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VEDIC MATHEMATICSVEDIC MATHEMATICS
R.SANKARR.SANKARASSISTANT PROFESSOR,ASSISTANT PROFESSOR,
DEPARTMENT OF MATHEMATICS,DEPARTMENT OF MATHEMATICS,PROFESSONAL GROUP OF INSTITUTIONS,PROFESSONAL GROUP OF INSTITUTIONS,
PALLADAM-641662.PALLADAM-641662.
BYBY
INTRODUCTIONINTRODUCTION Vedic Mathematics is the ancient system Vedic Mathematics is the ancient system
of Mathematics which was rediscovered of Mathematics which was rediscovered early last century by “ early last century by “ Sri Bharati Krishna Sri Bharati Krishna Tirtaji (1884-1960)Tirtaji (1884-1960)”.”.
Vedic Mathematics is a new approach to Vedic Mathematics is a new approach to mathematics, direct, one-line and mental mathematics, direct, one-line and mental solutions to mathematical problem.solutions to mathematical problem.
The following chapters explains how easy The following chapters explains how easy the mathematics.the mathematics.
TOPICS TOPICS Left to Right Calculation:Left to Right Calculation:
1. Addition1. Addition2. Multiplication2. Multiplication3. Subtraction3. Subtraction
Digit Sum (Checking)Digit Sum (Checking)Special MethodsSpecial MethodsAll from 9 and Last from 10 All from 9 and Last from 10 Squaring Squaring DivisibilityDivisibilityRemaindersRemaindersSpecial NumbersSpecial Numbers
Left to Right Left to Right CalculationCalculation
ADDITIONADDITIONNormal Form:Normal Form:58+77=?58+77=?5858
7777 135135
Vedic FormVedic Form58+77=?58+77=?5858
7777 135135Sum of Sum of Ten-thTen-th place digits place digits 5+7=120 5+7=120Sum of Sum of UnitUnit place digits place digits 8+7= 8+7=
1515 135135
Example : 2Example : 297+96=?97+96=?97979696 193193Sum of Sum of Ten-thTen-th place digits 9+9= 180 place digits 9+9= 180Sum of Sum of Unit Unit place digits 7+6= 13 place digits 7+6= 13 193193
3-digit and more….3-digit and more….789+999=?789+999=?Sum of Sum of 100-th 100-th place digits 7+9=1600place digits 7+9=1600Sum of Sum of Ten-thTen-th place digits 8+9= 170 place digits 8+9= 170Sum of Sum of Unit Unit place digits 9+9= 18place digits 9+9= 18
17881788
We can continue this process for any We can continue this process for any digit valuedigit value
Example : 2Example : 2578+764=?578+764=?Sum of Sum of 100-th100-th place digits 5+7=1200 place digits 5+7=1200Sum of Sum of Ten-th Ten-th place digits 7+6= 130place digits 7+6= 130Sum of Sum of UnitUnit place digits place digits 8+4= 12 8+4= 12
13421342We can continue this process for any digit We can continue this process for any digit
valuevalue
Practice 1Practice 1 875 + 746 875 + 746 = ?= ? 375 + 96375 + 96 = ?= ? 1565 + 17451565 + 1745 = ?= ? 4688 + 4651 4688 + 4651 = ?= ? 685 + 592685 + 592 = ?= ? 88525+ 47469 = ?88525+ 47469 = ? 87848+ 12345 = ?87848+ 12345 = ? 68598+ 99999 = ?68598+ 99999 = ? 9484899+9 9484899+9 = ?= ?
Ans:Ans: ??
MULTIPLICATIONMULTIPLICATIONFind 584 × 8 = ?Find 584 × 8 = ?Normal Form :Normal Form : 584 × 8584 × 8
46724672We can use Vedic method to solve We can use Vedic method to solve
the above problem easily.the above problem easily.
Vedic FormVedic Form Find 584 × 8 = ? Find 584 × 8 = ? Multiplication of Multiplication of 100-th100-th place 5 × 8 = 4000 place 5 × 8 = 4000 Multiplication of Multiplication of Ten-thTen-th place 8 × 8 = 640 place 8 × 8 = 640 Multiplication of Multiplication of UnitUnit place 4 × 8 = 32 place 4 × 8 = 32
46724672
Practice 2Practice 2, , Try this: Try this: 989 × 9=?989 × 9=? 68778 × 5=?68778 × 5=?
