This document discusses Vedic mathematics and methods for addition, subtraction, and multiplication based on ancient Indian techniques. Some key points: - Vedic mathematics uses direct, mental approaches to solve problems in one line. - Addition is done by adding the place values from right to left and carrying over if needed. - Multiplication involves multiplying the place values and carrying over similar to standard algorithms. - Subtraction borrows from the next place value when the top number is smaller, working from right to left. - Special methods are described for multiplying near a base of 10 or 100 by subtracting/adding the amounts above or below the base.

1. VEDIC MATHEMATICSVEDIC MATHEMATICS
R.SANKARR.SANKAR
ASSISTANT PROFESSOR,ASSISTANT PROFESSOR,
DEPARTMENT OF MATHEMATICS,DEPARTMENT OF MATHEMATICS,
PROFESSONAL GROUP OF INSTITUTIONS,PROFESSONAL GROUP OF INSTITUTIONS,
PALLADAM-641662.PALLADAM-641662.
BYBY

2. INTRODUCTIONINTRODUCTION
 Vedic Mathematics is the ancient system ofVedic Mathematics is the ancient system of
Mathematics which was rediscovered early lastMathematics which was rediscovered early last
century by “century by “ Sri Bharati Krishna TirtajiSri Bharati Krishna Tirtaji
(1884-1960)(1884-1960)”.”.
 Vedic Mathematics is a new approach toVedic Mathematics is a new approach to
mathematics, direct, one-line and mentalmathematics, direct, one-line and mental
solutions to mathematical problem.solutions to mathematical problem.
 The following chapters explains how easy theThe following chapters explains how easy the
mathematics.mathematics.

3. TOPICSTOPICS
Left to Right Calculation:Left to Right Calculation:
1. Addition1. Addition
2. Multiplication2. Multiplication
3. Subtraction3. Subtraction
Digit Sum (Checking)Digit Sum (Checking)
Special MethodsSpecial Methods
All from 9 and Last from 10All from 9 and Last from 10
SquaringSquaring
DivisibilityDivisibility
RemaindersRemainders
Special NumbersSpecial Numbers

4. Left to Right CalculationLeft to Right Calculation

5. ADDITIONADDITION
Normal Form:Normal Form:
58+77=?58+77=?
5858
7777
135135

6. Vedic FormVedic Form
58+77=?58+77=?
5858
7777
135135
Sum ofSum of Ten-thTen-th place digitsplace digits 5+7=1205+7=120
Sum ofSum of UnitUnit place digitsplace digits 8+7= 158+7= 15
135135

7. Example : 2Example : 2
97+96=?97+96=?
9797
9696
193193
Sum ofSum of Ten-thTen-th place digits 9+9= 180place digits 9+9= 180
Sum ofSum of UnitUnit place digits 7+6= 13place digits 7+6= 13
193193

8. 3-digit and more….3-digit and more….
789+999=?789+999=?
Sum ofSum of 100-th100-th place digits 7+9=1600place digits 7+9=1600
Sum ofSum of Ten-thTen-th place digits 8+9= 170place digits 8+9= 170
Sum ofSum of UnitUnit place digits 9+9= 18place digits 9+9= 18
17881788
We can continue this process for any digitWe can continue this process for any digit
valuevalue

9. Example : 2Example : 2
578+764=?578+764=?
Sum ofSum of 100-th100-th place digits 5+7=1200place digits 5+7=1200
Sum ofSum of Ten-thTen-th place digits 7+6= 130place digits 7+6= 130
Sum ofSum of UnitUnit place digitsplace digits 8+4= 128+4= 12
13421342
We can continue this process for any digitWe can continue this process for any digit
valuevalue

10. Practice 1Practice 1
 875 + 746875 + 746 = ?= ?
 375 + 96375 + 96 = ?= ?
 1565 + 17451565 + 1745 = ?= ?
 4688 + 46514688 + 4651 = ?= ?
 685 + 592685 + 592 = ?= ?
 88525+ 47469 = ?88525+ 47469 = ?
 87848+ 12345 = ?87848+ 12345 = ?
 68598+ 99999 = ?68598+ 99999 = ?
 9484899+99484899+9 = ?= ?
 Ans:Ans:
 ??

11. MULTIPLICATIONMULTIPLICATION
Find 584 × 8 = ?Find 584 × 8 = ?
Normal Form :Normal Form :
 584 × 8584 × 8
46724672
We can use Vedic method to solve theWe can use Vedic method to solve the
above problem easily.above problem easily.

