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Wednesday, June 2, 2010

Vedic Mathematics Lesson 51: Recurring Decimals 3

In the previous lesson, at the very end, I left you with a teaser, talking about how it is possible to compute multiples of a reciprocal (such 4/7, 3/13, etc.) without computing them, but by simply computing the reciprocal and using the reciprocal to derive these multiples without any addition, multiplication, division, etc. We will see how to do that in this lesson.

You can find all my previous posts about Vedic Mathematics below:

Introduction to Vedic Mathematics
A Spectacular Illustration of Vedic Mathematics
10's Complements
Multiplication Part 1
Multiplication Part 2
Multiplication Part 3
Multiplication Part 4
Multiplication Part 5
Multiplication Special Case 1
Multiplication Special Case 2
Multiplication Special Case 3
Vertically And Crosswise I
Vertically And Crosswise II
Squaring, Cubing, Etc.
Subtraction
Division By The Nikhilam Method I
Division By The Nikhilam Method II
Division By The Nikhilam Method III
Division By The Paravartya Method
Digital Roots
Straight Division I
Straight Division II
Vinculums
Divisibility Rules
Simple Osculation
Multiplex Osculation
Solving Equations 1
Solving Equations 2
Solving Equations 3
Solving Equations 4
Mergers 1
Mergers 2
Mergers 3
Multiple Mergers
Complex Mergers
Simultaneous Equations 1
Simultaneous Equations 2
Quadratic Equations 1
Quadratic Equations 2
Quadratic Equations 3
Quadratic Equations 4
Cubic Equations
Quartic Equations
Polynomial Division 1
Polynomial Division 2
Polynomial Division 3
Square Roots 1
Square Roots 2
Square Roots 3
Recurring Decimals 1
Recurring Decimals 2

The key to the process of computing multiples of a reciprocal without actually doing any additional computations is the concept of a cyclic number. Essentially, you rewrite a number in circular fashion (along the rim of a circle), and you can create different numbers out of it by changing the starting position from which you choose to read the cyclic number.

The simplest example of this, and the one that most commonly causes amazement among people not familiar with vedic mathematics and cyclic numbers, is the number 142857. There have been several articles written about this number, and here is one such article.

The simple fact of the matter is that 1/7 is equal to 0.142857 . . ., repeated endless times. The remarkable thing is that the computation of 2/7 requires no effort at all. You simply take the digits of 1/7, and find out a way to start from the second lowest digit in the number and complete the cycle. Thus, instead of starting from 1 and writing 142857, you would start with 2 and write 2857, and then cycle to the beginning of the number to complete it, getting 285714. You could then say that 2/7 is 0.285714 . . .. Similarly, for 3/7, you would start with the 3rd smallest digit and get 3/7 = 0.428571 . . ., and so on. Finally, you would get 6/7 = 0.857142 . . ., and we all know that 7/7 = 1.

But this property is certainly not limited to when the denominator is 7. 1/7, 2/7, etc., do not have a monopoly on some special property that no other numbers have. In fact, quite the opposite. All numbers whose reciprocals result in purely recurring decimals (i.e. the numbers that do not have 2 or 5 as one of their factors) have this cyclic property. 7 is simply the most commonly known one.

Before we go any further, let us review one very important property of recurring decimals that we will use extensively in this lesson. That property concerns the number of digits in a recurring decimal. This number of digits is also called the period of a recurring decimal. It is the number of digits you have to write to fully represent the recurring decimal before you can put ellipsis (. . .) at the end of it, or put a bar on top of it, or put dots on top of the first and last numbers (depending on your preferences, these are all valid ways of representing a recurring decimal).

Thus 1/7 is 0.142857 . . ., and the period of this recurring decimal is 6. 1/9 is 0.111 . . ., and the period is 1. You can similarly find the period of any recurring decimal simply by counting the number of digits until the pattern starts repeating again.

The important property of recurring decimals is that the period of any recurring decimal is always between 1 and one less than the denominator. The period can not be greater than or equal to the value of the denominator. Thus the period of 1/7 is 6, which is one less than the denominator, 7. The period of 1/9 is 1, which is much less than one less than the denominator. The period of 1/29 is 28, which is, once again, one less than the denominator.