Practice 2Practice 2 Try this: Try this: 989 × 9=?989 × 9=? 68778 × 5=?68778 × 5=? 899474 × 8 =?899474 × 8 =? Sometimes the Sometimes the
addition of the No’s addition of the No’s is 10 r more, here is 10 r more, here we carry the one to we carry the one to the previous no and the previous no and continue the continue the process.process.
989 × 9= 8100989 × 9= 8100 720720 81 81 89018901
989 × 9 = 8901 989 × 9 = 8901
SUBTRACTIONSUBTRACTIONFind 625 – 183 = ?Find 625 – 183 = ?Normal Form:Normal Form: 625 –625 –
183183 442442
Here in the unit digit 5-3=2 and in Here in the unit digit 5-3=2 and in the 10the 10thth place borrow 10 from 6 and place borrow 10 from 6 and make 2 as 12 and then subtract 8 we make 2 as 12 and then subtract 8 we get 4 and 100get 4 and 100thth place 5-1=4 place 5-1=4
Vedic FormVedic Form Find 625 – 183 = ?Find 625 – 183 = ? We subtract in each column on the left, but before We subtract in each column on the left, but before
we put an answer down we look in the next column.we put an answer down we look in the next column. IfIf the top is greater than the bottom we put the figure down the 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 we reduce the figure by 1, put that down and give the
other 1 to smaller number at the top of the next columnother 1 to smaller number at the top of the next column If the figures are the same we look at the next column to If the figures are the same we look at the next column to
decide whether to reduce or notdecide whether to reduce or not
625 -625 -183183442442
Here in the unit digit 6-1=5. Now we look at the next Here 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 column, here the top 2 is less than the bottom 8, so we put down 4 in the 1we put down 4 in the 1stst column and carry 1 in the column and carry 1 in the next column top so 12-8=4 in that column and look next column top so 12-8=4 in that column and look at the next column 5 is greater than 3 so 5-3=2at the next column 5 is greater than 3 so 5-3=2
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: ??
DIGIT SUM (CHECKING)DIGIT SUM (CHECKING) This is an interesting and also very useful to This is an interesting and also very useful to
check our answer.check our answer. The The digit sumdigit sum of a number is found by adding of a number is found by adding
the digits in a number and adding again if the digits in a number and adding again if necessary until a single figure is reached.necessary until a single figure is reached.
Example:Example:consider the no:78158consider the no:78158Sum 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 Any pair or group of digits which add up to 9 can be deleted.can be deleted.
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 There fore from & , our answer is
correct.correct. Check the answers from practice 1,2 and 3Check the answers from practice 1,2 and 3
1
21 2
Special MethodsSpecial Methods
MULTIPLICATION NEAR A MULTIPLICATION NEAR A BASEBASE
Numbers just below the baseNumbers just below the baseNumbers just above the baseNumbers just above the baseAbove and Below the baseAbove and Below the baseWith different baseWith different base
NUMBERS JUST BELOW THE NUMBERS JUST BELOW THE BASEBASE
Find 98 × 94 = ?Find 98 × 94 = ?Normal Form:Normal Form: 98 × 94 98 × 94
392392 882882 92129212
In Vedic form here we use In Vedic form here we use 100100 as a as a basebase
98 × 94 = ?98 × 94 = ?98 - 0298 - 02
94 - 0694 - 0692 / 1292 / 12
Subtract 98-06 or 94-02 we get 92Subtract 98-06 or 94-02 we get 92Multiply 02 × 06 we get 12Multiply 02 × 06 we get 1298 × 94 =921298 × 94 =9212
Vedic FormVedic Form
Example:2Example:2 88 × 89 = ?88 × 89 = ? 88 - 1288 - 12
89 - 1189 - 1177 / 13277 / 132
Subtract 88-11 or 89-12 we Subtract 88-11 or 89-12 we get 77get 77
Multiply 12 × 11 we get Multiply 12 × 11 we get 132132
We can’t put the answer We can’t put the answer like this 77132like this 77132
Here from 132, 1 carry to Here from 132, 1 carry to 77 and it becomes 78, then77 and it becomes 78, then
88 × 89 = 783288 × 89 = 7832
To Multiply 12 and 11,we To Multiply 12 and 11,we can use 10 as a base.can use 10 as a base.
