.H.S. is 8+7 = 15 -> 1+5 -> 6 and R.H.S. is 6+8+8+9 -> 31 -> 3+1 -> 4)
     We get 6² -> 4, which is not correct.
     But 83² = 6889 becomes 2² -> 4, so the answer must be 83.
     The other method is to recall that since 85² = 7225 and 6889 is below this. 6889 must be below 85. So it must be 83.

Note: To find the square root of a perfect 4-digit square number we find the first figure by looking at the first figures and we find two possible last figures by looking at the last figure. We then decide which is correct either by considering the digit sums or by considering the square of their mean.

(2). √5776
     The first 2-digit(i.e. 57) at the beginning is between 49 and 64, so the first figure must be 7.
     The last digit (i.e. 6) at the end tells us the square root ends in 4 or 6.
     So the answer is 74 or 76.
     74² = 5776 becomes 2² -> 7 which is not true in terms of digit sums, so 74 is not the answer.
     76² = 5776 becomes 4² > 16 -> 7, which is true, so 76 is the answer.
     Alternatively to choose between 74 and 76 we note that 75² = 5625 and 5776 is greater than this so the square root must be greater than 75. So it must be 76.

Second technique is useful for bigger numbers and in this method, we use "Duplex". In the next article, we shall continue to discuss this second technique. Until then, good luck and happy computing!!


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Vedic Math - Fourth Power of 2 Digit Numbers

We discussed the cube of 2-digit number in previous article. In this article, we shall describe the fourth power of 2-digit numbers using the same formula.

The Algebraic Expression of (a + b)⁴

 (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³+ b⁴

 We can rewrite the above equation as:
           a⁴       a³b          a²b²           ab³        b⁴

                   3a³b        5a²b²          3ab³
So, apply the same rule which we applied in previous article, while finding cubic of the number. Consider the first term as a⁴ and the remaining terms get multiplied by b/a with the previous term.
The Difference comes in second row, in fourth power, we multiply 2nd and 4th term by 3 and 3rd term by 5.

Example: 11⁴

            1    1    1    1    1
                 3    5    3
          -------------------------
            1    4    6    4    1
          -------------------------

Example: 32⁴
         
            81     54      36     24     16
                   162    180     72
          -------------------------------------
          104      8       5        7       6
          -------------------------------------

The "Binomial Theorem" is thus capable of practical application more comprehensively in Vedic Math. Here it is been utilised for splendid purpose as described above, with Vedic Sutras.

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Vedic Math - Cube of 2 Digit Numbers

Cube of numbers plays an important role in mathematics calculations, like in finding cube root of the numbers. So it is useful if we can do the cube of numbers quickly. We are trying for the same here using 'Anurupya Sutra' of Vedic Math. Let us learn how we can do it.

Cubes of the single digits i.e. from (1 to 9) are given below:
 1³  = 1,              2³  = 8,              3³  = 27,            4³  = 64,            5³  = 125,
6³  = 216,          7³  = 343,          8³  = 512,           9³  = 729,          10³  = 1000

If we observe closely, the last digit of every cubic number is unique i.e. numbers from (1 to 9) does not repeat. This observation will be very helpful while calculating cube roots of any 2-digit numbers.

Let us first see the Algebraic Expression for Cube root:

(a + b)³ = a³ + 3a²b + 3ab²+ b³

Above expression for cube root of (a + b) contain 4 terms in total.

• 1st term is  a³
• 2nd term is a²b =  a³  x (b/a)  = 1st term x (b/a)
• 3rd term is ab² = a²b x (b/a) = 2nd term x (b/a)
• 4th term is b³   = ab² x (b/a) = 3rd term x (b/a)

Here (b/a) is the common ratio

Also, as the whole, 2nd term is 3a²b = a²b + 2a²b          {split as sum of two terms}
                      and, 3rd term is 3ab² = ab² + 2ab²           {split as sum of two terms}

So to find the cube, we have to compute a³ and b/a.

In Vedic Math, same formula can be used in a different way to find the cube of 2-digit numbers i.e. ab. Apply formula on 'ab' like (a+b)³ as stated above, and add the results of different rows in vertical columns. You will be able to do the cube of any two digit numbers quickly.

We shall use 'Anurupya Sutra' of Vedic Math for this cube calculation, which states:
"If you start with the cube of first digit and take the next three numbers (in the top row) in a Geometrical Proportion (in the ratio of original digits themselves), you will find that the fourth figure on the right hand will be just the cube of second digit".

Following is the step by step description of finding the cube of 2-digit number:

• Step 1: In the first row, start with a^3 as 1st term and multiplying 1st term by (b/a) to get 2nd term.
• Step 2: Repeat the multiplication till 4th term.
• Step 3: In the second row, double the two middle terms (i.e. 2nd term and 3rd term) and write just below 2nd term and 3rd term.
• Step 4: Add them vertically in columns. Carry forward the 10th place digit to next column.

