Kawri Maths
8
Kawri Mathematics Introduction India is famous for mathematics from the ancient past. To denote the numbers 5 systems were present here - 1) Sabda Sankhya Method 2) Boota Sankhya method 3) KTPY system (Katapayadi; Paralper) 4) Aryabhta System 5) Sankhya System (Decimal system) Many have described in detail about all these five methods. But even after understanding these methods used for notating numbers the doubt will remain for the knowledgeable that “All these methods are just for NOTATING numbers. That is symbolic representation methods for numbers. What was the method used by ancient Indian people TO DO the calculations such as Addition, Subtraction, Multiplication and Division?” Even though helpful in notating and thus remembering the numbers involved in mathematical questions and answers all the five methods discussed above is not helpful in actually DOING the calculations. It was only after the advent of writing method that the 5th method mentioned above became helpful in actually doing the calculations. There for the question is “Prior to the advent of writing system how the Indian people used to do the calculations?” It is only the Kawri system of doing calculation that answers this question. Therefore instead of boasting about the various ancient system of notion methods and the greatness of indian mathematics, let us try to understand the unique Kawri method used by the ancient Indians for doing calculations. This system is still prevalent and well practiced by the astrologers of Kerala. The efficient users of this Kawri system of mathematics at times do the calculations faster than calculators using Kawri alone! It is wonderful system, which may give unique insights to the seekers of the history and methods of ancient mathematics, though not of much practical usefulness in this era of well-developed Decimal system and the use of Calculators and Computers. As told earlier astrologers of Kerala use the Kawri method even today. The best method to master this method is to observe an efficient person using this method and learn it from him directly in his presence. It is not theory but practice and understanding which is important. Even though this method is in use for thousands of years in India, it is pretty difficult to describe this method in paper. May be this is what caused the absence of books that describe this method in detail. Many wrote to me asking where there is a book available that describes Kawri method of doing calculation in detail. I don’t know about any one who wrote any detailed books or articles on the same. Therefore I though it would be beneficial to all if I wrote a series of articles that describe the Kawri method of calculation in detail, in a way understandable and beneficial to the modern readers. Usually orthodox astrologers advice that don’t put such things in paper - may be also due to the difficulty of describing such practical methods efficiently in writing. But if some one is not describing this method in detail in writing, like the many thousands of ancient knowledge streams and methods we lost, this Kawri system may also get forgotten and lost. That is why I though of putting it in writing. But still the advice is that try to learn this method directly from a teacher in his presence, so that you could understand it clearly, the doubts would get clarified, and you could gain mastery in this method. Another point to note is that, if we haven’t learned the modern methods of primary calculations, it would have been easy for us to understand this method fast - because the method used is at times diametrically opposite to that of modern mathematics even for primary calculations! Initial Information I have used numbers to denote the values instead of Kawris. If I am writing 2 in a place, as per Kawri system that location will contain 2 Kawris and if I write 3 that place will contain 3 Kawris and so on it should be understood. The individual doing the calculation would be sitting on a seat and before him a flat wooden board would be placed on earth. He will place all the Kawris in it and will start doing the calculations using the Kawris - that is the actual method followed. In Kawri system the initial numbers used for calculation will hold the answers after the calculations. But here in this article for the sake of understanding I have denoted the intermediate results in steps. First observe someone doing the calculations using Kawris, then by reading this article and also by doing it yourself using Kawris understand and practice the method. With some days of practice you too will become efficient in using Kawris for doing even complex arithmetical calcula- tions with them.
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kawri
Transcript of Kawri Maths
Kawri MathematicsIntroduction
India is famous for mathematics from the ancient past. To denotethe numbers 5 systems were present here - 1) Sabda Sankhya Method2) Boota Sankhya method 3) KTPY system (Katapayadi; Paralper) 4)Aryabhta System 5) Sankhya System (Decimal system)
Many have described in detail about all these five methods. Buteven after understanding these methods used for notating numbers thedoubt will remain for the knowledgeable that “All these methodsare just for NOTATING numbers. That is symbolicrepresentation methods for numbers. What was the method usedby ancient Indian people TO DO the calculations such asAddition, Subtraction, Multiplication and Division?” Even thoughhelpful in notating and thus remembering the numbers involved inmathematical questions and answers all the five methods discussedabove is not helpful in actually DOING the calculations. It was onlyafter the advent of writing method that the 5th method mentioned abovebecame helpful in actually doing the calculations. There for the questionis “Prior to the advent of writing system how the Indian peopleused to do the calculations?” It is only the Kawri system of doingcalculation that answers this question. Therefore instead of boastingabout the various ancient system of notion methods and the greatnessof indian mathematics, let us try to understand the unique Kawri methodused by the ancient Indians for doing calculations. This system is stillprevalent and well practiced by the astrologers of Kerala. The efficientusers of this Kawri system of mathematics at times do the calculationsfaster than calculators using Kawri alone! It is wonderful system, whichmay give unique insights to the seekers of the history and methods ofancient mathematics, though not of much practical usefulness in thisera of well-developed Decimal system and the use of Calculators andComputers.