12. Vedic FormVedic Form
 Find 584 × 8 = ?Find 584 × 8 = ?
 Multiplication ofMultiplication of 100-th100-th place 5 × 8 = 4000place 5 × 8 = 4000
 Multiplication ofMultiplication of Ten-thTen-th place 8 × 8 = 640place 8 × 8 = 640
 Multiplication ofMultiplication of UnitUnit place 4 × 8 = 32place 4 × 8 = 32
46724672
Practice 2Practice 2,, Try this:Try this:
 989 × 9=?989 × 9=?
 68778 × 5=?68778 × 5=?

13. Practice 2Practice 2
Try this:Try this:
 989 × 9=?989 × 9=?
 68778 × 5=?68778 × 5=?
 899474 × 8 =?899474 × 8 =?
 Sometimes theSometimes the
addition of the No’s isaddition of the No’s is
10 r more, here we10 r more, here we
carry the one to thecarry the one to the
previous no andprevious no and
continue the process.continue the process.
 989 × 9= 8100989 × 9= 8100
720720
8181
89018901
 989 × 9 = 8901989 × 9 = 8901

14. SUBTRACTIONSUBTRACTION
Find 625 – 183 = ?Find 625 – 183 = ?
Normal Form:Normal Form:
 625 –625 –
183183
442442
Here in the unit digit 5-3=2 and in the 10Here in the unit digit 5-3=2 and in the 10thth
place borrow 10 from 6 and make 2 as 12place borrow 10 from 6 and make 2 as 12
and then subtract 8 we get 4 and 100and then subtract 8 we get 4 and 100thth
place 5-1=4place 5-1=4

15. Vedic FormVedic Form
 Find 625 – 183 = ?Find 625 – 183 = ?
 We subtract in each column on the left, but before we putWe subtract in each column on the left, but before we put
an answer down we look in the next column.an answer down we look in the next column.
 IfIf the top is greater than the bottom we put the figure downthe top is greater than the bottom we put the figure down
 If not,If not, we reduce the figure by 1, put that down and give the otherwe reduce the figure by 1, put that down and give the other
1 to smaller number at the top of the next column1 to smaller number at the top of the next column
 If the figures are the same we look at the next column to decideIf the figures are the same we look at the next column to decide
whether to reduce or notwhether to reduce or not
 625 -625 -
183183
442442
 Here in the unit digit 6-1=5. Now we look at the nextHere in the unit digit 6-1=5. Now we look at the next
column, here the top 2 is less than the bottom 8, so wecolumn, here the top 2 is less than the bottom 8, so we
put down 4 in the 1put down 4 in the 1stst
column and carry 1 in the nextcolumn and carry 1 in the next
column top so 12-8=4 in that column and look at the nextcolumn top so 12-8=4 in that column and look at the next
column 5 is greater than 3 so 5-3=2column 5 is greater than 3 so 5-3=2

16. Practice 3Practice 3
 68 – 25 = ?68 – 25 = ?
 813 – 489 = ?813 – 489 = ?
 986 – 584 = ?986 – 584 = ?
 6226-2662= ?6226-2662= ?
 5161-1838 = ?5161-1838 = ?
 35567-12346=?35567-12346=?
 486645 – 359 = ?486645 – 359 = ?
 Ans:Ans:
 ??

17. DIGIT SUM (CHECKING)DIGIT SUM (CHECKING)
 This is an interesting and also very useful to checkThis is an interesting and also very useful to check
our answer.our answer.
 TheThe digit sumdigit sum of a number is found by adding theof a number is found by adding the
digits in a number and adding again if necessarydigits in a number and adding again if necessary
until a single figure is reached.until a single figure is reached.
 Example:Example:
 consider the no:78158consider the no:78158
 Sum of the digit is7+8+1+5+8=29Sum of the digit is7+8+1+5+8=29
=2+9=2+9
=11=11
=1+1=1+1
=2=2
 Any pair or group of digits which add up to 9 can beAny pair or group of digits which add up to 9 can be
deleted.deleted.