The example given below will describe this method well.

Example: 11³
Here a = 1 , b = 1 ,  a³ = 1 ,  b/a = 1/1 = 1          (Here common ratio is equal to 1)

Now see the formation of the table:
First Row             1     1     1     1
Second Row               2     2
                    -------------------------
Add                     1     3     3     1
11³ = 1331

Example: 13³
Here a = 1 , b = 3 ,  a³ = 1 ,  b/a = 3/1 = 3          (Here common ratio is greater than 1)

Now see the formation of the table:
First Row              1     3     9     27       (Note: 4th term is just  b³  as shown above in algebraic  
Second Row                6     18               expression)
                          -------------------------
Add                      1     9     27    27
                          -------------------------
                         =   1     9      7     7       (Apply carry over rule)
                                    2     2
 =   1      9      9     7
                    2
                 =   1      1     9     7
                                    1
 =   2      1     9     7
13³= 2197

Example: 52³
Here a = 5 , b = 2 ,  a³ = 125 ,  b/a = 2/5            (Here common ratio is less than 1)

Now see the formation of the table:
First Row                 125     50     20     8
Second Row                     100     40
                           ---------------------------
Add                         125   150     60     8
                           ---------------------------
                           =  125       0        0     8        (Apply carry over rule)
                                        15        6
  =  140      6    0    8
52³= 140608

Some more examples are as follows:
(1) 16³ =  1    6    36    216
                       12    72
               -----------------------
                   4    0     9      6
               -----------------------

(2) 32³ =  27    18    12     8
                           36    24
                -----------------------
                   32     7      6      8
                -----------------------

(3) 97³ =  729    567      441      343
                           1113      882
                     -----------------------------
                    912       6          7          3
                     ----------------------------- 
or better way for number near base, 97³ = (100-3)³    where a=100, b= -3 and b/a= -3/100
                                                                       = 1000000    -30000      900    -27
                                                                       =                   -60000    1800                                                                                                                                                                      
                                                                           ------------------------------------
  1000000    -90000    2700    -27
   ------------------------------------
                                                                     = 912673

Hope it will help!!                        
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Vedic Math - General Squaring

Today, the topic which we are going to discuss is the 'General Procedure to Square any number'. Earlier we discussed about the squaring of numbers near base, however, general procedure is another nice formula to do the squaring and is applicable universally. The method or sutra is "Vertically and Crosswise", but here it is used in a different sense; based on a procedure known as 'Dwandwa Yoga' or 'Duplex Combination Process' or 'Duplex'; denoted as (D).

'Duplex' term is used in two different sense; for squaring and for multiplication. And for current formula, it will be used in both the senses. If we are having a single or central digit, then 'Duplex' means squaring that digit (a² ). Secondly it can be used for even digits number or on numbers having equidistant digits, then 'Duplex' means to double of cross multiplication of the equidistant numbers (2ab). This concept is very important to understand the current formula and will be used in future articles also. Let us see few example to understand it more:

For 1 digit  – D(a) = single digit = a²
                     e.g. D(5) = 5²  = 25
For 2 digits – D(ab) = even digits number = twice the product of the digits (2ab)
                     e.g. D(26) = 2(2)(6) = 24
For 3 digits – D(abc) = product of equidistant digits from center and square of center digits
     = twice the product of the outer digits (2ac) + the square of the middle digit (b² )
                     e.g. D(734) = 2(7)(4) + 3²
                        = 56 + 9 = 65
For 4 digits – D(abcd) = product of equidistant numbers 
                    = twice the product of the outer digits (2ad) + twice the product of the inner digits (2bc)
                     e.g. D(1034) = 2(1)(4) + 2(0)(3)
                           = 8 + 0 = 8
For 5 digits – e.g. D(10345) = product of equidistant digits and square of center digits
= 2(1)(5) + 2(0)(4) + 3² 
                        = 10 + 0 + 9 = 19

and so on. This is called Duplex.


Now, let us come to original question i.e. how to square a number. And the square of any number is just the total of its Duplexes.

For Example,
34² = 1156
= D(3) = 9, D(34) = 24, D(4) = 16

Combining these three results in the usual way, we get
= 9 | 24 | 16

Now add these results as
= 9 |   4 |   6
        2     1    
= 9 |   5 | 6
        2
= 11 | 5 | 6
= 1156

56²  = 3136
     D(5) = 25, D(56) = 60, D(6) = 36
by combining, we get 25 / 60 / 36 = 3136

Equivalent Algebraic Expression is: (10a + b)²  = 100(a² ) + 10(2ab) + b² .

This method can also be explained by multiplying a number by itself using the general multiplication method.

Note :- If a number consists of n digits, its square must have 2n or 2n-1 digits.