As told earlier astrologers of Kerala use the Kawri method eventoday. The best method to master this method is to observe an efficientperson using this method and learn it from him directly in his presence.It is not theory but practice and understanding which is important. Eventhough this method is in use for thousands of years in India, it is pretty
difficult to describe this method in paper. May be this is what causedthe absence of books that describe this method in detail. Many wrote tome asking where there is a book available that describes Kawri methodof doing calculation in detail. I don’t know about any one who wroteany detailed books or articles on the same. Therefore I though it wouldbe beneficial to all if I wrote a series of articles that describe the Kawrimethod of calculation in detail, in a way understandable and beneficialto the modern readers. Usually orthodox astrologers advice that don’tput such things in paper - may be also due to the difficulty of describingsuch practical methods efficiently in writing. But if some one is notdescribing this method in detail in writing, like the many thousands ofancient knowledge streams and methods we lost, this Kawri systemmay also get forgotten and lost. That is why I though of putting it inwriting. But still the advice is that try to learn this method directly froma teacher in his presence, so that you could understand it clearly, thedoubts would get clarified, and you could gain mastery in this method.Another point to note is that, if we haven’t learned the modern methodsof primary calculations, it would have been easy for us to understandthis method fast - because the method used is at times diametricallyopposite to that of modern mathematics even for primary calculations!Initial Information
I have used numbers to denote the values instead of Kawris. If I amwriting 2 in a place, as per Kawri system that location will contain 2Kawris and if I write 3 that place will contain 3 Kawris and so on itshould be understood. The individual doing the calculation would besitting on a seat and before him a flat wooden board would be placedon earth. He will place all the Kawris in it and will start doing thecalculations using the Kawris - that is the actual method followed.
In Kawri system the initial numbers used for calculation will holdthe answers after the calculations. But here in this article for the sakeof understanding I have denoted the intermediate results in steps. Firstobserve someone doing the calculations using Kawris, then by readingthis article and also by doing it yourself using Kawris understand andpractice the method. With some days of practice you too will becomeefficient in using Kawris for doing even complex arithmetical calcula-tions with them.
In this method the whole calculation depends on the use of comple-mentary digits (complements of 10). The list of complementary digits isgiven below
Complement of 1 = 9 ; Complement of 9 = 1Complement of 2 = 8 ; Complement of 8 = 2Complement of 3 = 7 ; Complement of 7 = 3Complement of 4 = 6 ; Complement of 6 = 4Complement of 5 = 5To denote complement I will use C. That is C1 = 9; C9 = 1; C2 = 8;
C8 = 2; C3 = 7; C7 = 3; C4 = 6; C6 = 4; and C5 = 5 for abbreviatednotation.
Kawri system is nothing but “Doing the decimal system of calcula-tion using Kawris”; the most ancient practice of DOING CALCULA-TIONS used throughout Ancient India, and is still well preserved andpracticed in Kerala by the astrologers.
AdditionAddition is simple. In Kawri system as well you just have to add the
digits from right to left as per place value, similar to the modern method.But the difference is that-
* Here you are using only Kawris for doing the calculation* The complementary numbers are used for doing the calculation* Complementary numbers are always REMOVED from the digit
when ever the digit above is smaller and 1 would be added to the nexthigher place (i.e. to left).
Some examples will clarify the method. I have represented the num-bers considered in each step of the calculation using red color so as toidentify the steps clearly.
1) Add 24 to 12
1 22 4--------1 62 0--------3 6=====
2) Add 45 with 234
2 3 44 6
----------------2 4 0
4 0----------------2 8 0==========
To denote complement I will use C form here onwards. Thatis C1 = 9; C9 = 1; C2 = 8; C8 = 2; C3 = 7; C7 = 3; C4 = 6; C6 =4; and C5 = 5 for brevity.
