Sutra 1: "By one more than the one before"

provides a simple way of multiplication of numbers with the same first digit and the sum of their last unit digits is 10.

1 An interesting application of this formula is in computing squares of numbers ending in five. Consider

35 × 35 = ((3 × 3) + 3)*10 + 25 = 1225 or
= (3 x 4) *10 +(5 x 5) = 1225
125 x 125 = (12 x 13)*10 + 25 = 15625

Explanation: The latter portion is multiplied by itself (5 by 5) and the previous portion is multiplication of first digit and the digit higher to the first digit resulting in the answer 1225

Example 2 :
----------------------

Method 1
45*45 = (( 4*4) + 4 )*10 + 25
200
25
------
2025
or

Method 2 :
45 * 45 = (4*5 ) * 100 + 5*5
= 20 * 100 + 25
= 2000 + 25
= 2025


2 It can also be applied in multiplications when the last digit is not 5 but the sum of the last
digits is the base (10) and the previous parts are the same. Consider

37 × 33 = (3 × 4)*100 + (7 × 3) = 1221


Example 2 :

29 × 21 = (2 × 3)*100 + (9 × 1) = 609


Example 3 :

32 * 38 = ( 3 * 4) * 100 + 8 * 2

1200
16
-------
1216

3 The Sutra is very useful in its application to convert fractions into their equivalent
decimal form.

Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.

a) ( Multiplication Method )


Using multiplication. The sutra "one more than the one before" provides a simple way of calculating values like 1/x9 (e.g 1/19, 1/29, etc). Let's take one 1/x9 and calculate e.g. 1/19. In this case, x=1. To convert 1/19 to decimals, since 19 is not divisible by 2 or 5, the fractional result is a purely circulating (recurring) decimal.

Example 1

Steps :

1) start with the numerator digit.Ie 1 apply the sutra 1. ie By one more than the one before
multiply the first digit by the number one more than before.
let the number be x = 1 , this number should be multiplied by numbr 1 more than before. ie x+1 = 2.

b) Place the number before the original number.

2x = 21 ( where x=1 ). Here is should not be consired to moultiply. Just prefix 2 before the original number.

3) multiply the first number from 2 & prefix the number to preivios number.
now the number is
(2*2)21
= 421
(4*2)421
= 8421

(8*2) 8421
= (16)8421 , if the number obtained after multiplying by 2 is greater than 10.keep the number as carryover & add it to the next digit
= 68421 ( carryover is 1 )

(6*2)68421
= ( 12 ) 68421 , add the carryover (1) to the number 12
= ( 13 ) 68421
= 368421 carryover 1

= (3*2) 68421
= 768421

= (7*2) 768421
= 4768421 carry over 1

= 9768421

Now we have 9 digits of the answer. There are a total of 18 digits (= denominator − numerator i.e 19-1 = 18) in the answer. The last 9 digits can be computed by complementing the lower half (with its complement from nine i.e number + complement = 9):

052631578
947368421

Thus the result is 1/19 = 0.052631578,947368421 repeating.

If you picked up 1/29, you'll have to do it till 28 digits (i.e. 29-1). You'll get the following
1/29 = 03448275862068,
96551724137931
Run this on your calculator and check the result!

Example 2 :

1/19 decimal value =

x = 1

1+1 = 2

= 1
= 21
= 421
= 8421
= 68421
= 368421 1
= 7368421
= 47368421 1
= 947368421
= 8947368421 1
= 78947368421 1
= 578947368421 1
= 1578947368421 1
= 31578947368421
= 631578947368421
= 2631578947368421 1
= 52631578947368421
= 052631578947368421 1
= 1052631578947368421
= 21052631578947368421
= 421052631578947368421
= 8421052631578947368421
= 68421052631578947368421 1
= 368421052631578947368421 1
= 7368421052631578947368421

7368421
052631578947368421

The value od 1/29 = 0.052631578947368421

b) ( Division Method )

Using division. The earlier process can also be done using division instead of multiplication.

For A/X9, We divide A by (1+X). Incase of 1/19, we divide 1/(1+1), the answer is 0 (lets say N) with remainder 1 (lets say D)


1 Divide the number "A" by "1+X".where X value is the first number in the denominator.
2 0.0: D (in this case 1) carries forward and become (D*10 + N) i.e. 10. This is then divided by
2 for N = 5. postfix the value of N to the existng value.
3 0.05 : next dive 5 by 2 , the N = 2 & D =1
4 0.052 Next 12 ( D*10 + N = 1*10 + 2 , 1 from earlier remiander and 2 from the answer) is
divided by 2 for answer 6 ie N = 3 , D = 0
5 0.0526 Next divide 6 by 2 , N = 3 ,D = 0
6 0.05263 Next Divide 3 by 2 , N = 1 D = 1
7 0.052631 next ( D*10 + N = 1*10 + 1 , 1 from earlier remiander and 1 from the answer) is
divided by 2 , ie N = 5 , D = 1
8 0.0526315 next ( 1*10 + 5 = 15 ) by 2 , ie N = 7 D =1
9 0.05263157 next ( 1*10 + 7 ) N 8 , D = 1
10 0.052631578

Value of 1/19 = 0.052631578 , use the calculator to verify the answer.


