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Vedic Mathematics

Speed and joyful calculus for children

Lesson 4 specific vedic math techniques.

Dr. Tiziano VALENTINUZZI

Introduction on vedic techniques.

In Vedic Mathematics there are two different kind

of techniques, the general and the specific ones.

The specific ones relate to very fast and

effective ways to solve mathematical operations,

but can be applied only to a specific combination

and/or collection of numbers.

On the other hand, the general techniques have a

much wider scope of application, as the

criss-cross system of multiplication which can be

used to multiply any possible combination of

numbers-

As a first example we will see how to square

numbers ending with 5 this technique can be used

only for that numbers, and can never be applied

to other type of numbers.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Squaring numbers ending with 5.

If we want to obtain the square of a number

ending with 5, we just use the following rule,

which comes from the by one more than the one

before sutra

We ignore the 5 digit, take the remaining digits

and multiply them by itself incremented by 1.

Then join a 25 (5x5) at the end of the obtained

number.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Squaring numbers ending with 5.

The same rule applies also for bigger numbers

Algebraic proof (ax5)2a(a1)x225 where

x10

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

A corollary to the previous rule.

The same rule (Ekadhikena purvena) applies also

if we are multiplying to numbers that have the

same numbers in each position digits, and the

last two numbers sum up to 10

Algebraic proof (axb)(ax10-b)a(a1)x2b(10-b)

where x10

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

A corollary to the previous rule.

This can be effective also with a different

separation of digits. For example 397 and 303

have 97 and 3 which add up to 100, so 397x303

12 0291 where 123x4 and 029197x3. Special care

has to be taken to the digits of the second

multiplication, as we are multiplying 2 digit

figures, we need four digits in the answer.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Squaring of numbers between 50 and 60.

Just add 25 to the units digit and put it to the

left end. Then square the units digit and put the

result to the right end

Algebraic proof (50a)2100(25a)a2

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Multiplication of numbers with a series of 9.

Using the following method we can instantly

multiply any given number by another one made

only by 9s.

Case 1 multiply a number with an equal number of

digits 9.

Subtract 1 from the first number and put it at

the left hand of the result, then subtract each

of the digits from 9, and write them to the right

hand of the result. And here you are!

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Multiplication of numbers with a series of 9.

Case 2 multiply a number with an higher number

of digits 9.

Just follow the previous rule, but add as many 0

before the first number as needed to match the

same number of 9 digits.

The previous cases come under the Ekanyukena

Sutra-by one less than the one before is used, in

combination with the Nikhilam Stura-all from 9

and the last from ten.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Multiplication of numbers with a series of 9.

Case 3 multiply a number with a lower number of

digits 9.

354 x 99 -------- 35400 354

35046

1547 x 999 ------------1547000

1547 1545453

1547x(9991)

354x(991)

In this case the rule is not very effective, but

the result is easily recovered any way

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Multiplication of numbers by 11.

Multiply numbers by 11 is a easier as the 11

times table

52 x 11 5 52 2 572

To multiply a two figures number by 11 we put

down the two extremes, and put in the middle the

sum of the two figures. The case of a carry over

is treated in the usual way 57 x 11 5 57

7 627

1

To multiply bigger numbers is just a an extension

of the same rule

132x11 1 13 32 2 1452 1563 1

15 56 63 3 17193

1

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

A corollary to multiplication by 11.

We can apply the same rule to augment of a 10

any number, by using the sutra Anurupyena-proporti

onally

Increase the number 345 by 10 345 x 11 3

34 45 5 3795 So the result is 379.5

We just multiply the number by 11 and then divide

by 10!

It is worth noticing that the usual

multiplication by 11 is easy done too.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Cube roots of perfect cubes.

Note that this method works only for perfect

cubes, otherwise it will lead to incorrect answer.

This method implies the memorization of a

key-list. This will stimulate also the mnemonic

skills of the students. It has to be said that in

vedic times, there werent written books, but all

the knowledge were pass to the students out of

memory, and out of memory were learned.

The key-list establish a one-to-one relation

between the numbers fro 1 to 10 and the

bolded-underlines numbers of the cubes.

Thus if any given cube ends with 2, its cube root

will end with 8, if the cube ends with 3, its

cube root will end with 7, and so on.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Cube roots of perfect cubes.

Find the cube root of 205379!