3) Add 46 to 285
2 8 54 6
----------------
Add 4 Kawris to the 2 Kawrisplaced above. Answer = 6 Kawris
Add 2 Kawris to the 1 Kawriplaced above. Answer = 3Kawris.
Result : 3 6 (Thirty six)
The complement of 6 = 4. There-fore, remove four Kawris fromabove and move a single Kawrito the next higher place. (i..e Re-move 4 from the digit above 6, andAdd 1 to 3). The result is shownin the next step.(Since we cannot remove the 6,which is complement of 4) Add 4to 4.Answer: 280 (Two hundred andeighty)
Remove C6 (=4) from 5 and Add1 to the next place.
2 9 14 0
----------------3 3 1=========
Now onwards instead of saying “move 1 to the next higher place (tothe left)”, we will say “Move Up 1” for brevity.
4) Add 859 to 785
7 8 58 5 9----------------7 9 48 5 0----------------8 4 48 0 0----------------
1 6 4 4============
Hope the method used for Addition is clear by now. Practice it byyour own using Kawris or pebbles. I will continue this series bydescribing Substation, Multiplication, Division etc in coming lessons.
Ask your doubts, and I will try to clarify as far as the written wordscould convey. The written discussion of such practical methods thatshould be taught in the direct presence of teacher, shows us the limita-tion of conveying easy to understand ideas in writing, in the absence ofthe direct presence of teacher. That is why it is usually told that all suchpractical knowledge branches should be learned from teachers (= gu-rus) and not from books. :)
Remove C4 (=6) from 9 and Add 1to the next place.
Remove C9 (=1) from 5 and MoveUp 1
Remove C5 (=5) from 9 and MoveUp 1.
Remove C8 (=2) from 8 and MoveUp 1.
SubtractionSubtraction is also very simple - similar to addition. But in addition
we were removing the complement from the digit above, but here thecomplement should be always added. Thus the difference from normaladdition would be -
* Here you are using only Kawris for doing the calculation* The complementary numbers are used for doing the calculation* Complementary numbers are always ADDED to the digit above
when ever the digit above is smaller and 1 would be removed from thenext higher place (i.e to left) and added to the current digit.
The following examples will clarify the method of doing subtractionwith cawries.
1) Subtract 12 from 57
5 71 2---------5 51 0---------4 5=====
The same calculation is repeated again below with brief description,to clarify the terminology that would be used in further examples.
5 71 2--------5 51 0--------
Remove 2 Kawris from the 7Kawries placed abve. Answer= 5 Kawris
Remove 1 Kawri from the 5Kawries placed abve. Answer= 4 Kawries
Result : 4 5 (Fourty Five)
Remove 2 from 7.
Remove 1 from 5.
4 5=====
In the above calculation all the digits above were big and the digitsbelow small. Now let us do subtraction with numbers for which theabove digits are small. Then complements come in to play.
2) Subtract 87 from 235
2 3 58 7
----------------2 2 8
8 0----------------1 4 8==========
The same calculation is given below with another terminology ofdescription in the next example. Here instead of saying “Add Cx to the‘above digit’ Remove 1 from the next higher place (to the left)”, Iwould be saying “ Add Cx -1 to the ‘above digit’ and Move down 1".i.e. ‘Move down’ will mean ‘Remove 1 from the next higher place (tothe left) and add it here” Comparing with the above calculation andreading the below description will clarity this terminology. This is donefor brevity of terminology used.
2 3 58 7
----------------2 2 8
8 0----------------
1 4 8==========
I hope by now both the method and the terminology used for de-scription is clear to you. If not the following examples and the corre-sponding description will help you in understanding the method better.
3) Subtract 48 from 124
1 2 44 8
----------------1 1 6
4 0----------------
7 6==========
4) Subtract 256 from 728
7 2 82 5 6----------------7 2 22 5 0----------------6 7 22 0 0----------------4 7 2==========
Result : 4 5 (Fourty Five)
Add C7 (=3) to 5 and Remove 1from next higher place (to theleft).Add C8 (=2) to 2 and Remove 1from next higher place (to theleft).
Answer: 148 (One hundred andfourty eight)
Add C7 -1 (=2) to 5 and Movedown 1.
Add C8 -1 (=1) to 2 and Movedown 1.
Answer: 148 (One hundredand fourty eight)
Add C8 -1 (=1) to 4 andMove down 1.
Add C4 -1 (=5) to 1 andMove down 1.
Answer: 76 (Seventy Six)
Remove 6 from 8.
