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Nikhilam1.Base10 9 1 13 +3 12 +2

X 7 3 X14 +4 8 2

93| 3 17| 12 10 4=96.Or 71| 3 18| 2=182.Or9+710| 3Or1013| 3

6| 3=63.

2.Base100 93 7 88 12 107 +797 3 94 6 112 +12

90 21=9021 82 72=8272 119 84=11984

112 +12 63 +13 89 +79 887 1384 16 54 +4 13 +3 907 +7 96 192=9408 2/67 52 92 237=1157 9*894 91

33 52=3402 8046 91804509

3.Base1000 887 113 1047 +47 888 12997 3 991 9 888 12 884 339=884339 1038 423=1037577 9*876 144

7884 144=788544

9997 0003 988 012 99979 000219997 0003 998 002 99999 00001 9994 0009=99940009 986 024=986024 99978 00021

249 001 532 +32 635 +135246 004 472 28 502 +002 4/245 004 5*504 896 2/637 27061 004=61254 2520 896=251104. 318 270

=318770389 111516 +16

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FuturiSM2/405 1776202 1776=202 1276=201 276=200724

Squaring:1.AddthepositivedifferencefromthebasetothequantityorSubtractthenegative

differencefromthebasefromthequantity.2.Appendthesquareofdifferencetotheresultobtainedfrom1above.3.Takecarryoverasperrulesfollowedearlier,thatiscarryovertensincaseofbase10,

andhundredsincaseofbase100.4.Remembertomultiplyordividetheresultobtainedfrom1above,ifbaseisotherthan

anypowerof10.Ex.

19 +9 28/81 =361.19 1 18*2/1 =361.91 9 82/81 =8281.108 +8 116/64 =11664989 11 978/121 =97812159 +9 68y2/81 =348143 +3 46*4/9 =1849

SquareofNumberendingin51.Multiplynumberoftenswithnexthigherintegerandannex25totheproduct.

MultiplicationofCognatenumbers1.Thisisapplicabletonumberswithnumberoftensidenticalandunitstotalingtogether

10.e.g.23&27,73&77,84&86,62&68,154&156,173&177,etc.2.Multiplynumberoftenswithnexthigherintegerandannextheproductofunitstothe

roductsoarrived.Ex.

23*27=2*3/21=621.153*157=15*16/21=24021

Multiplicationwithanumberof9sCase:1Whenthenumberofdigitsaresameinmultiplicandandinmultiplier.1.Arithmetically1lessthanthemultiplicandformsleftpartoftheanswer.2.Complementofleftpartformsrightpartoftheanswer.Ex.

7*9 = 6/315*99= 14/85673*999= 672/327

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FuturiSM9876*9999=9875/0124

Case:2Whenthedigitsinmultiplicandarelessthanthedigitsinmultiplier.1.Prefixesasmanyzerosasrequiredtomakebothmultiplicandandmultiplierofsame

numberofdigits.2.Proceedasincase1above.Ex.

6*99=06*99=05/9473*999=073*999=072/927798*9999999=0000798*9999999=0000797/9999202=7979999202.

Case:3Whenthenumberofdigitsinmultiplicandaremorethanthenumberofdigitsinthemultiplier.

1.Dividethemultiplicandverticallysoastohavesamenumberofdigitsasinmultiplierontherightsideofthedivision.

2.Subtract1morethanthenumberonleftofdivisionfromthemultiplicand,whichgivesleftpartoftheanswer.

3.Righthandpartshouldbesubtractedfromthemultiplier+1togetrightsidepartoftheanswer.

Ex.43*9=>4|3 122*9=>12|2 111011*99=>1110|11

5|9+13 13|9+12 1111 38/7=387 109/8=1098 109900/89

UrdhvaTiryakThisisageneralmethodapplicabletoanykindofmultiplication.Inthismethodstartingfromfirstdigitonleft,wegoonmultiplyingcrosswaysalldigitsandaddthemtogetindividualdigitsoftheanswer.Eachpositionisallowedonlyonedigit,sodigitsinexcessofonearcarriedovertonextmultiplicationadditionontheleftside.

Ex.abc

fghaf|ag+bf|ah+bg+cf|bh+cg|ch

1.Firstjustfirstdigitonleftismultiplied.2.ThenFirsttwodigitsfromtheleftaremultipliedandadded.3.Similarlywegoongettingdigitstillwereachuptofirstandthelastdigitsfromtheleft.4.Thenwestartmovingtowardsrightandgoonreducingdigitstobemultipliedandget3FuturiSM

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FuturiSMsuccessivedigitsoftheanswer.111 108 582111 108 231 1|2|3|2|1 1|0|6|0|4 10|1|3|4|2=12321 16 331

1|1|6|6|4=11664 13|4|4|4|2=134442

785 321362 052 21|6|7|6|0 0|5|6|9|26|7|4|1 11 28|4|1|7|0=284170 1|6|6|9|2=16692

872653211724|7|8|7|2|7|5|7|53239624328|0|2|6|9|0|0|0|5

Ex.(a+b)(a+9b) a+b x5+3x4+5x3+3x2+x+1a+9b 7x5+5x4+3x3+x2+3x+5

a2+10ab+9b2 7x10+26x9+53x8+56x7+33x6+40x5+41x4+38x3+19x2+

8x+5Whenacertainpowerisnotpresentinanyoftheexpression,weputzeroinitsplaceandproceedsameasabove.Studentshouldusehis/herprudenceinselectingthemethodtomultiplydifferentnumberseitherwiththehelpofNikhilamorwiththehelpofUrdhvaTiryak.When,allthedigitsaremorethan5,viniculammethodisalsouseful.Ex.889*898 1111

11021|2|0|2|3|2|2798322

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