9

1. Introduction to Vedic MathematicsThe term Vedic mathematics indicates that mathematics which

is related to the Vedas.Then, what is Veda? Veda means knowledge,which includes both the spiritual knowledge and worldlyknowledge. The Holy Vedas are the principal and primary sacredtexts of the followers of Sanatana dharma. They are believed to beeternal truths of the cosmos revealed to the sages during theirpenance.They are believed to be of great antiquity dating back tobeginning of time itself. The Vedas are the grand heritage- theorigin of all sciences and knowledge systems.MATHEMATICS IN VEDAS & VEDANGAS

In addition to the intellectual debates of highest level, likeUpanishads, on various aspects of spiritual knowledge, the Vedasextensively deal with the rituals also for the upliftment of thehumanity. These rituals involve several topics which are directlyrelated to the mathematics. Hence a thorough knowledge ofmathematics is essential for proper performance of the rituals.

The books of Vedas and Vedangas involve mathematicsextensively. The Kalpa, dealing with the preparation of altars, oryajnavedikas, utilizes several geometrical principles, which arecalled Sulba sutras. Presently few of them traced are in the namesof Boudhayana, Apastamba, Katyayana and others.

Because of the vast coverage of mathematics and itssignificance, the Jyotisha is adored from times immemorial.Lagadha, the great mathematician of India of about 1500b.c., statesin his book of Vedanga Jyotisha that “Like the crest of a peacock,and like the gems on the hoods of serpants, the mathematics liesat the top of all knowledge systems.”

Similarly the Chandas has lot of knowledge related to themathematics related to the permutations and combinations.

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3. Digital Roots of NumbersDescription

It is well known that a number can have any number ofdigits. The process of addition of individual digits of thegiven number results in generating a smaller number, which mayhave more than one digit. The digits of this smaller numberobtained, again when added individually, will result in generatingfurther smaller number. This process, when repeatedly carried out,will ultimately result in generating a single digit. That ultimatesingle digit is called the digital root of the original given number.Example -1:Find the digital root of the number 6974Answer :6 + 9 + 7 + 4 = 262 + 6 = 8The digital root of the given number 6974 = 8

Example-2 :Find the digital root of the number 123456Answer :1 + 2 + 3 + 4 + 5 + 6 = 212 + 1 = 3The digital root of the given number 123456 = 3

Example-3 :Find the digital root of the number 7869431Answer :7 + 8 + 6 + 9 + 4 + 3 + 1 = 383 + 8 = 111 + 1 = 2The digital root of the given number 7869431 = 2

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9. Multiplications 3 (Eka Nyunena Purvena)

Eka Nyunena PurvenaMeaning : By one less than the previous one.

Example 1 : Method 1 :1. Put four blanks in the result.Step 1 :i) The number preceding 99 = 68ii) Complement of 68 = 32iii)Put this complement in the two blanks on RHS of the re-sult.Present Status :

Step 2 :i) The number preceding 99 = 68ii) A number that is less by 1 than the given number= iii) Put this 67 in the two blanks on LHS of the result.

∴

Sutra :

12

Positional Notation in Valmiki RamayanaIt may be appropriate to review the decimal number system

mentioned in Valmiki Ramayana, which describes a mindbogglingextension of its terminology. The names were assigned high rangingvalues starting from 107 (that is, one crore), up to 1062, in successivemultiples of one lakh times of the previous number. Their referencewas noticed in the context of expressing the army strength ofSri Rama, who was preparing for an assault on Lanka.

Koti (one crore) 107

Sanku (one lakh crores) 1012

Maha Sanku 1017

Vrinda 1022

Maha Vrinda 1027

Padma 1032

Maha Padma 1037

Kharva 1042

Maha Kharva 1047

Samudra 1052

Ogha 1057

Mahaugha 1062

(Each one of them is one lakh times the previous number)

In addition to the above, some more are noticed in various booksof Sanskrit literature as given below:

Utsanga – 1021

Tithilamba – 1027

Hetuhilam – 1031

Nitravadyam – 1041

Sarvabala – 1045

Tallakshanam – 1053

These are only few references for illustration purpose.

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Ex 1 : 2 5 7 x 9 9 9

We divide our answer into two parts.

LHS (Left hand side) - We write 1 less than multiplicand.

2 5 6 /

Step 2: RHS - We write the complement of multiplicand by using allfrom nine and last from ten.

Complement of 2 5 7 is 7 4 3 .

2 5 6 / 7 4 3

Ans : 2 5 7 x 9 9 9 = 2 5 6 7 4 3Ex 2 : 1 3 6 x 9 9 9

LHS = 1 3 6 - 1 = 1 3 5

1 3 5 /

RHS complement of 1 3 6 is 8 6 4.

1 3 5 /

Ans : 1 3 5 8 6 4 .Rules : We get our answer in the following steps. 1. LHS is one less than multiplicand.

2. RHS is complement of multiplicand.3. This rule is applicable when digits in multiplicand should be equal to number of digits in multiplier.

Exercise : Solve the sums1. 4 6 x 9 9 2. 5 9 x 9 9 3. 7 8 x 9 94. 1 8 6 x 9 9 9 5. 7 4 8 x 9 9 9 6. 6 4 8 2 x 9 9 9

Multiplications - Eka Nyunena PurvenaIn the whole number system 9 is very interesting number.

Therefore multiplication by 9 is also very interesting. If we have anumber to be multiplied by a number consisting of only 9’s we usethe sub sutra.

"By one less than the previous" and"All from nine and the last from ten".

First Type :When the number of digits in the multiplie and multiplicand are

same and all the digits of multiplier are 9’s.

