# Vedic Mathematics IV

__Base Method__

For MultiplicationThis is very suitable when numbers are close to a base like 10, 100, 1000 or so on. Let's take an example:

106 x 108

Here the base is 100 and the 'surplus' is 6 and 8 for the two numbers. The answer will be found in two parts, the right-hand should have only two digits (because base is 100) and will be the product of the surpluses. Thus, the right-hand part will be 6 x 8, i.e. 48. The left-hand part will be one multiplicand plus the surplus of the other multiplicand. The left part of the answer in this case will be 106 + 8 or for that matter 108 + 6 i.e. 114. The answer is 11448.

12 X 14.

10 would the most suitable base. In the current example, the surplus numbers are +2 and +4.

If 8x7 were to be performed and base of 10 were chosen, then -2 and -3 would have been the deficit numbers.

Try the following numbers

(a) 13 X 16 (b) 16 X 18 (c) 18 X 19 (d) 22 X 24

Once you get comfortable, do not use any paper or pen.

__USING OTHER BASES__** **

In 46 X 48, the base chosen is 50 and multiplication of 44 by 50 is better done like this: take the half of 44 and put two zeros at the end, because 50 is same as 100/2. Therefore, product will be 2200. It would be lengthy to multiply 44 by 5 and put a zero at the end. In general, whenever we want to multiply anything by 5, simply halve it and put a zero.

Multiply 32 by 25. Most of the students would take 30 as the base. The method is correct but nonetheless lengthier. Better technique is to understand that 25 is same as one-fourth. Therefore, one-fourth of 32 is 8 and hence the answer is 800.

An application of Base Method to learn multiplications of the type 3238, where unit's digit summation is 10 and digits other than unit's digit are same in both the numbers. In the above example, 2 + 8 = 10 and 3 in 32 is same as 3 in 38. Therefore method can be applied. The method is simple to apply. The group of digits other than unit's digit, in this case 3, is multiplied by the number next to itself. Therefore, 3 is multiplied by 4 to obtain 12, which will form the left part of the answer. The unit's digits are multiplied to obtain 16 (in this case), which will form the right part of the answer. Therefore, the answer is 1216.

Try these now

53 X 57 91 X 99 106 X 104 123 X 127

The rule for squares of numbers ending with 5. e.g., 652. This is same as 65 X 65 and since this multiplication satisfies the criteria that unit's digit summation is 10 and rest of the numbers are same, we can apply the method. Therefore, the answer is 42 / 25 = 4225.

Try these:

352 952 1252 2052

__CUBING__** **

Finding the cubes of numbers close to the powers of 10. e.g., cubes of 998, 1004, 100012, 10007, 996, 9988, etc. Some of the numbers are in surplus and others are in deficit. Explain the method as given below.

Find (10004)^{3}

Step (I) : Base is 10000. Provide three spaces in the answer. The base contains 4 zeros. Hence, the second and third space must contain exactly 4 digits.

1 0 0 0 4 = ?/ ?/ ?

Step (II) : The surplus is (+4). If surplus is written as 'a', perform the operation '3a' and add to the base 10000 to get 10012. Put this in the 1st space.

1 0 0 0 4 = 1 0 0 1 2 /?/?

Step (III) : The new surplus is (+12). Multiply the new surplus by the old surplus, i.e. (+4)(+12) = (+48). According to the rule written in the step (I), 48 is written as 0048.

1 0 0 0 4 = 1 0 0 1 2 / 0 0 4 8 /?

Step (IV) : The last space will be filled by the cube of the old surplus (+4). Therefore, 43 = 64, which is written as 0064.

1 0 0 0 4 = 1 0 0 1 2 / 0 0 4 8 / 0 0 6 4

Therefore, the answer is 1001200480064.

Find (998)3

Step (I) : Base = 1000. Hence, exactly 3 digits must be there in the 2nd and 3rd space.The deficit = (+2)

9 9 8 = ?/?/?

Step (II) : Multiply the deficit by 3 and subtract (because this is the case of deficit) from the base.

9 9 8 = 9 9 4 /?/?

Step (III) : (old deficit) x (new deficit) = 2 x 6 = 12

9 9 8 = 9 9 4 / 0 1 2 /?

