by Andrew Stewart-Brown
In his book Vedic Mathematics1, H.H. Bharati Krishna Tirthaji, gives many succinct sutras or aphorisms for making light work of mathematics. Some of them have been explored from an educational perspective by George G. Joseph in his essay Multiplication Algorithms in the collection of essays Multicultural Mathematics2. Other aphorisms certainly repay study. For instance, 'ekadhikena purvena' or 'by one more than the one before' can be applied to recurring decimals, the decimal form of fractions whose denominator has a prime other than 2 or 5 as a factor. By means of this sutra and some simple arithmetic, it is possible to write down the 'period' of any recurring decimal with a minimum of effort.
Write the number 1 on the right hand side of the piece of blank paper which every mathematician always has to hand. Multiply it by 2 and write the product on the left of the 1. Multiply the 2 by 2 and write the product on the left of the 2. Multiply the 4 by 2 and write the product on the left of the 4. So far we have
8 4 2 1
Multiply the 8 by 2 and write down the 6 on the left of the 8 and carry the 1 into the next column.
16 8 4 2 1
Multiply the 6 by 2, add the 1 to get 13, write down the 3 on the left of the 6 and carry the 1,
13 16 8 4 2 1
Proceed until the cycle begins to repeat when the number 1 reappears with no carry.
(1) .0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1
You have written out the decimal expansion of 1/19. ' One more than the one before' here means one more than the number before the 9 in the denominator. One more than 1 is 2. So we repeatedly multiply by 2.
If you please, you can work it the other way and divide by 2 from the front. So 1, the numerator, divided by 2 gives you 0 remainder 1. Write it down as shown: 10 and divide by 2 giving 5, write that down as 1.0 5 and proceed: 1.0 5 12 6 ...
If you wish to reduce the labour further you may notice that the number 18 or 19 - 1, turns up in the middle. After this, the rest of the recurring decimal can be found by subtracting the digits from 9, starting from 0. So 0 from 9 gives 9, 5 from 9 gives 4, 2 from 9 gives 7 and so on.
Supposing the numerator to be 3 instead of 1, then the number to write on the right hand side would be 3, then proceed in the same way multiplying by 2: .......1263 then halt when 1 you get to 3 without remainder. If the fraction is 17/19, then we begin on the right with 1 7, multiply the 7 by 2 and add in the 1: .............11 15 17.
Similarly, if you want the decimal expansion of 1/29, one more than the number before the 9 in the denominator is 3. So starting with 1 on the left of the page, multiply leftwards in the same way as before but with 3 instead of 2, and you will produce the decimal expansion of 1/29 . Again if you wish to halve your work, and who wouldn't with a period of 28 places, you can spot the appearance of 28 (or 29 - 1) and work out the other digits by subtraction from 9. 8 is the last digit of the first half of the period and goes with the 1 at the end to make up 9.
A method which allows a numerate ten year old to write down the full decimal expansion of 1/29 must command some admiration for that reason alone. But what is going on behind the scenes ? Any rational number can be expressed as the sum of an infinite geometric series like 1, 2,4,8,16, 32 etc., but where the common ratio is less than 1. If we start from the formula for the sum of such series,
1+ r + r2 + r3 +r4 +...+ rn = (1- RN+1)/(1-r). So 1 + 2 + 4 + 8 = (1- 24)/(1- 2) = 15
If r is less than 1, and as n increases, RN+1 tends to 0, and the sum of the series tends to 1/(1-r) . For our purposes, it is easier to consider the sum of r + r2 + r3 .......= 1/(1-r) -1 = r/(r-1) So to find the infinite geometric series which equals any given rational number, we only need to find the common ratio, r, by equating that number to r/(1-r) and solving for r.
So if x/y = r/(1-r), x-rx = yr , so r = x/(x+y)
For example then if x / y = 1 / 19 then r = 1/20, so
1/19 = 1/ 20 + (1/20)2 + (1/20)3 ...... for ever. We can see a common ratio of 1/20 appearing.
If we rewrite this expansion as 1/2 × 1/10 + 1/4 × 1/100 + 1/8 × 1/1000 + ...... and make an unconventional use of the decimal system, we may say:
tenths hundredths thousandths thousandths thousandths millionths
1 = 0. 1 1 1 1 1 1 etc.,
19 2 4 8 16 32 64
that is, one half of a tenth, one quarter of a hundredth and so on. If one wishes, one can add these all up and the desired recurring decimal does indeed appear.
