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Title: Diapositiva 1

1
Vedic Mathematics
Speed and joyful calculus for children
Lesson 4 specific vedic math techniques.
Dr. Tiziano VALENTINUZZI
2
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
3
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
4
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
5
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
6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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
26
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
27
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
28
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
29
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
30
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
14 is divisible by 7, so 91 is too
Lesson 4 specific vedic math techniques.

Dr.
Tiziano Valentinuzzi
31
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
39 is divisible by 13, so 78 is too
Lesson 4 specific vedic math techniques.

Dr.
Tiziano Valentinuzzi
32
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
Multiply by 2
Multiply by 2
38 is divisible by 19, so 247 is too
Lesson 4 specific vedic math techniques.

Dr.
Tiziano Valentinuzzi
33
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
34
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
35
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
36
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