Following are some more examples:

263²  =
      D(2) = 4, D(26) = 24, D(263) = 48, D(63) = 36, D(3) = 9
      4 / 24 / 48 / 36 / 9 = 69169
4332²  =
      D(4) = 16, D(43) = 24, D(433) = 33, D(4332) = 34, D(332) = 21, D(32) = 12, D(2) = 4
      16 / 24 / 33 / 34 / 21 / 12 / 4 = 18766224

3247²  = 9 / 12 / 28 / 58 / 46 / 56 / 49 = 10543009

46325²  = 16 / 48 / 60 / 52 / 73 / 72 / 34 / 20 / 25 = 2146005625


We hope this method will help you in squaring of any number quickly. If you find this difficult, you may use another method which we have discussed earlier( Squaring numbers near base ). Every method will become easy with practice. In our next article, we shall discuss about the cubing of the number.


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Vedic Math - Multiplication of any Numbers

In the previous article, we have discussed few cases of Multiplication with "vertically and crosswise" sutra. In this article, we shall learn the remaining cases.

E. Multiplying three digit by three digit numbers
For example: abc x pqr

Multiply:
1) vertically                                              (a x p)
2) crosswise in both directions and add    (a x q) + (b x p)
3) crosswise in both directions and add    (a x r) + (b x q) + (c x p)
4) crosswise in both directions and add    (b x r) + (c x q)
5) vertically                                              (c x r)

Step2 and Step4 are same as we did in the previous sections i.e. multiplying crosswise with two columns at a time. But in Step3, we multiply crosswise using the outer columns, then multiply vertically in the middle column and add these numbers.

Example: 456 x 258
  4            5           6      
  2            5           8
--------------------------------------
  8 | 20+10 | 32+25+12 | 40+30 | 48
= 8 | 30 | 69 | 70 | 48
= 8 | 30 | 69 | (70+4) | 8
= 8 | 30 | 69 | 74 | 8
= 8 | 30 | (69+7) | 4 | 8
= 8 | 30 | 76 | 4 | 8
= 8 | (30+7) | 6 | 4 | 8
= 8 | 37 | 6 | 4 | 8
= (8+3) | 7 | 6 | 4 | 8
= 11 | 7 | 6 | 4 | 8
Hence 456 x 258 = 117648

The Algebraic Expression is:
Let the two numbers be (ax² + bx + c) and (dx² + ex + f).
Note that x=10

Now the product is
= (ax² + bx + c) (dx² + ex + f)
= ad.x⁴ + bd.x³ + cd.x² + ae. x³ + be.x² + ce.x + af.x² + bf.x + cf
= ad.x⁴ + (bd + ae). x³ + (cd + be + af).x² + (ce + bf)x + cf

The Vertically and Crosswise formula can also be Extended into 2 by 2 method. See the following example:
 123 × 132

We can split the numbers up into 12 | 3 and 13 | 2 , treating 12 and 13 as if they are single figures:
 12      3
 13      2
-------------
156 | 63 | 6
= (156+6) | 3 | 6
= 162 | 3 | 6
= 16236

F. Multiplying four digit by three digit numbers
For abcd x pqr

Multiply:
1) vertically                                              (a x p)
2) crosswise in both directions and add    (a x q) + (b x p)
3) crosswise in both directions and add    (a x r) + (b x q) + (c x p)
4) crosswise in both directions and add    (b x r) + (c x q) + (d x p)
5) crosswise in both directions and add    (c x r) + (d x q)
6) vertically                                             (d x r)

Example: 4562 x 258
  4          5            6           2      
             2            5           8
----------------------------------------------
  8 | 20+10 | 32+25+12 | 40+30+4 | 48+10 | 16
= 8 | 30 | 69 | 74 | 58 | 16
= 8 | 30 | 69 | 74 | (58+1) | 6
= 8 | 30 | 69 | 74 | 59 | 6
= 8 | 30 | 69 | (74+5) | 9 | 6
= 8 | 30 | 69 | 79 | 9 | 6
= 8 | 30 | (69+7) | 9 | 9 | 6
= 8 | 30 | 76 | 9 | 9 | 6
= 8 | (30+7) | 6 | 9 | 9 | 6
= 8 | 37 | 6 | 9 | 9 | 6
= (8+3) | 7 | 6 | 9 | 9 | 6
= 11 | 7 | 6 | 9 | 9 | 6
 4562 x 258 = 1176996

I hope this would help you in quick multiplication. If you have any doubt, you can welcome to post your queries.

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Vedic Math - Multiplication of any numbers

In this article, we shall discuss the Vedic Math sutra "vertically and crosswise". This is the General Formula which is applicable to all the cases of multiplication. You will found this method very useful in division also that we shall discuss later. In the previous sutras, you subtracted crosswise; now you will multiply crosswise.