A more obvious fact about reciprocals is that there are always one less than the denominator distinct multiples of the reciprocal of a number. Thus, 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 are the 6 (one less than the denominator, 7) unique multiples of the reciprocals of 7 (7/7 is just one, and 8/7 is equal to 1 and 1/7, and so on. They are not unique the way the first six multiples are). Similarly, 1/3 and 2/3 are the 2 distinct multiples of the reciprocal of 3 and so on.

The confluence of these two factors is what makes cyclic numbers possible. Essentially, when the period of a reciprocal is equal to one less than the denominator, the multiples of the reciprocal can be derived by cyclic combinations of the reciprocal without any further computation! That is the first and most useful result that comes out of cyclic numbers and recurring decimals.

Let us use this first and most useful result to do some computations. We have already seen how to use this in the case of the multiples of 1/7. Let us take 1/19 as our next example. Using the methods in the previous lesson (or the one before that), we can compute that 1/19 = 0.052631578947368421 . . .. The period of this recurring decimal is 18, which is one less than 19. Therefore, we can be confident that the number is a cyclic number.

How do we take advantage of that to compute 2/19, 3/19, etc. This is easy to do by just looking at the number. To compute 2/19, simply find the second smallest cyclic rearrangement of the numbers in 1/19. In the case of 1/7 to 2/7, we accomplished this simply by looking for the second smallest digit in the number. But when the number of digits is greater than 10, as in this case, there are bound to be repetitions of digits. Therefore, the more general method to doing this is simply to look for the second smallest, third smallest, etc. cyclic representations of the given number.

In this case, the second smallest cyclic representation is obtained by starting from the last digit of the number, cycling around to the first digit and then writing the rest of the digits in order (remember, wherever you start from, go to the end of 1/19, then cycle around to the beginning and write down the digits until you are at the digit just before where you started from). This then tells us that 2/19 = 0.105263157894736842 . . ..

For finding 3/19, we repeat the procedure once again, finding the next higher cyclic representation of the given number. This will then lead to 3/19 = 0.157894736842105263 .. .. As you can see it is trivially easy to follow this procedure to derive the entire series of multiples of 1/19 as below:

1/19 = 0.052631578947368421 . . .
2/19 = 0.105263157894736842 . . .
3/19 = 0.157894736842105263 . . .
4/19 = 0.210526315789473684 . . .
5/19 = 0.263157894736842105 . . .
6/19 = 0.315789473684210526 . . .
7/19 = 0.368421052631578947 . . .
8/19 = 0.421052631578947368 . . .
9/19 = 0.473684210526315789 . . .
10/19 = 0.526315789473684210 . . .
11/19 = 0.578947368421052631 . . .
12/19 = 0.631578947368421052 . . .
13/19 = 0.684210526315789473 . . .
14/19 = 0.736842105263157894 . . .
15/19 = 0.789473684210526315 . . .
16/19 = 0.842105263157894736 . . .
17/19 = 0.894736842105263157 . . .
18/19 = 0.947368421052631578 . . .

There, you see, I told you 142857 was not the only cyclic number. Obviously, finding the cyclic pattern in the multiples of the reciprocals of 19 is a lot harder, but the theory behind both sets of cyclic numbers is the same.

In fact, there are numerous such cyclic numbers, and we have just scratched the surface with 1/7 and 1/19. The period of 1/17 is 16, therefore we are assured that the recurring decimal representation of 1/17 is a cyclic number. The same is the case for 1/23, 1/29, and numerous other numbers whose period is one less than the denominator. Now you know how to derive the multiples of these reciprocals once you finish computing just the reciprocal, with no further addition, subtraction, multiplication or division!

However, what happens when the period is not one less than the denominator. For instance, 1/13 is 0.076923 . . ., whose period is only 6. Similarly, 1/9 has a period of just 1. What do we do in these cases? Unfortunately, in these cases, the computation of the multiples of the reciprocal is not as straight-forward and orderly as the computation we just performed to get the multiples of the reciprocal of 19.

The problem is that for numbers whose period is less than one less than the denominator, we need to compute additional multiples of the reciprocal before applying the method. This is the second important result to remember about the relationship between cyclic numbers and reciprocals and multiples of reciprocals. For instance 1/13 = 0.076923 . . ., and 2/13 = 0.153846 . . .. We see that the digit sequence in 1/13 is completely different from the digit sequence in 2/13 (i.e. 2/13 is not a cyclic rearrangement of the digits in 1/13). The combination of both digit sequences now has 12 digits, which is one less than the denominator, 13.