Step (IV) : The cube of the old deficit = 8. Since it is the case of deficit, -8 should be written. All that you need to do to write the negative number in the third space is to find the complement of the number, in this case 8. But since the third space must have exactly 3 digits, the complement of 008 must be calculated. The complement of 008 is 992. Don't forget to reduce the last digit of the second space number by 1

9 9 8 = 9 9 4 / 0 1 2 / 9 9 2

- 1

9 9 4 / 0 1 1 / 9 9 2

Therefore, the answer is 994011992

As an exercise, try the following :

999943 = 9 9 9 8 2 / 0 0 1 0 8 / 0 0 2 1 6 = 99982/00107/99784

100053 = 1 0 0 1 5 / 0 0 7 5 / 0 1 2 5 = 10015/0075/0125

1000253 = 1 0 0 0 7 5 / 0 1 8 7 5 / 1 5 6 2 5 = 100075/01875/15625

99999883 = 9 9 9 9 9 6 4 / 0 0 0 0 4 3 2 / 0 0 0 1 7 2 8

= 9999964/0000431/9998272

__Multiplying numbers just over 100. __

1. 103 x 104 = 10712

The answer is in two parts: 107 and 12,

107 is just 103 + 4 (or 104 + 3),

and 12 is just 3 x 4.

2. Similarly 107 x 106 = 11342

107 + 6 = 113 and 7 x 6 = 42

__Few more Ideas __

Again, just for mental arithmetic ,quick way to square numbers that end in 5 using the formula BY ONE MORE THAN THE ONE BEFORE.

1. 752 = 5625

752 means 75 x 75.

The answer is in two parts: 56 and 25.

The last part is always 25.

The first part is the first number, 7, multiplied by the number "one more", which is 8:

so 7 x 8 = 56

2. 32 x 38 = 1216

Both numbers here start with 3 and the last figures (2 and 8) add up to 10.

So we just multiply 3 by 4 (the next number up) to get 12 for the first part of the answer.

And we multiply the last figures: 2 x 8 = 16 to get the last part of the answer.

__Method for diving by 9. __

1. 23 / 9 = 2 remainder 5

The first figure of 23 is 2, and this is the answer. The remainder is just 2 and 3 added up!

2. 43 / 9 = 4 remainder 7

The first figure 4 is the answer

and 4 + 3 = 7 is the remainder - could it be easier?

3. 134 / 9 = 14 remainder 8

The answer consists of 1,4 and 8.

1 is just the first figure of 134.

4 is the total of the first two figures 1+ 3 = 4,

and 8 is the total of all three figures 1+ 3 + 4 = 8

Till now, you were multiplying like this:

Question: Multiply 432 by 617.

Answer: 432 x 617

3024

432

2592

266544

More the number of digits in the numbers, more lines and time you consume. No more! Using the Sutra "Vertically and Crosswise", you have

Step 1 (mentally, don't write on notebook) : vertically (last digits) :

2x7=14; write 4 carry 1

Step 2 (mentally) : crosswise (last two digits) :

3x7 +2x1 = 23 +carry 1 = 24; write 4 carry 2

Step 3 : vertically and crosswise (three digits) :

4x7 + 3x1 +2x6 = 43 +carry 2 = 45; write 5 carry 4

Step 4 : (move left; first two digits) :

4x1 +3x6 = 22 +carry 4 = 26; write 6 carry 2

Step 5 : (move left; first digit of each number) :

4x6 = 24 +carry 2 = 26. End.

Write answer : 266544

This is how it appears on notebook :

432

x 617

266544

Multiply 2376 by 4060

2376

x 4060

6x0 = 0; write 0;

7x0 +6x6 = 36; write 6 carry 3;

3x0 +7x6 +6x0 = 42 +carry 3 = 45; write 5 carry 4

2x0 +3x6 +7x0 +6x4 = 42 +carry 4 = 46; write 6 carry 4

2x6 +3x0 +7x4 = 40 +carry 4 = 44; write 4 carry 4

2x0 +3x4 = 12 +carry 4 = 16; write 6 carry 1

2x4 = 8 +carry 1 = 9; write 9. End. Answer is 9646560

**Submitted by*** Hitesh.*