12 + 0212 + 0211 + 0111 + 0113/213/2
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: ??
NUMBERS JUST ABOVE THE BASENUMBERS JUST ABOVE THE BASE
Find 103 × 104 = ?Find 103 × 104 = ?Normal Form:Normal Form: 103 × 104 103 × 104
412412 000000
10310310712 10712
Vedic FormVedic Form In Vedic form here we use In Vedic form here we use 100100 as a as a
basebase103 × 104 = ?103 × 104 = ?103 + 03103 + 03
104 + 04104 + 04107 / 12107 / 12
Add 103+04 or 104+03 we get 107Add 103+04 or 104+03 we get 107Multiply 03 × 04 we get 12Multiply 03 × 04 we get 12103 × 104 = 10712103 × 104 = 10712
125 × 105 = ?125 × 105 = ?125 + 025125 + 025
105 + 005105 + 005130 / 125130 / 125
Add 125+005 or 105+025 we get 130Add 125+005 or 105+025 we get 130Multiply 25 × 5 we get 125Multiply 25 × 5 we get 125We can’t put the answer like this 130125We can’t put the answer like this 130125Here from 125, 1 carry to 130 and it Here from 125, 1 carry to 130 and it
becomes 131, thenbecomes 131, then125 × 105 = 13125125 × 105 = 13125
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 : ??
ABOVE AND BELOW THE ABOVE AND BELOW THE BASEBASE
Find 102 × 95 = ?Find 102 × 95 = ?Normal Form:Normal Form: 102 × 95 102 × 95
510510 918918
96909690
Vedic FormVedic Form In Vedic form here we use In Vedic form here we use 100100 as a base as 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
136 × 90 = ?136 × 90 = ? 136 + 36136 + 36
90 - 1090 - 10126 / 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 360 Multiply 36 × 10 we get 360 Therefore 12600-360 we get the answer Therefore 12600-360 we get the answer 136 × 90 = 12240136 × 90 = 12240
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: ??
WITH DIFFERENT BASEWITH DIFFERENT BASE• 9997 ×9997 × 98 = ? 98 = ?• Here the numbers are Here the numbers are
close to different close 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 02 is not subtracted from the last two digit (97) of the last two digit (97) of 9997, but from 99 of 9997.9997, but from 99 of 9997.
• And 03 is a deficiency from And 03 is a deficiency from 10,000 so we can’t subtract 10,000 so we can’t subtract it from 98,because it’s a it from 98,because it’s a base of 100base of 100
• Mulply 98 by 100 and Mulply 98 by 100 and subtract 3 also give the anssubtract 3 also give the ans
• 9997 ×9997 × 98 = 979706 98 = 979706
Practice 7Practice 7999 × 80999 × 80 = ?= ?9987 × 989987 × 98 = ?= ?99995 × 99899995 × 998 = ?= ?96 × 896 × 8 = ?= ?10004 × 10810004 × 108 = ?= ?9985 × 9969985 × 996 = ?= ?98889 × 99 98889 × 99 = ?= ?
Ans :Ans : ??
10,20,… As a Base10,20,… As a Base We can use 10 as a We can use 10 as a
base for single digit base for single digit numbers.numbers.
Ex: 7 Ex: 7 × 8=?× 8=? 7-37-3 8-28-2
5/65/6 7 7 × 8=56× 8=56
We can use 20 as a base We 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 + 0429/2029/20
Now Multiply 29 by 2, we Now Multiply 29 by 2, we get 58 again multiply by get 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
Practice 8Practice 8 8 8 × 9 × 9 =?=? 22 22 × 23 =?× 23 =? 18 18 × 24 =?× 24 =? 29 29 × 34 =?× 34 =? 54 54 × 56 =?× 56 =? 38 38 × 39 =?× 39 =? 45 45 × 42 =?× 42 =? 9 9 × 5× 5 =?=?
