In Vedic Mathematics there exists a way to write higher numbers (6, 7, 8, 9) in terms of lower numbers (0, 1, 2, 3, 4). Why is this important? It’s important because it makes “difficult” looking multiplication problems fairly simple. This is because it is much easier to work with the lower numbers, especially in multiplication. Using this method we only need to know multiplication table up to 5 x 5. This method of writing numbers is called vinculum numbers, and results in numbers containing both positive and negative digits.
How to get vinculum numbers
The secret to vinculum numbers lies in the “Nikhilam Navatashcaramam Dashatah” sutra, which translates as “All from 9 the Last from 10” and the “Ekadhikena Purvena” sutra which translates as “By one more than the previous one.” These sutra is used to obtain the vinculum numbers as I will show you below. Let us begin with an example…
Example #1: 26
Here we wish to convert the number ’26′ into a number that only consists of a mixture of (0, 1, 2, 3, 4, 5).
1) For this first step we will be using the “All from 9 the Last from 10” sutra. The first thing we do is identify the last digit that happens to be a higher number (6, 7, 8, 9). In this case that number is ’6′, the last digit.
2) We see that ’6′ is 4 less than 10 (we use ’10′ because of the sutra…”All from 9 the Last from 10”), so we write this as 4. (Side Note: In books on Vedic math, the line will be on top of the numbers, but since I’m typing this on a computer for simplicity’s sake its easier to write it below.)
3) Now we use the “By one more than the previous one.” The previous one is ’2′ in this case, and by one more means ’3′. Our number is now…
26 = 34
You can think of 34 as (30 – 4) which gives you 26.
Example #2: 183
183 = 223
1) Since 8 is the last “higher number”, we take it from 10….10-8 = 2. We write it as 2.
2) We then add one to the previous one…1 + 1 = 2.
3) The ’3′ is unaffected so it stays the same.
This should be thought of as 203 – 20.
Example #3: 169
169 = 231
1) The last in this example is the ’9′, so we take it from 10….10 – 9 = 1. We write it as 1.
2) In this example we have a second digit that is a “higher number”…’6′. For this we use the first part of the Nikhilam sutra…”All from 9.” So we take ’6′ from 9….9 – 6 = 3. We write this down as 3.
3) We add one to the previous one (’1′)…1 + 1 = 2.
This should be thought of as either “200 – 30 – 1” or “200 – 31.”
Example #4: 372962
372962 = 433042
1) Before our “higher numbers” were always grouped together so we had only one “last”, however now we will have 2 in the number “372962”….the ’7′ and the ’96′.
2) The “last” of the first group is 7…so 10 – 7 = 3. We write 3. This takes care of the first group.
3) Add one to the previous one…so 3 + 1 = 4.
4) Now look at the second group, the last is ’6′…10 – 6 = 4. We write 4. The other number, ’9′, we take from 9……9 – 9 = 0.
5) Add one to the previous one…2 + 1 = 3.
This number (433042) represents 403,002 – 30,040.
Here are a few more:
15627283 = 16433323
397968697 = 402031303
37 x 98 = 43 x 102
The great thing about this is that 0 & 1 are twice as likely to show up now, and these numbers are easy to multiply by! Look at the last example…we went from having to multipy by ’9′ and ’8′ which are hard, to ’0′, ’1′ and ’2′ which are much simpler.
Using vinculum numbers in subtraction
Example #5: 347657 – 238294
1) The first thing we do is write these numbers one on top of the other.
347657
238294
2) Now we just start subtracting vertically, and whenever our number is negative we represent it as a vinculum number.
347657
238294
11
3) This next subtraction is 7 – 8 = -1…so we just write this as 1 and continue on.
347657
238294
111443
4) The ‘1‘ and ‘4‘ here are considered as two separate groups (since theres a non-vinculum number in-between them), so each is subtracted from 10. This has the effect of reducing the “previous one” by one.
109363
Example #6: 3251282 – 1896853
3251282
1896853
2645631
Which in turn becomes…
2645631 = 1354429
Using Vinculum numbers in multiplication
Example #7: 78 x 67
78
67
1)Normally this would be somewhat difficult (since we have to multiply by higher numbers and add them), but now we can convert it…
122
133
_____
Now we just do vertically and crosswise.
1) 1 x 1 = 1.
122
133
_____
1
2) (1 x 3) + (1 x 2) = -3 + -2 = -5 or 5. Remember…vinculum numbers represent negative numbers, so 5 is the same as “-5”.
122
133
_____
15
3) (1 x 3) + (1 x 2) + (2 x 3) = -3 + -2 + 6 = 1.
122
133
_____
151
4) (2 x 3) + (2 x 3) = 6 + 6 = 12. We write down the ’2′ and carry the one to the right.
122
133
_____
1522
5) (2 x 3) = 6.
122
133
_____
15226
So our answer is 15226 or 5226.
The practice problems are going to be split up into three groups: converting to vinculum, subtracting, and multiplying.
