Revolving Numbers

WHILE experimenting with a system of counting by twelves,1 suddenly I came upon a series of numbers that behaved in the most extraordinary way. Discovery followed discovery. One day the whole beautiful system was complete — all except the ultimate Why.

Timely research disclosed that some of my ‘discoveries’ had been made by Ibn-el-Banna as early as the thirteenth century, and that there is now a large specialized literature on the subject. These strange numbers which in mediæval days were regarded as magical are being explored by modern mathematicians in terms of primitive roots, and with the hope that some of the final secrets of number may be wrenched from them. But since they are not generally known to the layman, and may afford amusement as well as a springboard for profound research, here is a brief discussion of them.

I

Let us define ‘revolving numbers’ (also called periodic fractions) by example, taking the number 142857. Multiplied by 2, it becomes 285714 — the same digits, but commencing at a different place in the series. Multiplied by 3, it is 428571 — again the same digits, but commencing at a still different place. Multiplied by 4, the answer is 571428 — still the same series. So we might continue until a perfect revolution is completed, with each of the six figures appearing in first place, and the others following in their established order.

Although this is demonstrably true, it seems against reason that any such number should exist. However, 142857 is by no means the only number with such extraordinary properties. To take another six-place number, 076923 can be multiplied by either 3, 4, 9, 10, or 12 and it will similarly ‘revolve’ its original figures. Or examine the number 0588 2352 9411 7647 in the table below, which can be multiplied by any number from 2 to 16 and always reproduces revolving forms of the original number: —

The number 0588 2352 9411 7647
Multiplied by 2 1176 4705 8823 5294
3 1764 7058 8235 2941
4 2352 9411 7647 0588
5 2941 1764 7058 8235
6 3529 4117 6470 5882
7 4117 6470 5882 3529
8 4705 8823 5294 1176
9 5294 1176 4705 8823
10 5882 3529 4117 6470
11 6470 5882 3529 4117
12 7058 8235 2941 1764
13 7G47 0588 2352 9411
14 8235 2941 1764 7058
15 8823 5294 1176 4705
16 9411 7647 0588 2352
17 9999 9999 9999 9999

Now while a six-place number with these peculiar properties could conceivably be constructed by simple experiment, it is obvious that to construct by trial and error a sixteen-place number such as we have examined would take something like a lifetime of experiment. They must be discoverable in some simpler way. What that way is will be found amply revealed, if it is not already known, in our seventeenth total for the sixteen-place number, which runs entirely to nines.

All the revolving numbers we have so far examined, and presumably all the existing numbers with similar properties, are in reality repeating decimals. For instance, the repeating decimal for 1/7 is .142857 142857 . . . which produces our first revolving number. Also, .076923 is the repeating decimal for fa, and .0588235294117647 for 1/17.

Repeating decimals, it will be remembered, are always produced when unity is divided by a prime number, excepting only 2 and 5, which, though prime, are also divisible into our number base, 10. The reader may construct revolving numbers for himself by simply working out the repeating decimals for 1/7, 1/13, 1/17, 1/19, 1/23, and so on. (A few prime numbers, as 1/3, 1/11, produce very short repeating decimals, which are revolving numbers only in a limited sense to be discussed later.) Since the number of prime numbers is infinite so far as has yet been demonstrated, it follows that these extraordinary revolving numbers, instead of being rare curiosities, can also be found in infinite number.

It might be hastily assumed that revolving numbers were products of decimal counting — the basing of a number system upon 10. This is not at all true. Any number system which employs a zero, and can therefore produce fractions in decimal form, produces a similar series of revolving numbers.

I have tested this statement by actually developing and working with number systems whose bases had respectively no factors and twice as many factors as our exist ing system. In duodecimal mathematics, I came upon a very interesting revolving number which can be added to itself up to the 102d time (which is the same as multiplying it by all numbers from 2 to 102) and will each time reproduce a revolving form of the original number.

For the curious, this number is: —

01493 X176X 59XX0 29678 3318ε 79805 71346 635ε3 740ε2 26910 6εX72 81X45 16211 ε9254 388X3 0423ε 04X87 55860 847ε0 9952X ε5

This number can be tested by adding the original number to successive totals as many times as desired. It is only necessary to remember that we are dealing with duodecimal (12-system) mathematics,2 and therefore nothing is carried to the next column until the sum is 12, instead of our customary 10. If the sum is what we now express as 10 or 11, it is to be set down as X and ε respectively.

But it is not necessary to work with a variety of number systems to discover many of the universal rules of revolving numbers, or repeating decimals. They may be arrived at by pure logic, perhaps aided by the careful examination of the production of one such number: —

7) 1.00000000(.14285714 . . .