We can see that to go from left to right from column to column we divide by 2 and vice-versa we multiply by 2. When the series is shown in this way, the geometrical ratio appears as 1:2 not 1:20, because the ratio 1:10 is already implicit in the denary place value system.
Why start with 1 on the right? The period of the decimal form of 1/19 must end in 1, because each period multiplied by 19 must give us a row of 9's of the same length as the period, and without any digits to carry. (1 = 0.9999.. for ever).
So we can take the period by its tail, so to speak, and starting with 1, work out the numbers by doubling towards the left. How neat!
What about fractions where the denominator does not end in nine ? Because every rational number can be expressed as the sum of an infinite geometrical series, we can be sure the common ratio of the series is in there somewhere, but how do we find it ? The first thing to do is to strip out any factors of 2 or 5 from the denominator and rewrite them a equivalent fractions with powers of 10 as denominator: 1/ 26 = 5/10 × 1/13. Then write 1/13 as an equivalent fraction with denominator ending in 9 : 1/13 = 3/ 39.
So 1/26 = 1/10 × 15/39 = 1/10 × 15 1/40n. So start with 15 at the right hand end, multiply
????? n=0 . .
by 4 until we get back to 15: ..... 33 18 24 6 21 15. Adjust for the 1/10 and we have 1/26 = 0.0384615.
????? 3 1 2 2 1
So by means of the aphorism ' by one more than the one before' - and the insights we recall when we remember it, we can write down with little labour the whole period of any fraction whose decimal form recurs ! Pupils are, to say the least, intrigued to find that these periods, when longer than their calculator displays, marry up with the front part which the calculator does give them. Anything which promotes the awakening of curiosity deserves a bit of attention !
This is an explanation of only one application of one aphorism given by the author and is only one of the methods given for producing the decimal form of rational numbers. The geometrical ratios, for instance, also turn up in the cycle of remainders obtained when dividing the numerator by the denominator and can be exploited in a wonderful way. There is also the interesting question of exactly why any particular fraction repeats after so many places.
Many claims are made by the author which appear fanciful, such that maths teachers would starve if his methods were adopted ! His world-view is quite different from the one which prevails in the West. But his 'eight years of concentrated contemplation in forest solitude' has borne remarkable fruit. His ideal of one-line at-sight mental methods for solving mathematical problems, which with current methods require many laborious and cumbersome steps of working. Has been achieved in several fields. We would call his methods algorithms. But they are algorithms for the mind rather than for a machine. Children like the methods because they find themselves suddenly empowered to solve problems and find answers to problems which appear difficult. A sense of wonder may be felt which easily leads to the inquiry, 'How does that work ?' The moment pupils genuinely want to know, the subject comes alive and they can be led, or find their way, to understanding. So these methods are well worth investigating.
Skemp, in his Psychology of Learning Mathematics (p. 135) writes: How effective an intrinsic motivation for learning mathematics can be is something which many teachers do not yet appreciate. On a number of occasions, teachers finding that children actually enjoy mathematics when it is intelligently taught and learnt have reported this to me with a mixture of surprise and pleasure, but also of doubt; as if something must be wrong with an approach to mathematics which children enjoyed. But until this intrinsic motivation is better comprehended and put to work, mathematics will remain for many a subject to be endured, not enjoyed; and dropped as soon as the necessary exam results have been achieved.'
Sri Bharati Krishna Tirthaji, 1965, Vedic Mathematics, Motilal Barnasidas, Delhi.
ISBN 812 08 01 644
Available from Lavis Marketing, 73 Lime Walk, Headington, Oxford @ £5.75 + £2 p&p.
Nelson D., Joseph G.G., Williams J., 1993, Multicultural Mathematics, OUP, 1993.
ISBN 0 19 282241 1
Skemp, Richard R., 1971, Psychology of Learning Mathematics, Penguin Books, 1971
ISBN 0 14 02.1310 4
Andrew Stewart-Brown is teaching mathematics at Bicester Community College in Oxfordshire and is planning to offer an extra session at this year's Easter Conference on the Vedic approach to division. You may be interested in the website