## 藏書亭

#### 第一種情況：個位數互補 十位數相同的乘法

a, b = {1, 2, 3, 4, 5, ...,9}

a x 10 + b

a x 10 + (10 - b)

(a x 10 + b) x {a x 10 + (10 - b)}

= (a x 10) x (a x 10) + (a x 10) x 10 - (a x 10) x b

+ (a x 10) x b + b x (10 - b)

= (a x 10) x {(a + 1) x 10} + b x (10 - b)

= a x (a + 1) x 100 + b x (10 - b)

#### 第二種情況：十位數互補個位數相同的乘法

a, b = {1, 2, 3, 4, 5, ...,9}

a x 10 + b

(10 - a) x 10 + b

(a x 10 + b) x {(10 - a) x 10 + b}

= (a x 10) x {(10 - a) x 10} + (a x 10) x b

+ (b x 10 x 10) - (b x a x 10) + b x b

= {a x (10 - a)} x 100 + (b x 10 x 10} + b x b

= {a x (10 - a)} x 100 + (b x 100) + b x b

= {a x (10 - a) + b} x 100 + b x b

#### 第三種情況： 互補數乘以疊數

a, b = {1, 2, 3, 4, 5, ...,9}

a x 10 + (10 - a)

b x 10 + b

{a x 10 + (10 - a)} x (b x 10 + b)

= (a x 10) x (b x 10 + b) + 10 x (b x 10 + b)

- a x (b x 10 + b)

= (a x 10 + 10) x (b x 10 + b)

- a x (b x 10 + b)

= {(a + 1) x 10} x (b x 10 + b)

- a x (b x 10 + b)

= {(a + 1) x 10} x (b x 10) + {(a + 1) x 10} x b

- a x (b x 10) - a x b

= (a + 1) x b x 100 + a x 10 x b + 1 x 10 x b

- a x b x 10 - a x b

= (a + 1) x b x 100 + 10 x b - a x b

= (a + 1) x b x 100 + (10 - a) x b

#### 第四種情況：兩位數乘法，二個數字離100不遠，補數計算

(100 - a)      a

(100 - b)      b

(100 - a)(100 - b) = 10000 - 100 * (a + b) + ab

= 100 (100 - a - b) + ab

#### Indian Multiplication Table 19 x 19

14 x 12 = ?

step 1：  14 +  2 =  16

step 2：  16 x 10 = 160

step 3：    4 x   2 = 8

step 4： 160 +  8 = 168

10 + a

10 + b

(10 + a) x (10 + b) = (10 + a) x 10 + 10 x b + a x b

= (10 + a + b) x 10 + a x b

#### Vedic Mathematics – Multiplication – Nikhilam Method

Steps to be followed for multiplication

1. The two numbers to be multiplied are written one below the other. Their deviations are written in front of the respective numbers.
2. Answer space is divided into left hand side (LHS) and right hand side (RHS) by a slash.
3. RHS of the answer is the product of the deviations of the numbers and will contain the digits equal to the number of zeroes of the base.
4. LHS of the answer is the sum of one number with the deviation of the other.
5. If RHS contains less number of digits than the number of zeroes in the base , the remaining digits are filled by giving zero or zeroes on the left side of RHS. If the number of digits are more than the number of zeroes in the base , the excess digit or digits are to be added to LHS of answer.
6. Remove the slash to get the answer.

Example 1
Step I : Here numbers (105 , 107) are near base 100. So RHS will have two digits. D1 x D2 = 05 x 07 = 35 (RHS)
Step II : (N1+D2) = 105 + 07 = 112 OR (N2+D1) = 107 + 05 = 112 (LHS)
Step III : Product 11235

Example 2
Step I : Here numbers (98 , 93) are near base 100. So RHS will have two digits. D1 x D2 = (-02) x(-07) = 14 (RHS)
Step II : (N1+D2) = 98 + (-07) = 91 OR (N2+D1) = 93 + (-02) = 91 (LHS)
Step III : Product 9114

Example 3
Step I : Here numbers (992 , 996) are near base 1000. So RHS will have three digits. D1 x D2 = (-008) x(-004) = 032 (RHS)
Step II : (N1+D2) = 992 + (-004) = 988 OR (N2+D1) = 996 + (-008) = 988 (LHS)
Step III : Product 988032

Example 4
Below is the example of 2 digit number multiplication which has base other than 100 and this base need to be modified. Consider example of multiplying 46 x 45. Here base is 50.
Step I : 46 x 45
Step II : A = 46 - 50 = -4
Step III : B = 45 - 50 = -5
Step IV : C = -4 x -5 = 20
Step V : D = 46 - 5 = 45 - 4 = 41
Step VI : Result = 50 x 41 + 20 = 2070

Example 5
Consider example of multiplying 39 x 44. Here base is 40.
Step I : 39 x 44
Step II : A = 39 - 40 = -1
Step III : B = 44 - 40 = 4
Step IV : C = -1 x 4 = -4
Step V : D = 44 - 1 = 39 + 4 = 43
Step VI : Result = 40 x 43 - 4 = 1716

#### Vedic Mathematics – Division – Nikhilam Method

Division by Vedic methods - PDF
1. Nikhilam Sutra (Specific Technique)
2. Paravartya Sutra (Specific Technique)
3. Anurupyena Sutra (Specific Technique)
4. Direct Flag Method (General Technique)
##### Nikhilam Sutra (Specific Technique)
Nihilam Sutra - slideshow
Fastest Vedic methods for Division - Nikhilam Sutra - youtube
Fastest division tricks for Division by 9 - youtube

Nihilam Sutra is a Specific Method to Divide Numbers using Vedic Mathematics. This Vedic Maths Division Method can be applied when Divisor is closer to power of 10 BUT less than that of it. Using Nikhilam Sutra, you can easily divide when divisor is like 98, 92, 995, 89997, etc.

##### Paravartya Sutra (Specific Technique)
Fastest Vedic methods for Division - Paravartya Sutra - youtube

Paravartya Sutra is a Specific Method for division in Vedic Maths. This Vedic Maths Division Method can be applied when Divisor is closer to power of 10 BUT greater than that of it. Using Paravartya Sutra, you can easily divide when divisor is like 123, 104, 1112, etc.

##### Anurupyena Sutra (Specific Technique)
Fastest Vedic methods for Division - Anurupyena Sutra - youtube

Anurupyena Sutra is another Specific Vedic Maths Division Tricks which shows how to divide numbers when Nikhilam and Paravartya are not applicable. Using Anurupyena Sutra, we multiply Divisor by a factor so that either Nikhilam or Paravartya Sutra can be applied.

###### Vinculum Process of Division

Vinculum is another division in vedic maths tricks which can be applied when Divisor has digits greater than 5. Using Vinculum Process, convert those bigger digits to smaller digit and then apply Nikhilam Sutra or Paravartya Sutras of Division.