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### vedic mathematics

1. 1. VEDICVEDICMATHEMATICSMATHEMATICS
2. 2. What is Vedic Mathematics ? Vedic mathematics is the namegiven to the ancient system ofmathematics which wasrediscovered from the Vedas. It’s a unique technique ofcalculations based on simpleprinciples and rules , with whichany mathematical problem - be itarithmetic, algebra, geometry ortrigonometry can be solvedmentally.
3. 3. Why Vedic Mathematics?Why Vedic Mathematics? It helps a person to solve problems 10-15 times faster. It reduces burden (Need to learn tables up to nine only) It provides one line answer. It is a magical tool to reduce scratch work and fingercounting. It increases concentration. Time saved can be used to answer more questions. Improves concentration. Logical thinking process gets enhanced.
4. 4. Base of Vedic MathematicsBase of Vedic Mathematics VedicMathematics nowrefers to a set ofsixteenmathematicalformulae or sutrasand theircorollaries derivedfrom the Vedas.
5. 5. Base of Vedic MathematicsBase of Vedic MathematicsVedicMathematics nowrefers to a set ofsixteenmathematicalformulae or sutrasand theircorollaries derivedfrom the Vedas.
6. 6. EKĀDHIKENA PŪRVEŅAEKĀDHIKENA PŪRVEŅAThe Sutra(formula)EkādhikenaPūrvena means:“By one more thanthe previous one”. This Sutra isused to the‘Squaring ofnumbers endingin 5’.
7. 7. ‘Squaring of numbers endingin 5’.Conventional Method65 X 656 5X 6 53 2 53 9 0 X4 2 2 5Vedic Method65 X 65 = 4225( multiply theprevious digit 6 byone more thanitself 7. Than write25 )
8. 8. NIKHILAMNAVATAS’CHARAMAMDASATAHThe Sutra (formula)NIKHILAMNAVATAS’CHARAMAM DASATAHmeans :“all from 9 and thelast from 10”This formula canbe very effectivelyapplied inmultiplication ofnumbers, which arenearer to bases like10, 100, 1000 i.e., tothe powers of 10(eg: 96 x 98 or 102x 104).
9. 9. Case I :When both the numbers arelower than the base. Conventional Method97 X 949 7X 9 43 8 88 7 3 X9 1 1 8 Vedic Method9797 33XX 9494 669 1 1 89 1 1 8
10. 10. Case ( ii) : When both theCase ( ii) : When both thenumbers are higher than thenumbers are higher than thebasebase ConventionalMethod103 X 105103X 1055 1 50 0 0 X1 0 3 X X1 0, 8 1 5 Vedic MethodFor Example103 X 105For Example103 X 105103103 33XX 105 551 0, 8 1 5
11. 11. Case III: When one number isCase III: When one number ismore and the other is lessmore and the other is lessthan the base.than the base. Conventional Method103 X 98103X 988 2 49 2 7 X1 0, 0 9 4 Vedic Method103103 33XX 98 -21 0, 0 9 4
12. 12. ĀNURŨPYENAThe Sutra (formula)ĀNURŨPYENAmeans :proportionality orsimilarly This Sutra is highlyThis Sutra is highlyuseful to finduseful to findproducts of twoproducts of twonumbers whennumbers whenboth of them areboth of them arenear the Commonnear the Commonbases like 50, 60,bases like 50, 60,200 etc (multiples200 etc (multiplesof powers of 10).of powers of 10).
13. 13. ĀNURŨPYENAConventional Method46 X 434 6X 4 31 3 81 8 4 X1 9 7 8Vedic Method4646 -4-4XX 43 -7-71 9 7 8
14. 14. ĀNURŨPYENAConventional Method58 X 485 8X 4 84 6 42 4 2 X2 8 8 4Vedic Method5858 88XX 4848 -2-22 82 8 8 4
15. 15. URDHVA TIRYAGBHYAMThe Sutra (formula)URDHVATIRYAGBHYAMmeans :““Vertically and crossVertically and crosswise”wise”This the generalThis the generalformula applicableformula applicableto all cases ofto all cases ofmultiplication andmultiplication andalso in the divisionalso in the divisionof a large numberof a large numberby another largeby another largenumber.number.
16. 16. Two digit multiplication bybyURDHVA TIRYAGBHYAMThe Sutra (formula)URDHVATIRYAGBHYAMmeans :““Vertically and crossVertically and crosswise”wise” Step 1Step 1: 5×2=10, write: 5×2=10, writedown 0 and carry 1down 0 and carry 1 Step 2Step 2: 7×2 + 5×3 =: 7×2 + 5×3 =14+15=29, add to it14+15=29, add to itprevious carry overprevious carry overvalue 1, so we have 30,value 1, so we have 30,now write down 0 andnow write down 0 andcarry 3carry 3 Step 3Step 3: 7×3=21, add: 7×3=21, addprevious carry overprevious carry overvalue of 3 to get 24,value of 3 to get 24,write it down.write it down. So we have 2400 as theSo we have 2400 as theanswer.answer.
