This interesting article was forwarded by a friend. I wanted to share this with you.
Mathematics is the queen of subjects. Rightly so, then, Vedic Mathematics is the glowing crown that adorns its proud forehead. Very few of the masses today are aware today, of Vedic mathematics, the magnanimity of its profound implications, and of its origins which guided the rest of the world towards purer and more intricate branches of mathematics and which, laid the foundations for number theory and arithmetic, the teeny-weeny part of which we are taught during our alma-mater days with hardly any reference made to its rightful owners ? our very own ancestors ?who pursued the study of mathematics with no less finesse than that of a fine art.
A well-known fact it is, as
every one knows now, for he/she has seen himself/herself or his friend, being
answered by his teacher, during the primary years of his education, in response
to his/her query full of childish criticism “What has India given to
Mathematics? that the numeral 0 was indeed the creation of Indian
mathematicians. Introduction of zero brought about a new revolution into the
world of mathematics. It was zero that gave rise to the idea of representing
numbers using base 10, as it is commonly used today. And it is zero because of
which you are able to read this article. But why? How would a computer work
without zeros and ones!!! So that’s the zero there, right!
Though the Arabs are given the credit of taking mathematics into broader
frontiers, they had begun their work with the help of Indian manuscripts. The
story goes something like this. It was in 773 that the Arabs were able to set
their eyes on the astounding developments of numerical methods Indians used when
one of the Indian palmist and fortune-teller happened to visit the Arabian
lands. So impressed were the Arab mathematicians with Indian inventions that the
Arab mathematician Muhammed-Ibna-Musa-Abu-Jafar-Al-Khwarizmi himself came to
India to study Indian mathematics. After stating here for some time after
learning the subjects to his satisfaction, he wrote his manuscript “Algebra
’–b-e-Mukabla? This is how ‘Algebra?was born. His works, which were nothing but
a translation of his Indian studies, left the European mathematicians
spell-bound, especially by the use of base 10 to represent numbers. The idea of
representing numbers by base 10, is thus, originally Indian.
The trend then caught on. From Arabs to Greeks, from Greece to Spain and from
Spain to Europe. Europeans however, initially reluctant to use base 10 to
represent numbers, inevitably began to use it in 1202 (during the time of
Bhaskaracharya). Though the mathematical works went on improving as it changed
hands, the world availed the first systematically documented use of base 10 only
in Bhaskaracharya’s manuscript “Lilavati? 4th century mathematician Diaphantus,
who is also known as ‘Father of Arithmetic? has his works in his book
‘Arithmetica?coinciding closely with the Indian manuscripts of earlier age.
These are but a few evidences that arithmetic and basic mathematics that has
evolved today in various forms, is but the creation of those great olden Indian
minds.
In this article, we take a
glimpse of the very fantastic inventions of Vedic mathematics in the field of
number theory and arithmetic. Though cited here are examples of the methods
described in Vedas , Vedic mathematics has its frontiers expanding into many
more diverse areas and concepts like astrology, astronomy, geometry,
trigonometry, to name a few. The examples
cited here would be useful in performing calculations that you may be coming
across everyday. This article would be of course, a welcome change to those who
complain that they hardly get time to stretch their legs during those
interminably tightly timed aptitude tests! I will like to share some experts of
a Book called "DrutGanit"(Speed mathematics) by Mr. Shyam Marathe.... The book
is in Marathi discussing the techniques of doing mathematical operation of big
numbers very quickly using the techniques present in "Vedas" ....
We shall start with a method to multiply 2
numbers. It will reveal to you how innovative Vedic mathematics can be.
Lets deal, to begin with, with numbers that are close to POWERS of 10 i.e.
numbers near to 10, 100, 1000, etc.
Example 1:
Say, we have to multiply 98 and 97.
(First, try using conventional method and find out how much time it takes!)
Now we go for the Vedic method.
Write down the numbers in this manner:
98 -2
97 -3
In the above representation, -2 and –3 stand for the difference of 98 and 97
from 100. We may call these differences –2 and –3 as OFFSETS of 98 and 97 from
100. We may call 100 as the ‘BASE’ as differences are taken from 100.
Step 1) First, we multiply the offsets –2 and –3
We get 6.
Since our base is 100, which has 2 zeros, the product of offsets must also have
2 digits. Hence we write 6 as 06 and write it down as last 2 digits of our
answer.