A. Multiplication of two digit numbers
For ab x uv

The numbers are written from left to right.
Now multiply:
1) vertically                                              (a x u)
2) crosswise in both directions and add    (a x v) + (b x u)
3) vertically                                              (b x v)

The answer has the form:   au | av + bu | bv

Following is the example of two-digit multiplication will make this clear:
12 x 23
 1         2
 2         3
--------------
2 | 3 + 4 | 6
= 2 | 7 | 6
= 276

The previous examples involved no carry figures, so let us consider this case in the next example.
64 x 93
 6          4
 9          3
-----------------
54 | 18 + 36 | 12
=54 | 54 | 12
=54 | (54 + 1) | 2         (Carry over the 1)
=54 | 55 | 2
=(54 + 5) | 5 | 2          (Carry over the 5)
=59 | 5 | 2
64 x 93 =5952

The Algebraic Explaination is:
Let the two 2 digit numbers be (ax+b) and (ux+v). Note that x = 10. Now consider the product
(ax + b) (ux + v)
= au. x² + avx + bux + b.v
= au. x² + (av + bu)x + b.v
The first term i. e. the coefficient of  x² is got by vertical multiplication of a and u
The middle term i. e. the coefficient of x is obtained by the cross-wise multiplication of a and v and of b and u and the addition of the two products
The independent term is arrived at by vertical multiplication of 'b' and 'v'

B. Multiplying three digit numbers by two digit numbers
For abc x uv

The middle parts are obtained by adding the crosswise multiplications for a and b with v and u respectively, then b and c with v and u respectively. The outer parts are vertical multiplication of a with u on the left, c with v on the right.
Multiply:
1) vertically                                              (a x u)
2) crosswise in both directions and add    (a x v) + (b x u)
3) crosswise in both directions and add    (b x v) + (c x u)
4) vertically                                              (c x v)

The answer has the form:  au | av + bu | bv + cu | cv

Let us take an example: 236 x 53
   2       3       6
           5       3
-------------------------
  10 | 6+15 | 9+30 | 18
= 10 |  21  |  39  | 18

Work right to left carry over
= 10 | 21 | (39+1) | 8
= 10 | 21 | 40 | 8
= 10 | (21+4) | 0 | 8
= 10 | 25 | 0 | 8
= (10+2) | 5 | 0 | 8
= 12 | 5 | 0 | 8
236 x 53 = 12508

C. Multiplying four digit by two digit numbers
For abcd x uv

Multiply:
1) vertically                                              (a x u)
2) crosswise in both directions and add    (a x v) + (b x u)
3) crosswise in both directions and add    (b x v) + (c x u)
4) crosswise in both directions and add    (c x v) + (d x u)
5) vertically                                              (d x v)

The answer has the form:  au | av + bu | bv + cu | cv + du | dv

Example: 1348 x 74
  1      3      4      8  
                7      4
-------------------------------
 7 | 4+21 | 12+28 | 16+56 | 32
= 7 | 25 | 40 | 72 | 32
= 7 | 25 | 40 | (72+3) | 2
= 7 | 25 | 40 | 75 | 2
= 7 | 25 | (40+7) | 5 | 2
= 7 | 25 | 47 | 5 | 2
= 7 | (25+4) | 7 | 5 | 2
= 7 | 29 | 7 | 5 | 2
= (7+2) | 9 | 7 | 5 | 2
= 9 | 9 | 7 | 5 | 2
 1348 x 74 = 99752  

D. Multiplying five digit by two digit numbers
For abcde x uv

Multiply:
1) vertically                                              (a x u)
2) crosswise in both directions and add    (a x v) + (b x u)
3) crosswise in both directions and add    (b x v) + (c x u)
4) crosswise in both directions and add    (c x v) + (d x u)
5) crosswise in both directions and add    (d x v) + (e x u)
6) vertically                                              (e x v)

The answer has the form:  au | av + bu | bv + cu | cv + du | dv + eu | ev

Example: 12345 x 58
  1      2      3      4      5      
                       5      8
---------------------------------------
 5 | 8+10 | 16+15 | 24+20 | 32+25 | 40
= 5 | 18 | 31 | 44 | 57 | 40
= 5 | 18 | 31 | 44 | (57+4) | 0
= 5 | 18 | 31 | 44 | 61 | 0
= 5 | 18 | 31 | (44+6) | 1 | 0
= 5 | 18 | 31 | 50 | 1 | 0
= 5 | 18 | (31+5) | 0 | 1 | 0
= 5 | 18 | 36 | 0 | 1 | 0
= 5 |(18+3) | 6 | 0 | 1 | 0
= 5 | 21 | 6 | 0 | 1 | 0
= (5+2) | 1 | 6 | 0 | 1 | 0
= 7 | 1 | 6 | 0 | 1 | 0
 12345 x 58 = 716010

I hope it will help in quick multiplication. If there is any query, please post it here. In next article, we shall continue to discuss this technique for multiplying three digits by three digits numbers.