Now, we can use our previous procedure, or at least a slight modification to proceed further. The trick is to find the next smallest cyclic sequence from either of the sequences we have found so far, but be careful not to mix the two sequences up when you write the answer (thus each multiple will contain only a period of 6, none of them will contain a period of 12 or any number other than 6). Each multiple of 1/13 will use either the sequence we have found in 1/13 or what we found for 2/13, but not a combination of both. The order in which we produce numbers from each sequence is entirely dependent on the value of each cyclic sequence. The order always has to be from least value to greatest value. This may result in some regular alternating between the two sequences but in most cases, it will not. Thus:

1/13 = 0.076923 . . .
2/13 = 0.153846 . . .
3/13 = 0.230769 . . . (from the sequence for 1/13)
4/13 = 0.307692 . . . (from the sequence for 1/13)
5/13 = 0.384615 . . . (from the sequence for 2/13)
6/13 = 0.461538 . . . (from the sequence for 2/13)
7/13 = 0.538461 . . . (from the sequence for 2/13)
8/13 = 0.615384 . . . (from the sequence for 2/13)
9/13 = 0.692307 . . . (from the sequence for 1/13)
10/13 = 0.769230 . . . (from the sequence for 1/13)
11/13 = 0.846153 . . . (from the sequence for 2/13)
12/13 = 0.9230769 . . . (from the sequence for 1/13)

As you can see, the procedure is pretty straight-forward once we have a set of sequences such that the sum of the periods is one less than the denominator. As noted earlier, note that we picked cyclic sequences from both original sequences based purely on the resulting value. This resulted in 2/13 being used 4 times in a row rather than alternating evenly between 1/13 and 2/13. However, note that the number of times we derived a sequence from 1/13 is equal to the period of 1/13 (6), and the number of times we derived a sequence from 2/13 is equal to the period of 2/13 (6). This is the third important result to keep in mind when thinking about cyclic numbers, reciprocals and multiples of reciprocals.

But sometimes, getting to enough such independent sequences can become problematic. For instance, 1/39 = 0.025641 . . .. The period is only 6. We find that the following set of multiples have to be considered to get the sum of the periods up to 38 (without having any sequence be just a cyclic rearrangement of some previous one):

1/39 = 0.025641 . . . (period of 6)
2/39 = 0.051282 . . . (period of 6)
3/39 = 0.076923 . . . (period of 6)
6/39 = 0.153846 . . . (period of 6)
7/39 = 0.179487 . . . (period of 6)
13/39 = 0.3 . . . (period of 1)
14/39 = 0.358974 (period of 6)
26/39 = 0.6 . . . (period of 1)

How did I know which multiples to pick to get these unique sequences (unique in the sense that none of them is a cyclic rearrangement of any previous sequence)? I have to admit that it was entirely based on trial and error. By the time you get all the sequences, you have computed most of the multiples of the reciprocal, so the technique has minimal value. Also, sequences of period 1 are completely useless since the number of sequences based on them has to be 1 (their period), and therefore, they can not form any other sequences except themselves!

Fortunately for us, 39 is more of an exception than the rule. Most other numbers (particularly primes), either have a reciprocal whose period is one less than themselves or whose period is exactly half of one less than themselves. Finding twice this reciprocal will usually be sufficient to get two independent sequences whose periods add up to one less than the denominator, making the computation of other multiples trivial. This is true of 89 (1/89 has a period of 44, which is half of one less than 89), 83 (1/83 has a period of 41, which is half of one less than 83), and several other prime numbers. And in most, if not all, of these cases, the reciprocal and twice of the reciprocal will produce distinct sequences that will be sufficient to derive all the other multiples of the reciprocal.

Most composite numbers (like 39) are a lot more troublesome (the more factors, the more trouble, in general). Each factor times the reciprocal (3/39, 13/39, etc.) will usually produce a unique sequence in addition to various oddball multiples (like 2, 7, 14, etc., in the case of 39). In general, this technique is not very efficient at finding multiples of the reciprocals of these numbers, and you would be better of calculating them using the methods expounded in the previous lesson or the one before that.

Hope this lesson has been a good diversion from the usual dry fare I dish out here because of the interesting nature of cyclic numbers presented here. You can amaze your friends by talking to them about not only a well-known cyclic number like 142857, but more esoteric and exotic ones like 0.052631578947368421 (1/19), 0.0344827586206896551724137931 (1/29), etc. Good luck, and happy computing!

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