98 -2
97 –3
-------------------------
06
Step 2) Now for the previous digits, just add any 2 numbers in the above
figure crosswise i.e. either (98-3) or (97-2)
We get 95. This is written before 06 as follows: -
98 -2
97 –3
---------------------------
95 / 06
That gives us 9506. That’s our answer.
Hence 98 x 97 = 9506
Aint it simple!!
Example 2:
Once again, we solve 25 x 98.
25 -75
98 -2
---------------------------
Here, once again, our base is 100. -75 and –2 are offsets of 25 and 98 from 100.
Step 1) -75 x –2 gives 150
Since our base (100) has 2 zeros, the product of offsets must have 2 digits.
Hence we write down last 2 digits of 150 i.e. 50 as last 2 digits of our answer.
1 would be treated as carry.
25 -75
98 -2
---------------------------
50 ( 1 carry )
Step 2) Add the numbers crosswise. i.e. ( 25-2 ) or ( 98 – 75 ).
We get 23.
Add the carry from previous step.
We get 23 + 1 = 24
These are written as first 2 digits of our answer.
25 -75
98 -2
---------------------------
(23+1) / 50
i.e.
25 -75
98 -2
---------------------------
24 / 50
Answer: 25 x 98 = 2450
Example 3:
Now, lets try for 108 x 109
Let us write down the numbers as:
108 8
109 9
---------------------------
and carry out the same method as above.
Note that 8 and 9 are offsets of 108 and 109 from base 100.
Step 1) Product of offsets 8 x 9 = 72
Write down 72 as last digits of our answer.
108 8
109 9
--------------------------
72
Step 2) Add any 2 numbers cross wise ( 108 + 9 ) or ( 109 + 8 )
We get 108 + 9 = 117
These are the front digits of the answer.
108 8
109 9
---------------------------
117 / 72
Hence 108 x 109 = 11772
Example 4:
Lets try 98 x 104.
98 -2
104 4
---------------------------
Step 1 ) Product of offsets –2 x 4 = -8
We write it down as,
98 -2
104 4
---------------------------
-8
Step 2 ) Now, 98 + 4 or 104 –2 gives 102.
These are the earlier digits of our answer.
98 -2
104 4
---------------------------
102 / -8
Step 3) Since the number after “/ ” is as negative number we have to
decrease the number to the left of “/” by 1 ( i.e. 102 becomes 101 ). Next, the
number 8 (which has a negative sign) has to be deducted from the number base
times 1 i.e. deduct 8 from 100 x 1.
i.e. 100 – 8 = 92
92 is written in the place of –8. That makes our result as: -
98 -2
104 4
--------------------------
101 / 92
Hence 104 x 98 = 10192.
Lets see how it simplifies our calculation for huge numbers:
Example 5:
Multiply 888 by 998: -
888 -112
998 -002
---------------------------
886 / 224
Here, the base is 1000 i.e. offsets are differences from 1000
-112 x – 002 = 224 ( Simple enough )
And 888 – 002 = 886
Hence, 888 x 998 = 886224.
Example 6:
Multiply 9997 by 9998:-
9997 -3 ( base 10,000 )
9998 -2
------------------------------
9997 – 2 ) / 0006
( Here, base is 10,000, which has 4 zeros. Hence the product of offsets
–2
and –3 is padded with zeros to make it a 4-digit number )
Hence,
9997 -3
9998 -2
------------------------------
9995 / 0006 = 99950006 (Answer!!!)
Example 7:
Multiply 999979 by 999998: -
999979 -21 (base 1,00,000)
999998 -2
--------------------------------
( 999979 – 2 ) / 000042
Here, -21 x –2 = 42.
Base has 6 zeros. Hence pad 42 with 4 zeros to make 000042.
Then, 999979 – 2 = 999977. Those are the earlier digits of our answer.
999979 -21
999998 -2
------------------------------
999977 / 000042
Hence, 999979 x 999998 = 999977000042.
Ever wondered multiplying such big numbers could be that simple!!!
The methods discussed above are extremely effective for multiplying numbers near
the powers of 10. Similar methods can be applied to find the product of ANY 2
numbers (irrespective of whether they are near powers of 10) using ANY MULTIPLE
of 10 as base.