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Vedic Math - Squaring numbers near base

In the previous article, we have discussed about the method of multiplication by using the base value. In this article, we shall learn the squaring of numbers by using base value. Squaring numbers near base is much easier as there is no possibility of different cases that we discussed earlier for multiplication, like
1. One number is above the base and the other number is below it
2. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.

So it is comparatively simpler. Here, we can use the sub sutra “whatever the extent of its deficiency, lessen it still further to that extent; and also set up the square of that deficiency”.

In this, first corollary is “All from 9 and the last from 10”. This method will work for any type of squaring. There is another method by taking the sutra "Vertically and Crosswise" but that we will discuss later.

Suppose we have to find the square of 8. The following will be the steps for it:
1. We shall take the nearest power of 10 (10 itself in this case) as our base.
2. 8 is '2' lesser than 10, so we shall decrease 2 from 8 (i.e. 8 - 2 = 6). This will become the left side of answer.
3. And, for right part of answer, we write down the square of that deficiency i.e. 2 x 2 = 4
4. Thus 8 x 8 = 64

In exactly the same manner, we say
7² = (7-3)  |  3²
     =  4  |  9
     =  49

9² = (9-1)  |  1²
     =  8  |  1
     =  81

6² = (6-4)  |  4²
     =  2     |     6                (Here, since right side is 2 digit number, so '1' will be carried to its left)
                   1
     =  3     |   6        
     =  36

Now, if numbers are above base value, approach will be almost same.  The only difference will be that instead of reducing the number from its deficiency, we increase the number by the surplus. For example,square of 13:

Here working base is 10.
13² = (13+3)  |  3²
       =  16  |  9
       =  169

See few more examples:
14² = (14+4)   |   4²
       =  18  |  16            (Carry over 1)
       =  19  |  6
       =  196

15² = (15+5)   |   5²
       =  20  |  25            (Carry over 2)
       =  22  |  5
       = 225

19² = (19+9)   |   9²
       =  28  |  81            (Carry over 8)
       =  36  |  1
       =  361

And then, extending the same rule to numbers of two or more digits, we proceed further as:

98² = (98-2)  |  2²
       = 96  |  04
       = 9604

93² = (93-7)  |  7²
       = 86  |  49
       = 8649

106² = (106+6)  |  6²
         =  112  |  36
         =  11236

986² = (986-14)  | 14²
         =  972 | 196
         =  972196
 
9996² = (9996-4)  |  4²
           =  9992 |  0016
           =  99920016  

Note: Number of digits in right side part of the answer should always be equal to the zeros in the base value. Extra should be carried forward to left side answer. If number of digits in right side answer are less than the zeros, then it should be prefixed by zeros.
 
The Algebraic Expressions are as follows:

       (a + b)² = a² + 2ab + b² 

Thus,
97² = (100-3)²
       = 10000 - 600 + 9
       = 9409

107² = (100+7)²
         = 10000 + 1400 + 49
         = 11449
 
Another case arise here that if numbers are not near the base value (i.e. where base value is not power of 10). In that case we follow the same method as we discussed in our previous article. For example: 29² . Here working base is 30. So,
29²
 29    -1
 29    -1
-----------
 28  |  1
= 28 x 30  |  1       (multiply the left part of answer with the base)
= 840 |  1
= 840 + 1
= 841

Another example to understand it more: 786² . Here working base is 800.
786²
 786    -14
 786    -14
-------------
 772  |  196
= 772 x 800  |  196
= 617600 | 196
= 617600 + 196
= 617796

So this is all about squaring the numbers by using the base values. Isn't that quite simple and interesting approach.. In next article, we shall discuss about Vertical and Crosswise multiplication.

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Vedic Math - Base Multiplication Part-2

In the previous article, we learnt multiplication by taking base. We discussed following three cases:
A. Numbers are below the base number
B. Numbers are above the base number
C. One number is above the base and the other number is below it

And now, we shall discuss rest of the two cases here:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.

Before going further, let us take few examples of larger numbers with the same method, which we have discussed in previous article.

Example: 6848 x 9997

 6848    -3152
 9997    -0003
----------------
 6845  |  9456     (Refer to previous article for details of approach)
----------------

Example: 87654 x 99995

Let us discuss the fourth case:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.

As you know that this method is applicable in all the cases, but works best when numbers are close to the base. Here, we shall apply the same sutras  “All from 9 and the last from 10” and "Vertically and Crosswise"  discussed earlier and also the sub sutra i.e. "Proportionately".