18. CHECKING THE ANSWERCHECKING THE ANSWER
 58+77=?58+77=?
 5858
7777
135135
 The digit sum of 58=5+8=13=4The digit sum of 58=5+8=13=4
 The digit sum of 77=7+7=14=5The digit sum of 77=7+7=14=5
 Total digit sum is 4+5=9Total digit sum is 4+5=9
 From the answer,From the answer,
The digit sum of 135=1+3+5=9The digit sum of 135=1+3+5=9
 There fore from & , our answer is correct.There fore from & , our answer is correct.
 Check the answers from practice 1,2 and 3Check the answers from practice 1,2 and 3
1
2
1 2

19. Special MethodsSpecial Methods

20. MULTIPLICATION NEAR AMULTIPLICATION NEAR A
BASEBASE
Numbers just below the baseNumbers just below the base
Numbers just above the baseNumbers just above the base
Above and Below the baseAbove and Below the base
With different baseWith different base

21. NUMBERS JUST BELOW THENUMBERS JUST BELOW THE
BASEBASE
Find 98 × 94 = ?Find 98 × 94 = ?
Normal Form:Normal Form:
 98 × 9498 × 94
392392
882882
92129212

22. In Vedic form here we useIn Vedic form here we use 100100 as a baseas a base
98 × 94 = ?98 × 94 = ?
98 - 0298 - 02
94 - 0694 - 06
92 / 1292 / 12
Subtract 98-06 or 94-02 we get 92Subtract 98-06 or 94-02 we get 92
Multiply 02 × 06 we get 12Multiply 02 × 06 we get 12
98 × 94 =921298 × 94 =9212
Vedic FormVedic Form

23. Example:2Example:2
 88 × 89 = ?88 × 89 = ?
 88 - 1288 - 12
89 - 1189 - 11
77 / 13277 / 132
 Subtract 88-11 or 89-12 we getSubtract 88-11 or 89-12 we get
7777
 Multiply 12 × 11 we get 132Multiply 12 × 11 we get 132
 We can’t put the answer likeWe can’t put the answer like
this 77132this 77132
 Here from 132, 1 carry to 77Here from 132, 1 carry to 77
and it becomes 78, thenand it becomes 78, then
 88 × 89 = 783288 × 89 = 7832
 To Multiply 12 and 11,we canTo Multiply 12 and 11,we can
use 10 as a base.use 10 as a base.
 12 + 0212 + 02
11 + 0111 + 01
13/213/2

24. Practice 4Practice 4
 91 × 89 = ?91 × 89 = ?
 92 × 92 = ?92 × 92 = ?
 88 × 85 = ?88 × 85 = ?
 86 × 97 = ?86 × 97 = ?
 91 × 92 = ?91 × 92 = ?
 94 × 97 = ?94 × 97 = ?
 85 × 85 = ?85 × 85 = ?
 98 × 98 = ?98 × 98 = ?
 Ans:Ans:
 ??

25. NUMBERS JUST ABOVE THE BASENUMBERS JUST ABOVE THE BASE
Find 103 × 104 = ?Find 103 × 104 = ?
Normal Form:Normal Form:
 103 × 104103 × 104
412412
000000
103103
1071210712

26. Vedic FormVedic Form
In Vedic form here we useIn Vedic form here we use 100100 as a baseas a base
103 × 104 = ?103 × 104 = ?
103 + 03103 + 03
104 + 04104 + 04
107 / 12107 / 12
Add 103+04 or 104+03 we get 107Add 103+04 or 104+03 we get 107
Multiply 03 × 04 we get 12Multiply 03 × 04 we get 12
103 × 104 = 10712103 × 104 = 10712

27. 125 × 105 = ?125 × 105 = ?
125 + 025125 + 025
105 + 005105 + 005
130 / 125130 / 125
Add 125+005 or 105+025 we get 130Add 125+005 or 105+025 we get 130
Multiply 25 × 5 we get 125Multiply 25 × 5 we get 125
We can’t put the answer like this 130125We can’t put the answer like this 130125
Here from 125, 1 carry to 130 and itHere from 125, 1 carry to 130 and it
becomes 131, thenbecomes 131, then
125 × 105 = 13125125 × 105 = 13125

28. Practice 5Practice 5
 128 × 112 = ?128 × 112 = ?
 109 × 105 = ?109 × 105 = ?
 131 × 109 = ?131 × 109 = ?
 125 × 125 = ?125 × 125 = ?
 114 × 112 = ?114 × 112 = ?
 107 × 108 = ?107 × 108 = ?
 113 × 113 = ?113 × 113 = ?
 102 × 125 = ?102 × 125 = ?
 Ans :Ans :
 ??

