VEDIC MATH – MULTIPLICATION

“The Vedic Math Guy”
Lotus, Indian -Community Monthly Newspaper, Cleveland, February 2003 (www.fica-cleveland.org)
Veda, by definition, is ‘knowledge’. Hence
Vedic Math has a much ancient origin though attributed
to the techniques rediscovered between 1911-1918 (see
January 2003 column in Lotus). Mathematicians from
across the spectrum from Hindu, Buddha and Jaina sub-
cultures have contributed immensely to this body of
knowledge. To learn about Vedic Math in these columns,
I have two objectives.
I. First is to give a sense of the extent of
accomplishments of these scholars and rishis to
the readers. This I will do so by discussing
known works.
II. The second, and a continuing quest in these
columns is to illustrate one technique of Vedic
Math each time.
BHASKARA’S LILAVATI - A MATHEMATICAL
TREATISE
Of the many scholars Bhaskaracharya or
Bhaskara II (1114-1193 C.E.) stands out as a teacher and
poet. According to the description in his book
‘Philosophical Crown Jewel’ [Sidhantashiromani] he
lived either in Southern India – probably south of
modern day Bombay. Under the able tutelage of his
father and teacher Maheshwara a great astronomer,
young Bhaskara mastered mathematics, astronomy,
Panini (Sanskrit) grammar, and poetry. This treatise
written when Bhaskara was 36, consists of four parts:
Arithmetic (Lilavati), Algebra (Bijaganita), Celestial
Globe (Goladhyaya), and Planetary Mathematics
(Grahaganita).
Among these Lilavati stands out. The beauty of
Lilavati is that Bhaskara has been able to distill
mathematics into a poetry form with 261 slokas or
verses. This great mathematician was an excellent
teacher as well, as the two examples below illustrate:
1. In the XVIII’th stanza of Lilavati the author says:
O! you auspicious girl with enchanting eyes of a
fawn, Lilavati,
If you have well understood the above methods of
multiplication
What is the product of 135 and 12?
Also, tell me what number will you obtain when the
product is divided by 12.
2.
In the LIV’th stanza of Lilavati the author gives a
‘word problem’:
Of a group of elephants, half and one third of the
half went into a cave,
One sixth and one seventh of one sixth was
drinking water from a river.
One eight and one ninth of one eighth were
sporting in a pond full of lotuses
The lover king of the elephants was leading three
female elephants; [then], how many elephants
were there in the flock?
Now the reader may be wondering who in the
world was ‘Lilavati’? According to a 1587 translation
by Fyzi (an Arab translator) Lilavati was Bhaskara’s
daughter. A famed astronomer and astrologer,
Bhaskara foresaw that his daughter would not be
married and live happily if she is not wedded at an
auspicious moment. To find the moment, he
constructed a device - a cup with a small hole in its
bottom that was placed in a vessel filled with water.
The auspicious moment would be when the cup that
would sink having slowly filled-up. As fate would
have it, on the wedding day, a pearl from Lilavati’s
dress fell into the cup and blocked the hole and the
auspicious moment passed without her getting
married. Bhaskara then wrote Lilavati to console and
detract his grief stricken daughter to whom he taught
the mathematical techniques.
As the book demonstrates, Bhaskara though a
masterful mathematician, was also a rasika as his
poetry indicates. He teaches his pupil to be mindful of
her surroundings by formulating relevant contextual
word problems in arithmetic, algebra and geometry – a
clear expert in pedagogy. His book has been used as a
standard mathematical text in Indian Gurukulas
(traditional schools) for the last eight hundred years.
NIKHILAM
SUTRA
–
PRELUDE
TO
MULTIPLICATION
To fulfill my second objective, in this column
I will illustrate multiplication of two numbers using a
sutra from Vedic Math called “All from Nine and the
last from Ten” (Sanskrit - Nikhilam Navatashcaramam
Dashatah). I will choose a special case to illustrate
this. But, this can be expanded to any multiplication.
The sutra basically means start from the left most digit
and begin subtracting ‘9’ from each of the digits; but
subtract ‘10’ from the last digit.
1
Example 1: Let us choose the number 6. This has only
one digit, so it is also the last digit. Thus applying
the Nikhilam sutra we have 10 subtracted from 6 to
get ‘-4’.
Nikhilam Sutra
6
-4
Nikhilam Sutra
87
-13
Example 2: Given the number 87, it is clear that the
first digit is 8 and the last digit is 7. Using the sutra:
Subtract 9 from 8 to get ‘-1’; subtract 10 from the
last digit 7 to get ‘-3’.So on the application of the
Nikhilam sutra we get ‘-13’.
NIKHILAM APPLICATION: MULTIPLICATION -
SPECIAL CASE
In the following examples I will take two
numbers and illustrate how to multiply them in a very
quick way using Nikhilam sutra. Even though this
technique works for any pair of numbers, we will look at
a special case when the numbers are near a base such as
10, 100, 1000, etc. We start with a simple example.
Example 3: To multiply 8 and 7. Apply Nikhilam sutra
“All from nine and last from ten” to the number 8 to get
‘-2’ (since there is only one digit so subtract by 10), and
for the number 7 to get ‘-3’. Now write the following:
8
-2
One interesting observation, the origin of the
multiplication sign can be traced to the above ‘cross-
add’ing.
Now you may be wondering that ‘I knew the
answer all along- big deal’. Well, I used a baby
problem to illustrate. I will show you that such
multiplication can be done for two and higher digit
multiplication.
Example 4: To multiply 92 and 89. Apply Nikhilam
Sutra – “All from nine and last from ten” on both the
numbers.
Nikhilam Sutra
89
-11
Nikhilam Sutra
92
-08
•
•
Write this down side by side.
92
-08
X 89
-11
___________
___________
Multiply (-08) and (-11) to get ‘88’.
×
92
-08
× ×
•
•
89 -11
________88_
Now we cross-add. This is done by either adding
92 and -11 to get ‘81’or adding 89 and –08.
92 -08
89 -11
Note that in both operations you get the same answer
that is ‘81’ which is written below to get the solution.
92
-08
7
-3
___________
Multiply (-2) and (-3) to get ‘6’ and write it
down as below.
8
-2
7
-3
________ 6_
Next we ‘cross-add’. That is add 8 and -3 to get
‘5’ or add 7 and -2, to get ‘5’ as shown in the
picture below. Note that in either of the
operations you get the same answer that is ‘5’.
8
-2
×
7
–3
We find the solution by combining the numbers
we found by the above operations as:
8
-2
X 7
-3
__5_____6_
So the answer is ‘56’.
•
•
×
×
89
-11
__81____88_
So the answer to multiplying ’92 × 89’ is ‘8188’.
Again, this technique works very well if the
numbers to be multiplied are near a base. Upon slight
modification this also works very well for any pair of
numbers.
Homework For Fun: Try the “Nikhilam” sutra to
multiply: (i) 85×98, (ii) 995×988. (iii) Bonus problem
105x93. Send answers to vedicmath@hotmail.com.
All correct answers will be acknowledged.