Take example as 207 x 213 . Here the numbers are not near to any of the bases that we used before: 10, 100, 1000 etc. But these are close to '200'. So, when neither the multiplicand nor the multiplier is near to the convenient power of 10 then we can take a convenient multiple or sub-multiple of a suitable base (as our 'Working Base'). And then perform the necessary operation by multiply or divide the result proportionately. Like in this example, we take 100 as a 'theoretical base' and take multiple of 100 i.e. 200 (100 x 2) as our 'working base'.

207 x 213
  207    +007
  213    +013
  ------------
= 220  |  091
  ------------
= (220 x 200) | 91
= 44000 + 91
= 44091

As they are close to 200, therefore deviations are 7 and 13 as shown above. From the usual procedure (refer to previous article), we get, 220 | 91. Now since our base is 200, we multiply the left-hand part of the answer by 200 and add it to the right-hand part. That is, (220 x200) + 91 {(Left side x base) + right side}. The final answer is '44091'.

Let us take one more example: 406 x 417
Here Working base is 400 ,which is (100 x 4)

   406    +06
   417    +17
   ------------
 = 423  |  102  
   ------------
 = 423 x 400  |  102
 = 169200  |  102
 = 169200 + 102
 = 169302

Another example is, 46 x 43. We can solve it in two ways:
First, we take 10 as a theoretical base and take multiple of 10 i.e. 50 (10 x 5) as our working base.
Second, we take 100 as a theoretical base and take sub-multiple of 100 i.e. 50 (100/2) as our working base.

 46    -4
 43    -7
-----------
 39  |  28
-----------
From our usual procedure, we get 39 | 28. Now,
First way (10 x 5 = 50):
1. Multiply the left-hand part of the answer by 50 (39 x 50)and get 1950 as the first part.
2. Now add 1950 to 28 and get 1978.
3. Final answer is '1978'

Second way (100/2 = 50):
1. Divide the left-hand part of the answer by 2 and get 19.5
Note:-Here 39 being odd, its division by 2 gives us a fractional quotient or decimal number.So, number after decimal(i.e. 5) of the left hand side is carried over to the right hand (as 50).
2. Left-hand part is now '19' and right-hand part is 78(i.e. 50 + 28)
3. Final answer is: 1978

In the above discussed two examples, we can also apply the second method. Like in 207 x 213 , we can also take working base: 1000/4 = 250 or 1000/5 = 200. And in 406 x 417 , we can also take working base: 1000/2 = 500. We have to choose the best convenient multiple or sub-multiple.

Following an example in which you solve it in many ways:
 62 x 48

(1) Working Base 10 x 4 = 40          
        62    +22                                          
        48    + 8                                      
       -----------
        70  |  176                                       
       x40                                                    
       -----------                                          
       2800 |  176                                      
       -----------                                        
       = 2800 + 176                                                                                            
       = 2976          
                                                       
(2) Working Base 10 x 6 = 60        
        62    + 2                                         
        48    -12                                          
       ----------                                      
        50  | -24                                   
       x60                          
       -----------                                    
       3000 | -24                                        
       -----------                                          
       = 3000 - 24                                        
       = 2976                                           

(3) Working Base 10 x 5 =50
        62    +12                                      
        48    - 2                                          
       ----------                                        
        60  | -24                                 
       x50                            
       -----------                                            
       3000 | -24                                          
       -----------                                          
       = 3000 - 24                                          
       = 2976                                               

(4) Working Base 100/2 = 50
        62    +12                                      
        48    - 2                                          
       ----------                                        
        60  | -24                                 
        /2                            
       -----------                                            
        30  | -24                                          
       -----------                                          
       = (30 x 100) - 24                                          
       = 2976

Example: 43 x 47
In this, the base is 100/2 = 50 , so divide the left side by 2 to get the answer.
    43   -7                             (as above we subtract 50 from our numbers)
    47   -3
   ----------
   40  |  21                          

   /2                                     (calculate the left side, then divide the answer by 2)
  -----------
   20  |  21
  -----------
  =(20 x 100) + 21
  =2021
This gives: 43 x 47= 2021

E. Numbers near different bases i.e. 'multiplier' is near to one base and multiplicand is near to other base.
Many of times we need to multiply numbers that are not near to same base but are near to different bases.
In the example given below, you find that one number is close to 1000 and the other is close to 100.