29. ABOVE AND BELOW THE BASEABOVE AND BELOW THE BASE
Find 102 × 95 = ?Find 102 × 95 = ?
Normal Form:Normal Form:
 102 × 95102 × 95
510510
918918
96909690

30. Vedic FormVedic Form
 In Vedic form here we useIn Vedic form here we use 100100 as a baseas a base
 102 × 95 = ?102 × 95 = ?
 102 + 02102 + 02
95 - 0595 - 05
97 / 1097 / 10
 Add 95 + 02 or Subtract 102-05 we get 97Add 95 + 02 or Subtract 102-05 we get 97
 Multiply 97 with 100 =9700Multiply 97 with 100 =9700
 Multiply 02 × 05 we get 10Multiply 02 × 05 we get 10
 Finally we get the answer from 9700-10Finally we get the answer from 9700-10
 102 × 95 = 9690102 × 95 = 9690
10

31.  136 × 90 = ?136 × 90 = ?
 136 + 36136 + 36
90 - 1090 - 10
126 / 360126 / 360
 Add 90+36 or Subtract 136-10 we get 126Add 90+36 or Subtract 136-10 we get 126
 Multiply 126 with 100 =12600Multiply 126 with 100 =12600
 Multiply 36 × 10 we get 360Multiply 36 × 10 we get 360
 Therefore 12600-360 we get the answerTherefore 12600-360 we get the answer
 136 × 90 = 12240136 × 90 = 12240

32. Practice 6Practice 6
 146 × 80 = ?146 × 80 = ?
 97 × 145 = ?97 × 145 = ?
 98 × 125 = ?98 × 125 = ?
 139 × 95 = ?139 × 95 = ?
 141 × 90 = ?141 × 90 = ?
 128 × 96 = ?128 × 96 = ?
 126 × 89 = ?126 × 89 = ?
 130 × 99 = ?130 × 99 = ?
 Ans:Ans:
 ??

33. WITH DIFFERENT BASEWITH DIFFERENT BASE
• 9997 ×9997 × 98 = ?98 = ?
• Here the numbers areHere the numbers are
close to differentclose to different
bases:10,000 and 100bases:10,000 and 100
• The deficiencies areThe deficiencies are
-3 and -2.-3 and -2.
• Therefore: 9997 – 03Therefore: 9997 – 03
98 – 0298 – 02
9797/069797/06
• 02 is not subtracted from the02 is not subtracted from the
last two digit (97) of 9997,last two digit (97) of 9997,
but from 99 of 9997.but from 99 of 9997.
• And 03 is a deficiency fromAnd 03 is a deficiency from
10,000 so we can’t subtract it10,000 so we can’t subtract it
from 98,because it’s a basefrom 98,because it’s a base
of 100of 100
• Mulply 98 by 100 andMulply 98 by 100 and
subtract 3 also give the anssubtract 3 also give the ans
• 9997 ×9997 × 98 = 97970698 = 979706

34. Practice 7Practice 7
999 × 80999 × 80= ?= ?
9987 × 989987 × 98 = ?= ?
99995 × 99899995 × 998 = ?= ?
96 × 896 × 8 = ?= ?
10004 × 10810004 × 108 = ?= ?
9985 × 9969985 × 996 = ?= ?
98889 × 9998889 × 99 = ?= ?
 Ans :Ans :
 ??

35. 10,20,… As a Base10,20,… As a Base
 We can use 10 as a baseWe can use 10 as a base
for single digit numbers.for single digit numbers.
 Ex: 7Ex: 7 × 8=?× 8=?
 7-37-3
8-28-2
5/65/6
 77 × 8=56× 8=56
 We can use 20 as a baseWe can use 20 as a base
for single digit numbers.for single digit numbers.
 Ex: 25 × 24=?Ex: 25 × 24=?
 25 + 0525 + 05
24 + 0424 + 04
29/2029/20
 Now Multiply 29 by 2, weNow Multiply 29 by 2, we
get 58 again multiply byget 58 again multiply by
10,we get 580.10,we get 580.
 Add 580 with (5 × 4)=20Add 580 with (5 × 4)=20
 25 × 24=60025 × 24=600

36. Practice 8Practice 8
 88 × 9× 9 =?=?
 2222 × 23 =?× 23 =?
 1818 × 24 =?× 24 =?
 2929 × 34 =?× 34 =?
 5454 × 56 =?× 56 =?
 3838 × 39 =?× 39 =?
 4545 × 42 =?× 42 =?
 99 × 5× 5 =?=?
 Ans:Ans:
 ??