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Multiplication Table 3

TTable DDigital Sum 1

22

3

4

56

7

8

99

Multiplication Table 2

TTable DDigital Sum 1

22

3

4

56

7

8

99

Lesson : Geometrical Symmetry in Multiplication Tables

.1.

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Multiplications - Diagonal Method

Topic : Diagonal MethodDetails :1. The multiplication of two numbers can be carried out

easily with the diagonal method.

2. We have to draw number of columns, equal to the numberof digits in the first number. Put the digits above thecolumns.

3. We have to draw number of rows, equal to the number ofdigits in the second number. Put the digits along the rows.

4. Then we get a grid of boxes. Draw diagonal linesconnecting the corners.

5. Multiply the digits of the columns and the rows. Put thedigits of the products in the corresponding boxes on eitherside of the boxes suitably.

6. At the end of all multiplications, add all the numbers fallingbetween two diagonals.

7. Make necessary adjustments and get final answer.

Example 1:1. Given first number = 9

This number has a single digit. Hence draw one column.Put the digit 9 above the box.

2. Given second number = 5

This number has a single digit. Hence draw one row. Putthe digit 5 on the side of the box.

3. Ultimately there is one row only. That row has only onebox. Draw a diagonal connecting two corners.

Sutra :

32

Division by 25Hint = 2 5 x 4 = 100

Multiply the given number by 4.Move the decimal point from right to left by two places.

Ex 1 : 5 2 / 2 5 = ?

5 2 x 4 = 2 0 8

Ans : 2 .0 8

Ex2 : 7 8 3 / 2 5 = ?

7 8 3 x 4 = 3 1.3 2

Ans : 3 1.3 2

Ex 3 : 5 0 7 2 / 2 5 = ?

5 0 7 2 x 4 = 2 0 2 8 8

Ans : 2 0 2 .8 8

Division by 5

Hint = 5 x 2 = 10

Multiply the given number by 2.

Move Decimal point from right to left by one place.

Ex 1 : 6 8 / 5 = ?

6 8 x 2 = 1 3 6

Ans : 1 3.6

Ex2 : 3 1 7 / 5 = ?

3 1 7 x 2 = 6 3 4

Ans : 6 3 . 4

Ex 3 : 4 1 3 6 / 5 = ?

4 1 3 6 x 2 = 8 2 7 2

Ans : 8 2 7 . 5

Divisions by 5; 25 and 125

.110

Squares - 101 type

Topic: Finding the squares for numbers having 1’s on both sidesand zeros in the middle

Explanation:The numbers having 1’s on both sides and zeros in the middlewill be as follows :

It is required to calculate squares for the above numbers.

Procedure:

1. Let the given number be indcated by X.

2. It can be observed that the given number is obtained by puttingzeros in between 1 and 1 of the number 11.

3. The number of zeros in between 1’s may be indicated by N.

4. It is well known that the square of 11 is 121.

5. The result will be obtained by writing N number of zeros inbetween 1 and 2, and again N number of zeros in between2 and 1.

6. The Result gives the square of the given number.

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Squares - Technique 3TOPIC: To find the squares of numbersProcedure : Processing From left to right1. This method commences with the selection of the

following:.i) processing digit, andii) set of digits to the left of the processing digit.

3. The processing digit involves the following steps.Value 1: Calculate 2* (set of digits to the left of the processingdigit)* (processing digit)Value 2: Calculate square of the processing digit.iii) Write the obtained values in separate lines, with a shiftof one place.

4. The above procedure has to be repeated for all other digitsalso, proceeding from left to right of the given number.

5. Finally, the addition of all the values obtained above givesthe square of the given number.

Explanation1. Let ‘abc’ represent a given three digit number.2. Here ‘a’ represents the digit in hundreds place,

‘b’ represents the digit in tens place, and.‘c’ represents the digit in units place.

3 Now the processing has to start from left.The processing digit selected is ‘a’;There are no digits to the left of ‘a’.Value1 is taken as zero as there are no digits to the left of ‘a’.Value2 =a*a.

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8. Multiply the middle values by 2 to get the second row.

Second Row

9. Sum

Result =

Left to Right Method :

Given number = abab

2. In this case we select the ratio b/a.

b/a = 2/1 = 2

3. Initialy we start with the cube of the digit in the highestplace (a3).

1 x 1 x 1 = 1

4. For the next lower place, the above value is multipliedby b/a. This gets us a2b (=a3xb/a)

1 x 2 = 2

5. For the next lower place, the above value is multipliedby b/a. This gets us ab2 (=a2bxb/a)

2 x 2 = 4

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4. Cube root of 50653 =a b

Example 2: Find the cube root of 10648.Procedure:1. (a) Given number=10648 (b) Group the digits of the given number into sets of three,

starting from right to left. (that is, from units placeonwards towards left).Xgroup number =10Y group number=648

2. (a) The processing may be started with the number in theX group.

(b) Compare the X group number (=10) with the number inthe column of cube values.of Table 1.

(c) Identify from Table-1, the smaller number and biggernumber closer to X group number.

(d) Smaller number=8Bigger number =27

(e) Select the smaller number.(=8)(f) Take the smaller number as a cube value(=8), and note

its corresponding cube root (=2)from Table-1.Take this cube root value as ‘a’.a=2

3. (a) Now take up the processing of Y groupnumber(=648)

128

√1 3 3 1 √6 8 5 9 √ 3 3 7 5

√2 7 4 4 √5 8 3 2 √4 1 9 3

√6 8 9 2 1 √1 7 5 7 6 √1 0 0 0

√2 4 3 8 9 √9 7 3 3 6 √8 8 4 7 3 6

√7 2 9 0 0 0 √2 5 0 0 4 7 √1 2 5 0 0 0

3 3 3

3 3 3

3 3 3

3 3 3

3 3 3

Cube Roots

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