Example: 996 × 97
= 996 × (97 x 10) / 10            (can be rewritten like this)
= 966 x 970 / 10                     (which can futher be written like this)

Let us first solve 966 x 970
996   –004
970   –030
------------
966 |  120
------------
= 966 x 1000 + 120              (multiply left side result by base)
= 966000 + 120
= 966120

So the answer for 966 x 970 is 966120

Now we shall solve the complete equation i.e. will divide it by 10,
  = 966120 / 10
  = 96612

So 996 x 97 = 96612

Example: 10005 x 1002
Here the numbers are close to different bases: 10000 and 1000

So equation can be written as
= 10005 x (1002 x 10) / 10

Now let us solve 10005 X 10020 first

10005     + 005
10020     + 020
-----------------
10025   |   0100
= 10025 x 10000 + 100         (multiply left side result by base)
= 100250000+ 100
= 100250100

So 10005 X 10020 = 100250100
Now let us solve the original equation i.e. will divide the answer by 10
= 100250100/10
= 10025010

So 10005 x 1002 = 10025010

Example: 212 x 104
Rewrite as : 2 x (106 x 104)
            = 2 x 11024
            = 22048

Example: 192 × 44.
Here you can halve 192 and double 44.
= 192 x 44
= (192 / 2) x (44 x 2)              (divide by 2 and multiple by 2)
= 96 x 88

Now solve this equation by procedure defined above
= 96 × 88
= 8448.

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Vedic Math - Base Multiplication

Today, we are going to learn the GENERAL formula of multiplication. This is simple and easy technique and good part is that it is applied to almost all the cases. This method works better when numbers are near their base value. After going through the below discussed method, you will see that multiplication tables are not required for calculation above 5 x 5. You will be able to do all types of multiplication involving bigger multiplicands and multipliers quickly and easily; like for '789789 × 999997'. All the sutras (formulae) of Vedic Math are short and simple; and with the practice of the techniques, most of the calculations become a playful experience for you.

Following is the sutra that we will follow today:

The formulae (sutras) are : “All from 9 and the last from 10” and "Vertically and Crosswise"
The algebraical expression is :(x+a) (x+b) = x (x+a+b) + ab.

From the title of the article, you can understand that today we shall do the multiplication by taking the base of numbers. So, first we need to be familiar what is 'base'. The term ‘base’ in Vedic Math has a broader meaning than you may be used to. We work in a base 10 number system, but within Vedic Math the ‘base’ is the number you will use as a basis for calculation. The numbers taken can be either less or more than the base considered. The difference between the number and the base is termed as deviation. Deviation may be positive or negative.

Now observe the following table.

Number      Base         Number – Base       Deviation
   13             10                  13 - 10                      3

    7              10                     7 - 10                    -3

   89             100                  89 - 100                -11

 1110           1000             1110 - 1000              110

99998          100000       99998 - 100000            -2

So, the deviation obtained are from "All from 9 and the last from 10" sutra (formula).

Following are the cases which we shall discuss here:
A. Numbers are below the base number
B. Numbers are above the base number
C. One number is above the base and the other number is below it
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base

Let us discuss these cases one by one.

A. Numbers are below the base number

Working with Base 10
Let us take an easy and simple example to start this technique. Suppose we have to multiply 6 by 8.
Now the base is 10. Since it is near to both the numbers.
Place the two numbers 6 and 8 above and below on the lefthand side (as shown below). Subtract the base value (i.e. 10 in this case) from both of the numbers and write down the remainders (i.e. 4 and 2) on the right-hand side with their deviation sign (-).

6 x 8   

Left side (4 ways to calculate left side)
a) 6 + (- 2)= 4                   (add top left to bottom right)
b) 8 + (- 4)= 4                   (add bottom left to top right)
c) 6 + 8 - 10= 4                 (add numbers in the left column and subtract from base)
d) 10 + (-4)+(-2)= 4          (add numbers in the right column and the base)

Right side
(-4) x (-2) = 4                   (multiply numbers of right column)

Final answer: 6 x 8= 48

Some more examples of the same rule:

 9 x 9            9 x 8              8 x 7            8 x 8               7 x 9
 9    -1          9    -1             8    -2           8    -2             7    -3
 9    -1          8    -2             7    -3           8    -2             9    -1
---------          ----------        ----------         ----------         -----------
 8     1            7     2            5     6            6     4             6     3
---------          ----------        ----------         -----------        -----------

This comes under the "Vertically and Crosswise" Sutra.

Sometimes there can be a carry figure, so let’s have a look on the next example.
 6 x 6
6     -4
6     -4
---------
2   |   6
       1
---------
But here, as '16' is a two digit number, so you need to carry the 1 over to the left side i.e. '2'. Therefore, left hand side now becomes (2 + 1 = 3).
And Final answer : 6 x 6 = 36

Following are some of the examples:

 8 x 5            7 x 6          6 x 5            7 x 5
 8    -2          7    -3         6    -4           7    -3
 5    -5          6    -4         5    -5           5    -5
--------          ----------       ---------       -----------
3  |    0           3  |    2        1  |    0         2  |    5
      1                   1                 2                  1
--------           ----------      ----------      ------------
 = 40              = 42            = 30            = 35