17. 17. Two digit multiplication bybyURDHVA TIRYAGBHYAMVedic Method4 6X 4 31 9 7 8
18. 18. Three digit multiplication byURDHVA TIRYAGBHYAMVedic Method103X 1051 0, 8 1 5
19. 19. YAVDUNAMTAAVDUNIKRITYAVARGANCHA YOJAYETThis sutra meanswhatever the extentof its deficiency,lessen it stillfurther to that veryextent; and also setup the square ofthat deficiency.This sutra is veryhandy incalculating squaresof numbersnear(lesser) topowers of 10
20. 20. YAVDUNAMTAAVDUNIKRITYAVARGANCHA YOJAYET982= 9604 The nearest power of 10 to 98 is 100.The nearest power of 10 to 98 is 100.Therefore, let us take 100 as our base.Therefore, let us take 100 as our base. Since 98 is 2 less than 100, we call 2 asSince 98 is 2 less than 100, we call 2 asthe deficiency.the deficiency. Decrease the given number further by anDecrease the given number further by anamount equal to the deficiency. i.e.,amount equal to the deficiency. i.e.,perform ( 98 -2 ) = 96. This is the left sideperform ( 98 -2 ) = 96. This is the left sideof our answer!!.of our answer!!. On the right hand side put the square ofOn the right hand side put the square ofthe deficiency, that is square of 2 = 04.the deficiency, that is square of 2 = 04. Append the results from step 4 and 5 toAppend the results from step 4 and 5 toget the result. Hence the answer is 9604.get the result. Hence the answer is 9604.NoteNote :: While calculating step 5, the number of digits in the squared number (04)While calculating step 5, the number of digits in the squared number (04)should be equal to number of zeroes in the base(100).should be equal to number of zeroes in the base(100).
21. 21. YAVDUNAMTAAVDUNIKRITYAVARGANCHA YOJAYET1032= 10609 The nearest power of 10 to 103 is 100.The nearest power of 10 to 103 is 100.Therefore, let us take 100 as our base.Therefore, let us take 100 as our base. Since 103 is 3 more than 100 (base), weSince 103 is 3 more than 100 (base), wecall 3 as the surplus.call 3 as the surplus. Increase the given number further by anIncrease the given number further by anamount equal to the surplus. i.e., performamount equal to the surplus. i.e., perform( 103 + 3 ) = 106. This is the left side of( 103 + 3 ) = 106. This is the left side ofour answer!!.our answer!!. On the right hand side put the square ofOn the right hand side put the square ofthe surplus, that is square of 3 = 09.the surplus, that is square of 3 = 09. Append the results from step 4 and 5 toAppend the results from step 4 and 5 toget the result.Hence the answer is 10609.get the result.Hence the answer is 10609.NoteNote :: while calculating step 5, the number of digits in the squared number (09)while calculating step 5, the number of digits in the squared number (09)should be equal to number of zeroes in the base(100).should be equal to number of zeroes in the base(100).
22. 22. YAVDUNAMTAAVDUNIKRITYAVARGANCHA YOJAYET10092= 1018081
23. 23. SAŃKALANA –VYAVAKALANĀBHYAMThe Sutra (formula)SAŃKALANA –VYAVAKALANĀBHYAMmeans :by addition and byby addition and bysubtractionsubtractionIt can be applied inIt can be applied insolving a specialsolving a specialtype of simultaneoustype of simultaneousequations where theequations where thex - coefficients andx - coefficients andthe y - coefficientsthe y - coefficientsare foundare foundinterchanged.interchanged.
