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RECURRING DECIMALS
VEDIC STYLE
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
8421
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
8421
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
8421
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 :
ten
hundred
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 worldview 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.'
References
1. 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.
2. Nelson D., Joseph G.G., Williams J., 1993, Multicultural
Mathematics, OUP, 1993.
ISBN 0 19 282241 1
3. 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
[email protected]
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ACADEMY OF VEDIC MATHEMATICS
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