Working base 100
Example: 96 x 98

96    -04
98    -02
---------
94  |  08   =   9408
---------

Working base 1000
Example: 998 x 988

998    -002
988    -012
------------
986  |  024    =   986024
------------

B. Numbers are above the base number. Process will be same as described above.

Working with Base 10
Suppose we have to multiply 13 by 14.
Now the base is 10. Place the two numbers 13 and 14 above and below on the lefthand side (as shown below). Subtract  base (10) from each of the numbers and write down the remainders (3 and 4) on the right-hand side with their deviation sign (+).
 13 x 14
 13      +3                          (as before,subtract 10 from both numbers)
 14      +4
-----------
 17  |    2
          1           = 182
-----------

Left side (4 ways to calculate left side)

a) 13 + (+ 4)= 17                       (add top left to bottom right)
b) 14 + (+ 3)= 17                       (add bottom left to top right)

c) 13 + 14 - 10= 17                   (add numbers in the left column and subtract from base)
d) 10 + (+ 3) + (+ 4)= 17           (add numbers in the right column and the base)

Right side
(+3) x (+4) = 12                   (multiply numbers of right column)

Since 12 is  2-digit number, so carry the 1 over to the 7. Therefore, left hand side is now
(17 + 1 = 18). And Final answer : 13 x 14 = 182

Some more examples of the same rule :-

 12 x 11      16 x 12      13 x 15      14 x 12
 12   +2       16   +6       13   +3       14   +4
 11   +1       12   +2       15   +5       12   +2
---------        ---------       ---------        ----------
 13  |   2      18  |   2      18  |   5        16  |  8
                           1               1
---------        ---------        ---------        ----------
 =132          =192           =195          =168

Working base 100
Example: 102 x 107

102     02
107     07
-----------
109  |  14   = 10914
-----------

Working base 1000
Example: 1005 x 1003

1005     005
1003     003
-------------
1008  |  015   =  1008015
-------------

C. One number is above the base and the other number is below it
In this, one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore be subtracted i.e. right side answer get subtracted from (left side answer x base )

Working with Base 10
Suppose we have to multiply 13 by 4
The nearest base is 10.
 13 x 4
 13    +3                   (as before, subtract 10 from both numbers)
  4     -6
----------
  7  |  -18    = 7 x 10 - 18  =  70 - 18  =  52
----------

Left side(4 ways to calculate left side)

a) 13 - 6= 7                        (add top left to bottom right)
b) 4 + 3= 7                         (add bottom left to top right)

c) 13 + 4 - 10= 7                (add numbers in left column, subtract 100)
d) 10 + (+ 3) + (- 6)= 7      (add numbers in the right column and the base)
Right side
(+3) x (-6) = -18                 (multiply numbers of right column)
                         __
So, 13 x 4 = 7 | 18
           = (7 x 10) - 18          (right side answer get subtracted from (left side answer x base ))
           = 70 - 18
           = 52  
Final answer : 13 x 4 = 52

Following are some of the examples:-

 12 x 8            14 x 6             13 x 3          15 x 7
 12    +2         14    +4           13   +3        15    +5
  8     -2           6     -4             3    -7           7    -3
-----------         ----------         ---------         ----------
 10  | -4           10  |  -16         6  | -21       12  |  -15
-----------         ----------         ---------         ----------
 =100-4           =100-16         =60-21        =120-15
 =96                 =84                =39              =105

Working base 100
Example: 102 x 97

102     02
 97    -03
-----------
 99  | -06
----------
  = 9900-6
  = 9894

Working base 1000
Example: 1005 x 993

1005      005
 993     -007
--------------
 998  |  -035
--------------
 = 998000-35
 = 997965

We shall discuss about rest of the two cases in next article. Stay tuned!!

Next Article will cover:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.

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Vedic Math - Mathematical magic trick

Let us take a break from calculations and enjoy a small magical mathematics trick.

1. Think a number (any number). To keep it easy, think any number having 1 or 2 digit.
2. Now double this number i.e. multiple by 2.
3. Add 12 to the result
4. Divide the total by 2.
5. Subtract the original number from above result.
6. And you will always get '6' as final result.

Try this one with different numbers, and you will see that the sequence always produces the '6' as result ,no matter which number was originally selected.

For example, if the original number is 15.
1. 2 x 15 = 30
2. 30 + 12 = 42
3. 42 / 2 = 21
4. 21 - 15 = 6

You must be curious that how this magic trick is working. Following is the concept for this magic trick:

Let us take a number be x and the steps performed by us are:
1. 2x
2. 2x + 12
3.(2x + 12) / 2 = x + 6
4. x + 6 - x = 6

So, in last step, we see that whatever number we choose (as x) will finally result in '6' with above calculation. After knowing the formula, now you can change the answer by changing the formula. For example, if you want to change the answer to '4', then you add twice of that '4' i.e. '8' in the second step.

Enjoy this trick!!