37. ALL FROM 9 AND LAST FROMALL FROM 9 AND LAST FROM
1010
 Subtraction From aSubtraction From a
Base:Base:
 If we apply the formulaIf we apply the formula
to 854 we get 146to 854 we get 146
because 8 and 5 arebecause 8 and 5 are
taken from 9 and 4 istaken from 9 and 4 is
taken from 10.taken from 10.
 1000 – 46 = ?1000 – 46 = ?
 1000 -1000 -
046046
954954
 Subtract the unitsSubtract the units
digit from 10, thendigit from 10, then
each successive digiteach successive digit
from 9, then subtractfrom 9, then subtract
1 from the digit on the1 from the digit on the
left.left.
 60000 – 34843 = ?60000 – 34843 = ?
 60000 -60000 -
3484334843
2515725157

38. SQUARINGSQUARING
Digits ends with five and zeroDigits ends with five and zero
Two digit numbers (aTwo digit numbers (a22
+2ab+b+2ab+b22
) form) form

39. DIGITS ENDS WITHDIGITS ENDS WITH FIVEFIVE ANDAND
ZEROZERO
75 × 75 =?75 × 75 =?
Normal FormNormal Form
 75 × 7575 × 75
375375
525525
56255625

40. Vedic FormVedic Form
75 × 75 =?75 × 75 =?
Here we use n(n+1)/25 formulaHere we use n(n+1)/25 formula
Let n= 7Let n= 7
 n+1= 8n+1= 8
 n(n+1) = 7n(n+1) = 7 × 8 = 56× 8 = 56
Therefore 75 × 75 =5625Therefore 75 × 75 =5625

41. 3,4,… digits ends with 53,4,… digits ends with 5
135 × 135 = ?135 × 135 = ?
Let n=13Let n=13
n(n+1)=13(14)n(n+1)=13(14)
169+13=182169+13=182
135 × 135 =18225135 × 135 =18225
PracticePractice
151522
,25,2522
,etc..,etc..
 169+13=?169+13=?
 169169
1313
 11
0 70 7
1212
182182

42. DIGITS ENDS WITHDIGITS ENDS WITH ZEROZERO
 101022
=?, 100=?, 100
 202022
=?=?
22 × 2 = 4 and put 00× 2 = 4 and put 00
We get 400We get 400
 1250125022
=?=?
 125/0125/0
 125=12/5125=12/5
 12*13/5*5=156/2512*13/5*5=156/25
 1250125022
= 1562500= 1562500

43. TWO DIGIT NUMBERSTWO DIGIT NUMBERS (A(A22
+2AB+B+2AB+B22
))
FORMFORM
 36 × 36 =?36 × 36 =?
 A=3 & B=6A=3 & B=6
 3322
= 9= 9
 2*3*6 = 362*3*6 = 36
 6622
= 36= 36
 Therefore 9,36,36Therefore 9,36,36
12961296
 Left to RightLeft to Right
Calculation:Calculation:
 9+3=129+3=12
 6+3= 096+3= 09
 66 = 06= 06
12961296

44. Practice 9Practice 9
 89 × 8989 × 89 =?=?
 26 × 2626 × 26 =?=?
 93 × 9393 × 93 =?=?
 78 × 7878 × 78 =?=?
 61 × 6161 × 61 =?=?
 92 × 9292 × 92 =?=?
 44 × 4444 × 44 =?=?
 55 × 5555 × 55 =?=?
 Ans:Ans:
 ??

45. Left Side Same Digit and Addition ofLeft Side Same Digit and Addition of
Right Digit is 10Right Digit is 10
82 × 88 =?82 × 88 =?
Normal FormNormal Form
 88 × 8288 × 82
176176
704704
72167216

46. Vedic FormVedic Form
Here we are going to use n(n+1) formulaHere we are going to use n(n+1) formula
i.e, we take n=8 and get 8(8+1)=8(9)=72i.e, we take n=8 and get 8(8+1)=8(9)=72
and 2 × 8=16and 2 × 8=16
 Therefore 82 × 88 = 7216Therefore 82 × 88 = 7216

47. Practice 10Practice 10
 89 × 81 =?89 × 81 =?
 26 × 24 =?26 × 24 =?
 93 × 97 =?93 × 97 =?
 78 × 72 =?78 × 72 =?
 61 × 69 =?61 × 69 =?
 92 × 98 =?92 × 98 =?
 44 × 46 =?44 × 46 =?
 Ans:Ans:

48. Doubt?Doubt?
Keep Practice you will clear yourselfKeep Practice you will clear yourself
Contact:Contact:
R.SankarR.Sankar
98658176239865817623