Practice Problems
Converting to vinculum
1. 937
2. 672
3. 35908795
4. 18978697
Subtraction using vinculum
1. 546328 – 385977
2. 83947 – 63597
3. 475829828 – 63685
4. 28 – 0.67392
Multiplying using vinculum
1. 46 x 23
2. 58 x 19
3. 128 x 121
4. 473 x 398
If you see any errors, let me know!
Wow, I think this is getting WAY, WAY WAY too complicated for me….
Could I make a suggestion, especially for blog posts? Such a long, complicated blog post is extremely intimidating to readers (and I am not a reader who is easily intimidated, so imagine your typical reader)…..
I would suggest a post like this being broken into THREE, SHORTER posts.
I’m not sure I understand your Example Number One, much less your example number Two, and I felt so flustered I couldn’t even read the rest of the post. I feel flustered and frustrated because this is a subject I am REALLY, REALLY interested in.
For your FIRST example ONLY, I think we need a list of ten sample problems to work through, and below that (in the same post) a list of the ten ANSWERS to those sample problems.
The next blog post should go on to Example Two, and and use the same procedure (you have to allow readers to understand the idea and become competent in understanding it before moving to higher examples).
I’m just being honest here….I’m sure if you’re taking the time to write all this down for your readers, you DO want them to really understand and benefit from it. Please help! I’m LOST!
The information in your blog is SO, SO useful to me as not only do I want to know it for myself, I want to know it to make me a BETTER ELEMENTARY MATH TEACHER.
Best regards,
Eileen
Dedicated Elementary Teacher Overseas (in the Middle East)
elementaryteacher.wordpress.com
I understand what your saying about the posts being too long and such. I’ll work on shortening up the posts, adding more problems w/ answers.
I’m going to be reposting a lot of material as shorter posts, I’m going to try and go in a more structured order this time instead of just throwing stuff together.
I’ve also been in the process recently of making some PDF files, however I would like to have you take a look at them to see if they make sense before I post them. So if you want, send me an email and I’ll send those to you. (email: liberius1776@gmail.com)
It’s very useful.
You’ve done a good job
Many thanks
——————————————
moving overseas
Oh, Thanks! Really interesting. Big ups!
Oh, Thanks! Really amazing. Greets.
Nothing wrong with the length of this post, what are you all complaining about? Good job on posting this article, i’ve read copious amounts of information on this subject but have never understood this method. +1 props.
its really a good work,i wonder why this has not reached the masses.you are doing a great job in explaining it in detail.thank you first of all.and i dont know why u have locked some posts hope u unlock soon.
than u again,cheers.
A zero-start 2 Dimensional square matrix 0…9 is at bottom.
‘Matrix position virtues’ are summarized as “Vedic sutras”.
Vinculam is a manner of splitting ‘a matrix position value’ into two parts with respect to a related base number (in this case 100), which is (highest matrix row-column number 99 shown)+1
It implies Vedic manner uses as many bases as possible. 0, 10, 100, 1000, 10000…so on countless bases are used!
I think that a basic manner of dividing ‘a matrix position value’ as ‘plus’ and ‘minus’ numbers (in relation to base) is sense “vinculum”
87 is a matrix position that has been marked in matrix.
+87-13 relates a base 100.
As (+ ) signs are not usually prefixed, (87-13) will do!
87
13
9(10) <– a vertical adding of digits in each column is 9 and 10). A very useful vedic sutra 'all from 9 but last from 10' is ideal split 'numbers near a base' as plus value and minus value!
So what is a vinculam?
Vinculam is a differrence from 'base number' to 'number chosen to compute. It could be plus or minus.
87 is (-13 ) from 'a base 100'. Simply write as (87-13)
Similarly 103 is (+3) from a base 100. Write as (103+3)
What is (87*103)?
..87-13 <– Crosswise both (103 -13) and (87+3) are equal
103+3
———–
..90(-39) <– It has meaning 90 00 – 39 (because 90*100's).
..89 61 <– 8961 is answer!
To understand 'Vedic sutras' fully, knowledge of "Sanskrit" language greatly helps! (I know little sanskrit)!
Too many "matrix virtues" exist. Most useful virtues are "Vedic sutras"! Ancient Indians knew "zero" and "Vedic matrix" much earler to compiling four Vedas!
…0……..1……..2……..3……..4……..5…….6…….7…….8…….9
—— —— —— —— —— —— —— —— —— ——
0
—— —— —— —— —— —— —— —— —— ——
1
—— —— —— —— —— —— —— —— —— ——
2
—— —— —— —— —— —— —— —— —— ——
3
—— —— —— —— —— —— —— —— —— ——
4
—— —— —— —— —— —— —— —— —— ——
5
—— —— —— —— —— —— —— —— —— ——
6
—— —— —— —— —— —— —— —— —— ——
7
—— —— —— —— —— —— —— x87– —— ——
8
—— —— —— —— —— —— —— —— —— x99–
9
vinculum numbers are interesting but difficult to understand
but the way you have explained had made it easy for me
thanks
it’s greatly explained.. in vedic mathematics books.. it’s quite difficult to understand.. but you’ve shortened the concept upto very precise level.. good job done..! thank you so much..! :)