Let me start with some basics and an introduction.
Vedic Maths will allow you to carry out with great speed various multiplications, divisions as well as algebraic manipulations. To an extent it will also help along with Geometry. Reciprocals and ratios pitch in handily. It is simple, but like anything, you need to try it and do some practice.
We will be using base10 almost throughout. Another important concept is that of compliment. We will extensively use the concept of 9s and 10s compliment.
What is the 9s compliment of 7? 2
What is the 10s compliment of 7? 3
For some of the tricks/calculations, we will use 10s compliments of digits larger than 5 and will show them with a dash (to indicate compliment) above them (I will think of a way to do that).
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The most important formula aka sau~ is
}Qva-itya-gByaama which means vertical and crosswise. It is excellent for carrying out multiplications. It is based on the following algebraic identity:
(ax+b)*(cx+d)=acx*x+(bc+ad)*x+bd with x=10 for the decimal system that we use. Let us try the following:
037 (a=3, b=7)
x68 (c=6, d=8)
---
By conventional method it will be:
0037
x068
----
0296
222
-----
2516
(I wrote 068 etc. for formatting. How can one do uneven indenting?)
Let us see how it is with vertical and cross-wise
(1) First the vertical units place: 8*7=56
Put 6 down and carry 5 over
---6
(2) Add the carried over 5 to the cross multiplication of ad and bc i.e. 8x3+6x7+5=71
Put 1 down and carry 7 over
--16
(3) Finally do the vertical for the 10s place.
Add carried over 7 to a*c i.e. 3*6+7=25
Put that down
2516
We have arrived at the solution in a "single" step!
*** Let us do the discussions in a separate BB ***
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A longer example of the same:
000345
x00789
------
003105
02760
2415
------
272205 (doing it conventionally)
By our newly learnt method:
9*5=45 (write 5 carry 4)
9*4+8*5+4=80 (write 0 carry 8)
9*3+8*4+7*5+8=102 (write 2 carry 10)
8*3+7*4+10=62 (write 2 carry 6)
7*3+6=27 (write it all since that is the last part)
One Step Answer: 272205
(ax*x+b*x+c)*(dx*x+e*x+f)=
adx^4+(ae+bd)x^3+(af+be+cd)x^2+(bf+ce)x+cf
And this can of course be extended to any number of digits trivially.
(comments please: more examples? simpler examples? more explaination?)
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Are farach intersting ahe.thnx
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Ashish, he farach bhannat ahe yar, he tu shiklas kuthe ? asech division che pan kahi logic ahe ka ?
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Ani Ashish, samja 272 x 48 he kase karayche sang, I warche(3) ani khalche akade(2) same nastil tar kase karayche ?
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272
x048
----
Ase karave lagel na ? I mean, make both number equal in size by putting '0' ( zero ) at the left most digit so that digit's count will match.
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Ashish thanks for this BB. mi ata he mazya mulasathi save karun theven. Vaidik ganitachi pustake mahitat asatil tar please nave deshil ka?
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2*8=16 (write 6 carry 1)
8*7+4*2+1=65 (write 5 carry 6)
8*2+7*4+2*0+6=50 (write 0 carry 5)
4*2+7*0+5=13 (write 3 carry 1)
2*0+1=1 (write it all since that is the last part)
One Step Answer: 13056
Barobar na re ???
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are formula madhe kahitari zol aahe re...(ax*x+b*x+c)*(dx*x+e*x+f)=
adx^4+(ae+bc)x^3+(af+be+cd)x^2+(bf+ce)x+cf
chya aivaji...
(ax*x+b*x+c)*(dx*x+e*x+f)=
adx^4+(ae+bd)x^3+(af+be+cd)x^2+(bf+ce)x+cf pahije...
Vertical and cross multiplication karayache aahe na ? not horizontal... like 'bc' ?
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Hemant: perfect :)
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Hemant, barobar aahe, te 'bd' ch pahije. bahudha typo asavi
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It was indeed a typo. Thanx Hemant! I will correct it in the original posting too.
Try out the following for practice:
099 7899 4999 034565
x99 x777 x211 x45467
--- ---- ---- ------
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ha bb archive kara pls
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Viday : do not worry. Ha BB regularly archieve honar aahe.
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mahesh, if you email me, we can divide the work.
shekhar (and others) thanx. yes, there are formulae for division too, but we will come to them after we have seen some other relating to bases and compliments which play a central role. The book I use is: vedic mathematics by jagadguru swami sri bharati krsna tirthaji maharaj published by motilal banarasidas.
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people, I've posted the methods whatever I know of, here vaOidk gaiNat
I plan to update the site as and when I can. any volunteers to help me doing this?
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If you thought that the previous sutra makes multiplication easy, you aint seen nothin' yet.