Ans:Ans: ??
ALL FROM 9 AND LAST FROM ALL FROM 9 AND LAST FROM 1010
Subtraction From a Subtraction From a Base:Base: If we apply the If we apply the
formula to 854 we formula to 854 we get 146 because 8 get 146 because 8 and 5 are taken and 5 are taken from 9 and 4 is from 9 and 4 is taken from 10.taken from 10.
1000 – 46 = ?1000 – 46 = ? 1000 -1000 - 046046
954954
Subtract the units Subtract the units digit from 10, then digit from 10, then each successive each successive digit from 9, then digit from 9, then subtract 1 from the subtract 1 from the digit on the left.digit on the left. 60000 – 34843 = ?60000 – 34843 = ? 60000 -60000 - 3484334843
2515725157
SQUARINGSQUARING
Digits ends with five and zeroDigits ends with five and zeroTwo digit numbers (aTwo digit numbers (a22+2ab+b+2ab+b22) form) form
DIGITS ENDS WITH DIGITS ENDS WITH FIVEFIVE AND AND ZEROZERO
75 × 75 =?75 × 75 =?Normal FormNormal Form 75 × 7575 × 75 375375 52552556255625
Vedic FormVedic Form75 × 75 =?75 × 75 =?Here we use n(n+1)/25 formulaHere we use n(n+1)/25 formulaLet n= 7Let n= 7 n+1= 8n+1= 8 n(n+1) = 7 n(n+1) = 7 × 8 = 56× 8 = 56Therefore 75 × 75 =5625Therefore 75 × 75 =5625
3,4,… digits ends with 53,4,… digits ends with 5135 × 135 = ?135 × 135 = ?Let n=13Let n=13n(n+1)=13(14)n(n+1)=13(14)169+13=182169+13=182135 × 135 135 × 135
=18225=18225Practice Practice
151522,25,2522,etc..,etc..
169+13=?169+13=? 169169 1313 11 0 70 7 1212 182182
DIGITS ENDS WITH DIGITS ENDS WITH ZEROZERO 10102 2 =?, 100=?, 100 202022
=?=?22 × 2 = 4 and put 00× 2 = 4 and put 00We get 400We get 400
125012502 2 =?=? 125/0125/0 125=12/5125=12/5 12*13/5*5=156/2512*13/5*5=156/25
125012502 2 = 1562500= 1562500
TWO DIGIT NUMBERS TWO DIGIT NUMBERS (A(A22+2AB+B+2AB+B22)) FORM FORM
36 × 36 =?36 × 36 =? A=3 & B=6A=3 & B=6 332 2 = 9= 9 2*3*6 = 362*3*6 = 36 662 2 = 36= 36 Therefore 9,36,36Therefore 9,36,36 1296 1296
Left to Right Left to Right Calculation:Calculation:
9+3=129+3=12 6+3= 096+3= 09 66 = 06 = 06
12961296
Practice 9Practice 9 89 × 89 89 × 89 =?=? 26 × 26 26 × 26 =?=? 93 × 93 93 × 93 =?=? 78 × 7878 × 78 =?=? 61 × 6161 × 61 =?=? 92 × 92 92 × 92 =?=? 44 × 4444 × 44 =?=? 55 × 5555 × 55 =?=?
Ans:Ans: ??
Left Side Same Digit and Left Side Same Digit and Addition of Right Digit is 10Addition of Right Digit is 10
82 × 88 =?82 × 88 =?Normal FormNormal Form 88 × 82 88 × 82
176176 704704 72167216
Vedic FormVedic FormHere we are going to use n(n+1) Here we are going to use n(n+1)
formulaformulai.e, we take n=8 and get i.e, we take n=8 and get 8(8+1)=8(9)=728(8+1)=8(9)=72and 2 × 8=16and 2 × 8=16
Therefore 82 × 88 = 7216Therefore 82 × 88 = 7216
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:
Doubt?Doubt?Keep Practice you will clear yourselfKeep Practice you will clear yourselfContact:Contact:R.SankarR.Sankar
98658176239865817623