STEP 1 We always, and in any case, have to put a

slash before the last three digits, in this case

we will represent the number in the following way

3

205 379

STEP 2 the cube ends with 9, so the cube root

will end with 9 too.

STEP 3 the number 205 lays in between 125 and

216, corresponding to cube roots of 5 and 6,

respectively. We choose the lower one, 5.

3

205 379

5 9

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Cube roots of perfect cubes.

Find the cube root of 681472!

3

8 8

681 472

Cube ends with 2 so the cube root ends with 8

681 between 512(83) and 729(93) so take 8

For numbers of more than 6 digits, the method is

the same, the only thing to do is to extend the

key-list to subsequent numbers.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Digit sums.

The digit sum of a number in found by adding the

digits in a number and adding again till a single

figure is reached.

The digit sum of 32 is 325. The digit sum of

23502 is 2350212 --- 123.

So, every number, no matter how long it is, can

be reduced to a single digit number, called the

digit sum. If the digit sum is 9, than it is

equivalent to 0. Lets see visually how the digit

sums work

(0)

Adding 9 to a number, doesnt affect the digit

sum

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Digit sums.

This means that anytime we find the figure 9 or a

number of figures which summed gives 9, they can

just be deleted or removed from the digit sum.

The digit sum of 5344 is 347. The digit sum of

43652 is 2. The digit sum of 5454 is 0.

(0)

The digit sum is conserved during all the

operations, lets check this with some examples

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Checking results of operations.

The digit sum of the result of a multiplication,

is the product of the digit sums of the factors.

1236x4353148 (3)x(7)21 (3)

The digit sum of the result of an addition, is

the sum of the digit sums of the addends.

1236431279 (3)(7)10 (1)

The digit sum of the result of a subtraction, is

the subtraction of the digit sums of the

subtrahends.

1236-431193 (3)-(7)-49 (5)

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Subtraction from a base.

Applying the vedic sutra all from 9 and the last

from 10 to a number, we get the amount needed to

reach the next base number (power of 10).

36 64100-36

445 5551000-445

1427 8573 10000-1427

Using the sutra by one less than the one before

we can find the subtraction from a multiple of a

base.

36 164200-36

445 15552000-445

This is quite useful when calculating the change

when we pay something with a big note. When we

are paying some goods costing 5,34 euros with a

20,00 euros note, then me expect

As the change!

534 1466 20,00-5,3414,66

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Bar numbers.

The number 18 is very close to 20, so it is

possible to represent it as 20-2, graphically 22,

which we will pronounce two, bar two.

37 43

495 505

1921 2079

To convert a number in a bar form, you can use

the all from 9 and the last 10 ten sutra combined

with the by one less than the one before in a

sort of reverse mode.

Bar numbers can be found also in the middle of a

number, in which case you just split the number

and apply the sutras to the particular section of

the number you are looking at

534 534 474

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Advantages of bar numbers.

- Large numbers like 6,7,8,9 difficult to multiply

are removed. - Figures tend to cancel each other, or can be made

to cancel. - 0s and 1s occur much more frequently, simplifying

calculus.

A simple application is connected to subtraction.

Sometimes pupils subtract in each column

regardless of whether the top is greater than the

bottom or not.

4 5 4 2 8 6 --------- 2 3 2 168

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Anurupyena - proportionately.

Doubling and halving numbers is very easy to do,

and can help many times to rapidly solve

numerical problems.

For example if we want to halve 56, we just halve

50/225 and 6/23 and add the results 25328.

In the same way if we want to find out 35x4, then

we figure out 35x270 and the 70x2140.

If we want to divide 104 by 8, then we figure out

104/252, then 52/226 and the result 26/213,

that is 104/813.

This helps a lot the pupils to train their mental

capabilities, and speed in calculus.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Anurupyena - proportionately.

The same can be used to extend the multiplication

tables. You just need to know the multiplication

tables till 5, the other tables come right from

there

If you know 4x728, then immediately you get

8x756 (282).

Or more generally you can extend them in any way

6x14(6x7)x284 or 6x14(7x3)x484

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Division by 9.

The number 9 is special, as we saw in the digit

sum and in the numbers wheel. There is a very

straightforward way to divide by 9, even very

large numbers

9) 2 3 1 1 2 5 6 r 7

2311 / 9

You just bring down the first digit, and then sum

up in a cross fashion. This technique is just

summing, not even dividing anything.