When numbers are of a certain type, different sutras can be used to do the work even more quickly for you. We will see how
inaiKlaM navatXcarmaM dXatÁ
is used. The literal meaning of the sutra is:
all from nine, last from ten.
It is useful when at least one of the numbers is close to a power of 10 (our base) and is based on the algebraic identity:
(x-a)*(x-b)=x((x-a)-b)+ab
Let us take an example: 8x7
Both are single digits, and therefore last digits, so apply "last from 10" i.e. take 10s compliment and obtain 2 and 3
08 -2 (-2 because it is 2 less than 10)
x7 -3
-----
Now subtract 3 from 8 or 2 from 7 (cross) to get 5
as the 10s place (x-a-b or x-b-a). To get the units place, just multiply -2 and -3 (ab=6) and you have the answer viz. 56.
Another example:
95x95
095 -5
x95 -5
------
090/25 = 9025
(90 was obtained by subtracting 5 from 95, and 5x5=25)
98x89
098 -2
x89 -11
-------
087/22 = 8722
Thus when both numbers are close to and under 10, 100, 1000 etc., this sutra is very useful and much faster than the previous one
989x992
0989 -11
x992 -8
--------
0981/088 = 981088
Notice that we now wrote 11x8 as 088 (i.e. used three positions for it because our base this time was 1000, the third power of 10). You have to use a similar trick when the RHS overflows:
88x88
088 -12
x88 -12
-------
076/(1)44 = 7744
Now our base was 100, so of the 144 we got, we use only 44 and carry 1 over to the LHS.
Also note that you do not really have to memorize 12x12=144 because
(x+a)*(x+b)=x(x+a+b)+ab comes into play i.e.
012 +2
x12 +2
------
014/4 = 144
Similarly
17x17
017 +7
x17 +7
------
024/(4)9 = 289 (4 was carried since our base was 10)
103x105
0103 +3
x105 +5
--------
0108/15 = 10815
109x111
0109 +9
x111 +11
--------
0120/99 = 12099
Try things out yourself and yo will notice the magic in it.
Lastly (x-a)*(x+b)=x(x-a+b)-ab will work too.
103x98
103 +3
x98 -2
-------
101/-6 = 10100-6 = 10094
17x9
17 +7
x9 -1
-----
16/-7 = 153
1011x888
1011 +11
x988 -12 (notice that first 2 digits were
--------- removed from 9 and last from 10)
999/-132 = 998868
(remember that we can get the LHS part in two ways:
1011-12=999=988+11 i.e. you have the choice of whatwever you feel is easier).
Do practice witha few examples:
999x999
1011x1013
997x1004
99x99
103x103
Use the technoque twice for
1017x1017
1017x988
10103x10107
9899x9888
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ashish, this is really interesting :o) I did not remember this properly but now got reminded of it.
thanks for the detailed method and examples.
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sope try karto:
| 997 -3
x1004 +4
--------
1001/-12 = 1001000 - 12 = 1000988
calculator ne check karto ;o).......
yes perfect :o)
ashish, indentation sathi fixed width font vapar (Courier new) tyacha syntax aahe
\font{Courier new,
whatever you want to enclose
}
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dhanyawad Kiran. tuzhi site UNIX var aamhi paahu shakat naahi :-(
maaza paN vichar aahe ki yaa padhhatinche ek web page banawaawe.
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Ashish, I remember a person who calculated 2^32 in a flash. He must have used some method as u have suggested. At that time i was amazed but now i realise that he must have used some method like above. Do u know of any method that will calculate 2^32.
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prafull, 2^32 karane mala evadhe vishesh vatat nahi.
kahi goshti direct lakshat suddha rahu shaktat jase ki
2^8 = 256, 2^16 = 65536, ityadi
pan vaidik ganitat nakkich hyasathi procedure asel.
2 ankhi sankhyecha varg (^2) kasa karayacha te mi mazya site var sangitlech ahe.
ashach kahi techniquene
(2^16)^2 = 2^32 mhanaje (65536)^2 ase nakkich karata yeil.
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Excellent. Thanx for starting this BB Aschig.
--- Asmaadik
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ashish: thanks for opening this BB. in fact, this is my first time i am coming across this 'vedic mathematics'. really it is very helpful and intresting as well.
kiran: site infor is very good too.