Srinivasa Ramanujan Biography

Born: December 22, 1887
Died: April 26, 1920
Achievements: Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. His most famous work was on the number p(n) of partitions of an integer n into summands.

Srinivasa Ramanujan was a mathematician par excellence. He is widely believed to be the greatest mathematician of the 20th Century. Srinivasa Ramanujan made significant contribution to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Srinivasa Aiyangar Ramanujan was born on December 22, 1887 in Erode, Tamil Nadu. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. At the of five Ramanujan went to primary school in Kumbakonam. In 1898 at age 10, he entered the Town High School in Kumbakonam. At the age of eleven he was lent books on advanced trigonometry written by S. L. Loney by two lodgers at his home who studied at the Government college. He mastered them by the age of thirteen. Ramanujan was a bright student, winning academic prizes in high school.

At age of 16 his life took a decisive turn after he obtained a book titled" A Synopsis of Elementary Results in Pure and Applied Mathematics". The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. The book generated Ramanujan's interest in mathematics and he worked through the book's results and beyond. By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. He was given a scholarship to the Government College in Kumbakonam which he entered in 1904. But he neglected his other subjects at the cost of mathematics and failed in college examination. He dropped out of the college.

Ramanujan lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work. In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover.

On 14 July 1909 Ramanujan marry a ten year old girl S Janaki Ammal. During this period Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. In 191,1 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. He got the job of clerk at the Madras Port Trust with the help of Indian mathematician Ramachandra Rao.

The professor of civil engineering at the Madras Engineering College C L T Griffith was interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers. Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. Hardy wrote back to Ramanujan and evinced interest in his work.

University of Madras gave Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Right from the start Ramanujan's collaboration with Hardy led to important results. In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.

Ramanujan had problems settling in London. He was an orthodox Brahmin and right from the beginning he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research. He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. On February 18, 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and later he was also elected as a fellow of the Royal Society of London. By the end of November 1918 Ramanujan's health had greatly improved.

Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died on April 6, 1920.