Add C5 -1 (=4) to 2 andMove down 1.
Remove 2 from 6.
Answer: 472 (Four hun-dred and Seventy Two)
5) Subtract 4572 from 7938
7 9 3 84 5 7 2---------------------7 9 3 64 5 7 0---------------------7 8 6 64 5 0 0---------------------7 3 6 64 0 0 0---------------------3 3 6 6=============
I hope these examples must have clarified the doubts, and the methodis well understood by everyone.
MultiplicationLet us see how Bhaskaracharya describes this ancient method of
doing multiplication in his book Leelavati.MÉÖhÉÉxiªÉ¨ÉRÂóEÆò MÉÖhÉEäòxÉ ½þxªÉÉnÂùMÉÖhÉÉxiªÉ¨ÉRÂóEÆò MÉÖhÉEäòxÉ ½þxªÉÉnÂùMÉÖhÉÉxiªÉ¨ÉRÂóEÆò MÉÖhÉEäòxÉ ½þxªÉÉnÂùMÉÖhÉÉxiªÉ¨ÉRÂóEÆò MÉÖhÉEäòxÉ ½þxªÉÉnÂùMÉÖhÉÉxiªÉ¨ÉRÂóEÆò MÉÖhÉEäòxÉ ½þxªÉÉnÂù=iºÉɪÉÇ iÉäxÉè´É¨ÉÖ{ÉÉxiªÉ¨ÉÉnùÒxÉ**=iºÉɪÉÇ iÉäxÉè´É¨ÉÖ{ÉÉxiªÉ¨ÉÉnùÒxÉ**=iºÉɪÉÇ iÉäxÉè´É¨ÉÖ{ÉÉxiªÉ¨ÉÉnùÒxÉ**=iºÉɪÉÇ iÉäxÉè´É¨ÉÖ{ÉÉxiªÉ¨ÉÉnùÒxÉ**=iºÉɪÉÇ iÉäxÉè´É¨ÉÖ{ÉÉxiªÉ¨ÉÉnùÒxÉ**
- ±ÉÒ±ÉÉ´ÉÊiÉ
[Multiply the first digit of the number (from the right) to be multiplied,with the multiplier. Then multiply the next digit. Continue like this (andyou will get the final result)]
Of course you would be well aware that to do multiplication oneshould know the ‘multiplication table’ and also that multiplication is noth-
Remove 2 from 8.
Add C7 -1 (=2) to 3 and Movedown 1.
Remove 5 from 8.
Remove 4 from 7.
Answer: 3366
ing but repeated addition.The following special points should be noted about multiplication.* All multiplication should be done from ‘right to left’ (instead of the
‘left to right’ multiplication that we do now a days in modern math-ematics).
* The multiplier should be placed below the number to be multipliedin such a way that, the right most digit of the multiplier should comeright below the left most digit of the number to be multiplied. For themultiplication of the each subsequent digit in the number to be multi-plied, the multiplier is moved to one place to the right.
* The complementary numbers are not used in multiplication. Better than explanation an example will clarify multiplication method
used. Let us start with a small example.1) Multiply 26 with 7.
2 67
---------------------1 4 6
7---------------------1 8 2==========
To avoid confusion whenever new Kawris are to be brought andadded or placed in a position, i will try using the word, ‘bring’, to denotethe same. Let us repeat the above calculation using this terminology.
2 67
---------------------
2 x 7 = 14; So Place 1 Kawri toleft, Add 2 Kawaris to 2 (makingit 4), and move the multiplier (i.e.here 7) to one place right.
7 x 6 = 42; Remove 4 Kawries from6 making it 2, Add 4 Kawries (the10th place of 42) to 4 in 10thplace (making it 8)Answer = 182
2 x 7 = 14; Bring 1 and Move upthat 1. Bring 2 and add to 2 (mak-ing it 4). (Thus number 14 prop-erly placed) Move down multi-plier.
1 4 67
---------------------1 8 2==========
2) Multiply 13 with 12.
1 31 2---------------------1 2 3
1 2---------------------1 2 6
3---------------------1 5 6============
This description does not seem to be satisfactory, further it lacksclarity and brevity as well. Therefore I am changing the terminologyand describing the same calculation once again. Here onwards the words‘Place x’, will refer to ‘placing the numbers in proper position by addingor removing Kawries from proper locations’. The example repeatedbelow will clarify this.