24. 24. SAŃKALANA –VYAVAKALANĀBHYAMExample 1:45x – 23y = 11323x – 45y = 91 Firstly add them,Firstly add them,( 45x – 23y ) + ( 23x – 45y ) = 113 + 91( 45x – 23y ) + ( 23x – 45y ) = 113 + 9168x – 68y = 204    68x – 68y = 204    x – y = 3x – y = 3 Subtract one from other,Subtract one from other,( 45x – 23y ) – ( 23x – 45y ) = 113 – 91( 45x – 23y ) – ( 23x – 45y ) = 113 – 9122x + 22y = 2222x + 22y = 22x + y = 1x + y = 1 Rrepeat the same sutra,Rrepeat the same sutra,we getwe get x = 2x = 2 andand y = - 1y = - 1
25. 25. SAŃKALANA –VYAVAKALANĀBHYAMExample 2:1955x – 476y = 2482476x – 1955y = - 4913Just add,Just add,2431( x – y ) = - 24312431( x – y ) = - 2431x – y = -1x – y = -1 Subtract,Subtract,1479 ( x + y ) = 73951479 ( x + y ) = 7395x + y = 5x + y = 5Once again add,Once again add,2x = 42x = 4 x = 2x = 2subtractsubtract- 2y = - 6- 2y = - 6 y = 3y = 3
26. 26. ANTYAYOR DAŚAKEPIThe Sutra (formula)ANTYAYORDAŚAKEPImeans :‘‘ Numbers of whichNumbers of whichthe last digitsthe last digitsadded up giveadded up give10.’10.’ This sutra is helpful inThis sutra is helpful inmultiplying numbers whosemultiplying numbers whoselast digits add up to 10(orlast digits add up to 10(orpowers of 10). The remainingpowers of 10). The remainingdigits of the numbers shoulddigits of the numbers shouldbe identical.be identical.For ExampleFor Example: In multiplication: In multiplicationof numbersof numbers 25 and 25,25 and 25,2 is common and 5 + 5 = 102 is common and 5 + 5 = 10 47 and 43,47 and 43,4 is common and 7 + 3 = 104 is common and 7 + 3 = 10 62 and 68,62 and 68, 116 and 114.116 and 114. 425 and 475425 and 475
27. 27. ANTYAYOR DAŚAKEPIVedic Method6 7X 6 34 2 2 1 The same rule worksThe same rule workswhen the sum of the lastwhen the sum of the last2, last 3, last 4 - - - digits2, last 3, last 4 - - - digitsadded respectively equaladded respectively equalto 100, 1000, 10000 -- - - .to 100, 1000, 10000 -- - - . The simple point toThe simple point toremember is to multiplyremember is to multiplyeach product by 10, 100,each product by 10, 100,1000, - - as the case may1000, - - as the case maybe .be . You can observe that thisYou can observe that thisis more convenient whileis more convenient whileworking with the productworking with the productof 3 digit numbersof 3 digit numbers
28. 28. ANTYAYOR DAŚAKEPI892 X 808= 720736Try Yourself :Try Yourself :A)A) 398 X 302398 X 302= 120196= 120196B)B) 795 X 705795 X 705= 560475= 560475
29. 29. LOPANASTHÂPANÂBHYÂMThe Sutra (formula)LOPANASTHÂPANÂBHYÂMmeans :by alternateby alternateelimination andelimination andretentionretention Consider the case ofConsider the case offactorization of quadraticfactorization of quadraticequation of typeequation of typeaxax22+ by+ by22+ cz+ cz22+ dxy + eyz + fzx+ dxy + eyz + fzx This is a homogeneousThis is a homogeneousequation of second degreeequation of second degreein three variables x, y, z.in three variables x, y, z. The sub-sutra removesThe sub-sutra removesthe difficulty and makesthe difficulty and makesthe factorization simple.the factorization simple.
30. 30. LOPANASTHÂPANÂBHYÂMExample :3x 2+ 7xy + 2y 2+ 11xz + 7yz + 6z 2 Eliminate z and retain x, y ;factorize3x 2+ 7xy + 2y 2= (3x + y) (x + 2y) Eliminate y and retain x, z;factorize3x 2+ 11xz + 6z 2= (3x + 2z) (x + 3z) Fill the gaps, the given expression(3x + y + 2z) (x + 2y + 3z) Eliminate z by putting z = 0Eliminate z by putting z = 0and retain x and y andand retain x and y andfactorize thus obtained afactorize thus obtained aquadratic in x and y byquadratic in x and y bymeans ofmeans of AdyamadyenaAdyamadyenasutra.sutra. Similarly eliminate y andSimilarly eliminate y andretain x and z and factorizeretain x and z and factorizethe quadratic in x and z.the quadratic in x and z. With these two sets ofWith these two sets offactors, fill in the gapsfactors, fill in the gapscaused by the eliminationcaused by the eliminationprocess of z and yprocess of z and yrespectively. This givesrespectively. This givesactual factors of theactual factors of the
31. 31. GUNÌTA SAMUCCAYAH -SAMUCCAYA GUNÌTAHExample :3x 2+ 7xy + 2y 2+ 11xz + 7yz + 6z 2 Eliminate z and retain x, y ;factorize3x 2+ 7xy + 2y 2= (3x + y) (x + 2y) Eliminate y and retain x, z;factorize3x 2+ 11xz + 6z 2= (3x + 2z) (x + 3z) Fill the gaps, the given expression(3x + y + 2z) (x + 2y + 3z) Eliminate z by putting z = 0Eliminate z by putting z = 0and retain x and y andand retain x and y andfactorize thus obtained afactorize thus obtained aquadratic in x and y byquadratic in x and y bymeans ofmeans of AdyamadyenaAdyamadyenasutra.sutra. Similarly eliminate y andSimilarly eliminate y andretain x and z and factorizeretain x and z and factorizethe quadratic in x and z.the quadratic in x and z. With these two sets ofWith these two sets offactors, fill in the gapsfactors, fill in the gapscaused by the eliminationcaused by the eliminationprocess of z and yprocess of z and yrespectively. This givesrespectively. This givesactual factors of theactual factors of the
32. 32. Prepared By:NitinChhaperwal