Example 2 :

For A/X9For A/X9, We divide A by (1+X). Incase of 1/19, we divide 1/(1+1), the answer is 0 (lets say N) with remainder 1 ( lets say D)

x= 2
1/29 = 1/1+2 - = 1/3 N = 0 , D = 1

0.0 1*10 + 0 ( D*10 + N ) , Divide by 3 N=3 D=1

0.03 1*10 + 3 = 13 / 3 Divide by 3 , N = 4 , D = 1
0.034 1*10 + 4 = 14 / 3 ie N = 4 . D = 2
0.0344 2*10 + 4 = 24 / 3 ie N = 8 , D = 0
0.03448 0 + 8 = 8 / 3 = N = 2 , D = 2 0.034482 20 + 2 = 22 / 3 ie N = 7 , D = 1
0.0344827

Vedic Mathematics

Vedic mathematics is a system of mathematics consisting of a list of 16 basic sutras, or aphorisms , that allegedly encompass all mathematics. They were presented by a Hindu scholar and mathematician, Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century (Trivedi, 1965).

Tirthaji and the rediscovery of Vedic Mathematics

The ancient system of Vedic Mathematics was rediscovered from the Sanskrit texts known as the Vedas, between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). At the beginning of the twentieth century, when there was a great interest in the Sanskrit texts in Europe, Bharati Krsna tells us some scholars ridiculed certain texts which were headed 'Ganita Sutras'- which means mathematics. They could find no mathematics in the translation and dismissed the texts as rubbish. Bharati Krsna, who was himself a scholar of Sanskrit, Mathematics, History and Philosophy, studied these texts and after lengthy and careful investigation was able to reconstruct the mathematics of the Vedas. According to his research all of mathematics is based on sixteen Sutras, or word-formulae.

Bharati Krsna wrote sixteen volumes expounding the Vedic system but these were unaccountably lost and when the loss was confirmed in his final years he wrote a single book: Vedic Mathematics, currently available. It was published in 1965, five years after his death.

Background Information on the Vedas

The word "veda" has two basic meanings. The first, a literal translation of the Sanskrit word, is "knowledge" (Veda). The second, and most common meaning of the word, refers to the sacred ancient literature of Hinduism, the Vedas, a collection of hymns, poetry and Hindu ceremonial formulas (Veda). Believed to be one of the oldest human written records, the Vedas date back over 4000 years Traditionally, they were passed down orally and adapted from generation to generation by sacred sages called rishis, before eventually emerging written in Vedic, an ancient form of Sanskrit, during the 3rd century BCE (Veda).

The Vedas are divided into four main sections: the Rig-veda, Sama-veda, Yajur-veda and the Atharva-veda, known collectively as the Samhitas (Veda). The first three, the Rig-veda, Sama-veda, and Yajur-veda are basically ritual handbooks that were used by priests during the Vedic period (1500-500 BCE) (Veda). Vedic mathematics is apparently part of the fourth Veda, Atharva-veda, which is distinct from the others in several ways. First, unlike the religious focus of the other Vedas, the Atharva-veda contains hymns, spells and magical incantations for personal and domestic use (Veda). Also, the Atharva-veda, which was written later than the other Vedas, was not always considered authoritative, but only became so after being accepted by the Brahmans, the highest order of Hindu priests (Veda). Collectively, the Vedas do include information about a huge range of subjects, spanning religion, medicine, architecture, astronomy.

Applications of Vedic Mathematics

The most notable application of Vedic mathematics is in education. Vedic mathematical strategies may prove to be a useful resource for teachers and students, who may find elements of it easier and more accessible to teach and learn than conventional mathematics. In particular, these strategies may be an invaluable resource to students that already struggle with mathematics, and could benefit from alternative approaches.

Sutras of Vedic Mathematics

Merely 16 Sutras and 13 Sub-Sutra form the basis of Vedic Mathematics. These sutras can be used to solve all types of Mathematical Problems!!!

These sutras had been long lost in the cryptic Vedas where the mathematical solutions were hidden in Sanskrit Verses for centuries. Sri Bharati Krsna Tirthaji's "Vedic Mathematics" recently discovered these techniques again and given it to all of us to use.

The 16 sutras, or rules, and 14 sub-sutras can be used to solve complex mathematical problems mentally, leaving your fellow students and colleagues in awe

The 16 Sutras of Vedic Mathematics:

Sutra 1 : By one more than the one before

Sutra 2: All from 9 and the last from 10

Sutra 3: Vertically and crosswise

Sutra 4 : Transpose and apply

Sutra 5 : If the Samuccaya is the same it is zero

Sutra 6 : If one is in ratio the other is zero

Sutra 7: By addition and by subtraction

Sutra 8: By the completion or non-completion

Sutra 9: Differential calculus

Sutra 10: By the deficiency

Sutra 11: Specific and general

Sutra 12: The remainders by the last digit

Sutra 13: The ultimate and twice the penultimate

Sutra 14: By one less than the one before

Sutra 15: The product of the sum

Sutra 16: All the multipliers

The 13 Sub-sutras:

SS 1: Proportionately

SS 2: The remainder remains constant

SS 3: The first by the first and the last by the last

SS 4: For 7 the multiplicand is 143

SS 5: By osculation

SS 6: Lessen by the deficiency

SS 7: Whatever the deficiency lessen by that amount and set up the square of the deficiency

SS 8: Last totaling 10

SS 9: Only the last terms

SS 10: The sum of the products

SS 11: By alternative elimination and retention

SS 12: By mere observation

SS 13: The product of the sum is the sum of the products