1 31 2---------------------
1 2 31 2
---------------------1 5 6============
I know that the above kind of description achieves brevity, but lacksclarity for the beginner. Therefore to help the beginners the gist of it Iwill repeat the one more example with elaborate and brief descriptions.
3) Multiply 25 with 13.
2 51 3---------------------2 6 5
1 3---------------------2 6 5
51
---------------------3 2 5============
Let us give the same calculatin with brief description.
2 51 3---------------------2 6 5
7 x 6 = 42; Remove 4 from 6(making it 2), Bring 4 and addto 4 (making it 8). (Thus num-ber 42 properly placed)Answer = 182
1 x 1 = 1; Bring 1, and Place it.2 x 1 = 2; (1 placed) Bring 1and add to 1 (to make it 2). (2placed) Move multiplier oneplace right.1 x 3 = 3; Bring 3 and add to 2(3 placed); 3 x 2 = 6; Add 3 to 3to make it 6. (6 placed)
(Intermediate step displayed forclarity)
Answer = 156
1 x 1 = 1; Place 1. 2 x 1 = 2;Place 2. Move multiplier to right.
1 x 3 = 3; Place 3; 3 x 2 = 6;Place 6.
Answer = 156
1 x 2 = 2; Bring 2, and Place it.(2 placed). 3 x 2 = 6; Bring 4 andadd to 2 (to make it 6). (6 placed)Move multiplier one place right.1 x 5 = 5; Bring 5 and add to 6(5 placed); 3 x 5 = 15; 5 is al-ready there at left. Add 1 to left(15 placed).
(Intermediate step displayed forclarity)
Answer = 325
1 x 2 = 2; Place 2.3 x 2 = 6; Place 6.Move multiplier to right.1 x 5 = 5; Place 5.
1 3---------------------2 6 5
51
---------------------3 2 5============
Since the procedure is clear by now, here onwards I will provide thebrief description only, even though the intermediate steps would be givenfor clarity sake. In intermediate steps, removing Kawris will be dis-played as a -ve digit.
4) Multiply 234 with 143
2 3 41 4 3----------------------------2 8 6 3 4
1 4 3----------------------------2 8 6 3 4
3 2 61
----------------------------3 2 8 9 4
1 4 3----------------------------
3 2 8 9 44 6 -21 1
----------------------------3 3 4 6 2======================
I hope the intermediate results given, will help much in the actualprocess of moving the Kawris involved in the calculation. I have giventhe intermediate steps for helping the learners to compare the stepswith modern calculation method.
5) Multiply 234 with 567.
2 3 45 6 7
------------------------------------1 0 2 2 3 4
1 1 +2------------------------------------1 1 3 4 3 4
5 6 7------------------------------------1 1 3 4 3 4
1 5 8 -21 2
------------------------------------1 3 0 4 1 4
5 6 7------------------------------------
3 x 5 = 15; Place 15.
(Intermediate step displayed forclarity)
Answer = 325
1 x 2 = 2; Place 2.4 x 2 =8; Place 8.3 x 2 = 6; Place 6.Move multiplier to right.1 x 3 = 3; Place 3.4 x 3 =12; Place 12.3 x 3 = 9; Place 9.Move multiplier to right.
(intermediate results)
1 x 4 = 4; Place 4.4 x 4 =16; Place 16.3 x 4 = 12; Place 12.Move multiplier to right.
(intermediate results)
Answer = 33462
5 x 2 = 10; Place 10.6 x 2 =12; Place 12.7 x 2 = 14; Place 14.Move multiplier to right.(intermediate results)
5 x 3 = 10; Place 10.6 x 3 =18; Place 18.7 x 3 = 21; Place 21.Move multiplier to right.
(intermediate results)
5 x 4 = 20; Place 20.6 x 4 =24; Place 24.7 x 4 = 28; Place 28.
1 3 0 4 1 42 0 4 +4
2 2------------------------------------1 3 2 6 7 8========================
This method of multiplication is known as ‘Long multiplication’ or‘simple multiplication’. There is another method of multiplication called‘Fold and multiply’ in Kawri maths that is used to multiply numberswith decimal points, especially related to calculations related to cur-rency, units etc. I will discuss that method later.
I hope these examples must have clarified the method of simplemultiplication used in Kawri Kriya (Doing calculations using Kawri).
If you have any doubts, ask them, I will try to clarify. My advice toyou is instead of just reading this document, try to DO the same calcu-lations on top of your table using Kawris or pebbles. Doing once willclarify the idea better than reading it many times.
- Sreenadh [email protected]
(intermediate results)
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