7/30

28/20

14/60

56/40

Scrutiny of this example helps our logic to certain conclusions. With 7 as a divisor, the only possible remainders are 1, 2, 3, 4, 5, or 6. But as soon as any one of these appears a second time, the result from then on must repeat the former result. Therefore the outside limit of length for the period of the repeating decimal for 1/7 is six figures, and for any prime number it is one less than the prime number itself.

Again, we now see why .142857, or any repeating decimal, must also have revolving characteristics. Two times 1/7 must be the same as dividing 7 into 2.000 .. . This we have already done in the example of long division before us, if we omit the first two operations. Similarly, other multiples of 1/7 must be expressible by this same repeating decimal, simply beginning it at that operation where the given multiple appears as the ‘remainder’ in our long division.

II

From this analysis, and from considerable experimentation in several number systems, certain rules regarding the whole series of revolving numbers (or repeating decimals) can be announced with some confidence: —

1. In any number system, unity divided by a prime number (excepting the several prime numbers which may be factors of the number base) will produce a repeating ‘decimal’ with the revolving characteristics we have noted.

2. This number will run to a quantity of places which either is exactly one less than the prime number from which it is derived or bears an integral relation to that quantity.

For instance, the repeating decimal for 1/7 runs to six places, for 1/17 to sixteen places; but for 1/3 and 1/13 to one and six places respectively, or one half of ‘one less than the prime number,’ and for 1/103 to thirty-four figures, or one third of ‘one less than 103.’

3. If this number contains just one fewer places than the prime number from which it is derived, then it is a perfect revolving number; that is, one which in successive additions reproduces the original figures in the same general order, but with the series beginning at different places.

However improbable such numbers may seem at first thought, we have seen logically that they must exist, and we have demonstrated experimentally that they do exist, including one of 102 figures. Other and still larger ones can obviously be produced, up to the farthest patience of the experimenter. Moreover, these numbers not only ‘revolve’ until they reach the form .9999 . . . but, if that form is given its true value of 1. each time it appears, then they continue to revolve, by whatever number they are multiplied. For example, 22 × .142857 is 3.142854. The ‘3’ indicates that .9999 has been passed three times; this number is therefore added to the final figure, and the original revolving number appears.

4. If the repeating decimal does not run to just one place less than the prime number from which it is derived, but to a simple fraction of that number (see Ride 2), then it will prove to be a limited revolving number; that is, in some parts of the series it will reproduce its original figures, but in other parts it will produce a different set or sets.

We have already seen that the repeating decimal for 1/13 is .076923. This is a limited revolving number, for it reproduces its original figures when multiplied by 1, 3, 4, 9, 10, or 12, but it produces the quite different number, .153846 and its revolving variations, when mult iplied by 2, 5, 6, 7, 8, or 11.

5. Limited revolving numbers generate exactly as many new number sets as are necessary to complete the total of figures a perfect revolving number would have had.

Unity divided by 13 (1/13) produces a 6-place limited revolving number instead of the 12-place number which might be expected; therefore another 6-place limited revolving number appears in the complete series. Similarly, 1/103 produces a limited revolving number of only 34 places (.00970 87378 64077 66990 29126 21359 2233) instead of 102 places; therefore the series will generate two more 34-place limited revolving numbers, as can readily be proved.

6. From inspection of the prime number it is possible to predict not only the number of places the perfect revolving number (or the completed series of limited revolving numbers) will have, but even the numerals which will appear. There will be (a) as many complete sets of figures, 0-9, as there are tens in the prime number, plus (b) the figures included in the revolving number for the digits figure of the prime number.

For example, the revolving number for 17 is 0588 2352 9411 7647, as we have seen. This contains one complete set of the numbers 0-9 (representing the number of tens in 17) plus the revolving number for 7, or 142857. By this rule we should assume that the three limited revolving numbers for 1/103 would together contain 10 complete sets of all the numerals 0-9, plus 3 and 6, the limited revolving numbers for 1/3; it may be determined experimentally that this is the fact.

This perfect orderliness is true not only of 10-system mathematics but of other mathematical systems. For example, the prime number we represent as 103 is represented in the 12-system as 87 — 8 dozens plus 7 units. In the duodecimal system 1/87 produces the perfect revolving number of 102 figures we have elsewhere set down. Inspection of this long number reveals that it contains exactly 8 sets of all the numbers 0-ε (representing the number of dozens in 87) plus the duodecimal revolving number for 1/7, which is 186X35.

I leave to the mathematical expert the further exploration of these revolving numbers. A mathematical curiosity they certainly are; a key to the nature of number itself they may possibly be.

  1. See the Atlantic for October 1934. — EDITOR
  2. 35/50 49/10 7/30 28/2 2 October Atlantic, pp. 462 ff. — EDITOR