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Special cases of division by 9.

Lets see what happens with big remainders and

carry over

9) 3 1 7 2 3 4 11 r 13

3172 / 9

1

1

We get 11 at the 3rd digit, so we carry over a 1

to the previous digit. The remainder is divisible

per 9, so (13-9)4 is the remainder, and we carry

over a 1 to the previous digit.

The result is 352 remainder 4

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Division by 11.

Its the same principle, but we subtract instead

of summing.

11) 3 4 1 1 3 1 0 r 1

3411 / 11

You just bring down the first digit, and then

subtract 3 from4, put down the result 1, then

subtract 1 from 1, put down the resulting 0, and

at last subtract 0 from1, and the remainder is 1.

In the case of negative numbers, the bar numbers

can be used

11) 5 2 3 5 3 r 6

523 / 11

4 7 r 6

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Using the average.

This comes under the formula Specific General.

29x31

The average is 30

The difference from 30 of the numbers is 1

So the result is 302-12900-1 899

26x34

The average is 30

The difference from 30 of the numbers is 4

So the result is 302-42900-16 884

Algebraic proof (ab)(a-b)a2-b2

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Divisibility.

The Sopantyadvayamantyam sutra-the ultimate and

twice the penultimate tests for divisibility by

4 just add the last figure to the double of the

previous, if the result is divisible by 4, the

whole number will be. 12376 67x220 is

divisible, so 12376 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Divisibility.

The principle of the Ekadhika the Ekadhika for

the divisibility is one more than the one before

when the number ends in 9, it is also called

positive osculator. So the Ekadhika for 69 is 7

(61) The Ekadhika for 39 is 4 (31) The

Ekadhika for 13 is 4 (31) because 13 must be

multiplied by 3 to get a 9 in the last digit, so

13x339

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Divisibility.

Osculation we osculate a number by multiplying

its last figure by the osculator, and adding the

previous figure. Find if 91 is divisible by

7. The Ekadhika of 7 is 5, so we osculate 91

with 5 9 1 14 5

Multiply by 5

Add 9

14 is divisible by 7, so 91 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Divisibility.

We can go on osculating 14 with 5, we will get

21, also divisible by 7, and so on...

Find if 78 is divisible by 13. The Ekadhika of

13 is 4, so we osculate 78 with

4 7 8 39 32

Multiply by 4

Add 7

39 is divisible by 13, so 78 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Testing longer numbers.

In the testing number have more than 2 figure,

the procedure is easily extended

Find if 247 is divisible by 19. The Ekadhika of

19 is 2, so we osculate 247 with

2 2 4 7 36 14 38 18

Add 4

Multiply by 2

Add 2

Multiply by 2

38 is divisible by 19, so 247 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Testing longer numbers.

There is a short-cut for testing longer numbers

Ekadhika is 2 2 4 7 1 18 (8x212)

19

19 is divisible by 19, so 247 is too

Its a sort of carry over for osculators

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Other divisors.

If we want to test if 6308 is divisible by 38 we

just see (Vilokanam subsutra-by mere observation)

that 3819x2, so as soon as we have tested the

number is divisible by two, we just test if it is

divisible by 19.

Is 5572 divisible by 21? 217x3, digit sum of

5572 is 1, so it is not divisible by 21!

Is 1764 divisible by 28? 287x4, the number

formed by the last two figures of 5572 is

divisible by 4, so we just test if it is

divisible by 7. The Ekadhika is

5. 1 7 6 4 49 39 26

49 is divisible by 7, so 1764 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Negative osculators.

To get the negative osculator of a number, we

have to get a 1 at the end of the number, and

take the preceding number. The negative

osculator of 17 is 5, in fact 17x351

Is 3813 divisible by 31? The negative osculator

of 31 is 3.

We will bar to every other figure of the number,

we then osculate as normal except that any carry

figure is counted as negative 3 8 1 3 3 2 8

6 0

0 is divisible by 31, so 3813 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi

Negative osculators.

Is 59948 divisible by 14? 142x7, the negative

osculator of 7 is 2, the number is obviously

divisible by 2.

5 9 9 4 8 12 1 1 12 7 4 4 6 12

7 is divisible by 7, so 59948 is too

Lesson 4 specific vedic math techniques.

Dr.

Tiziano Valentinuzzi