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gaiNatacao maÜft vaga- Cana sau$ Jaalaolao Aahot. malaa ha baIbaI Kup AavaDlaa. maI sagaLI ]dahrNao kÉna paihlaI AaiNa malaa jamalaI pNa. malaa gaiNat ivaYaya Kup AavaDtÜ. ek kahItrI naivana iXakayalaa imaLto Aaho mhNauna AaiXaYacao manaÁpuva-k AaBaar² ikrNa tuJaI saaT malaa pNa naahI idsat AahoÊ maaJaa OS tr NT Aaho. malaa ikrNa ÔaÐT download krayalaa paihjao kaÆ
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... AaiXaYa tu dOnadIna vyavaharamaQyao vaOdIk gaiNat vaaprtÜsa kaÆ karNa baGa Aaplyaalaa XaaLot conventional methods iXakivalyaa gaolyaat AaiNa [tko vaYa- AapNa toca toca krt gaolaÜ mhNauna toca gaiNat AapNa vaaprtÜ. Aata sarava kÉna jar ho gaiNat ]pyaÜgaat AaNanyaacaa p`ya%na kolaa tr Aaplyaalaa nakÜ %yaa vaoLI vaOdIk gaiNat krt basauna Gaovaana dovaana laa vaoL nakÜ laagaayalaa. karNa sarava krNao vaogaLI gaÜYt Aaho Aaho AaiNa savaya laa]na Gaonao hI ek vaogaLI gaÜYT Aaho. sarava kaya paTI laoKNaI Gao}nahI krta yaola pNa dOnadIna vyavaharatlaI AakDomaÜDI vaOdIk gaiNatat krNao mhNajao jara kzINaca vaaTto. pNa malaa hI pwt AavaDlaI ho na@kI. vaOdIk gaiNat caalaU Qyaa² kahI saaT\sa AsatIla tr tohI saaMga.
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nice ones Ashish.
Kiran, changle kele aahes page. kasale volunteers pahije aahet tula ? ha mi aalo.. :o)
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Mitranno: if we can keep discussions separate of this BB, I guess that will be great....!!!
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Waa Ashish, ekdam maja ali.very interesting.
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yash, I do use it regularly. I have never owned a calculator. Knowing numbers intimately helps in general. Like Kiran did, check your calcs a first few times with the calculator, but try always to use these methods. That way you will quickly get acclaimatized. As I said before, we will come to division etc. a little later.
In a few mins I will be making another post.
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I told you yesterday that the inaiKlama ... sau~ can be used when the two numbers are close to a base of 10. That was oversimplification. It can be used in many more circumstances.
Consider 51x53
051 -49
x53 -47
--------
The 49*47 that we will have to do is as bad as 51*53. So instead we use not 100 as base, but 100/2=50. Then
051 +1
x53 +3
-------
054/03
Now comes an additional step: because we used 100/2=50 as base, divide the LHS by two too:
54/2=27 and the answer is 2703.
Alternately, do the same using base=10*5=50
051 +1
x53 +3
------
054/3 (note we have a single digit in RHS rather
than 03 since the basic base was 10 rather
100)
Now the last step: 54*5=270 and the answer is
2703.
Another example:
249x244 (base 1000/4=250)
0249 -1
x244 -6
--------
0243/006
----
4
60/756 (3/4 that remained after 243/4 becomes 750)
78x77
078 -2
x77 -3
------
75*8 6=6006
198*203
0198 -2
x203 +3
-------
201*2 -06 = 402 -06 = 40194
Sometimes of course the urdhvatiryakbhyam method is much quicker.
0198
x203
-----
8x3=(2)4
9x3+2=(2)9
3x1+2x8+2=(2)1
9x2+2=(2)0
2x1+2=4
Answer: 40194
You have yor choice.
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Rasik: Yes you need to download kiran fonts for that. See the top of the site for instructions or just go to http://www.kiranfont.com to get it.
I'm planning to use dynamic fonts, but not having enough time to do it right now.
Ashish: this is really great, I did not know about this method for base other tha 10's power.
But here's a question. This method still has a limitation. both the numbers to multiply need to be near a particular number what if we want to do something like 253 x 1456? Then we must use the urdhvatiryakabhyam method, right?
milindaa: I want volunteers, to convert these methods what ashish is posting into marathi and update the site contents. If we distribute the work it'll be lot easier.
Ashish: can you give a brief history also about the vaidik ganit? like when it was invented? what are the controlling shlokas, etc., thanks.
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Kiran, urdhva... will be better. However, if your practise is good, you could do the following:
1456x253
1456 +1206
x253 +0003
----------
1459/4 (3)618
Here we have used base=1000/4
Thats why 1459 needs to be divided by 4 and 3 needs to be carried over (1000 has 3 zeroes)
1459/4=364 3/4 i.e 364 750
364 750
+03 618
-------
368 368 is the answer.
So really, rather than knowing just the multiplications methods, it is good to know some other tricks too.
I will give the names for the other sutras too (available in the book I mentioned). But would it not be better if we do it as the sutras are revealed?
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Aschiq: yes it would be more beneficial if we do it as the sutras are revealed :)
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