FIGURE FOIBLES

New Zealand Listener, Volume 2, Issue 47, 17 May 1940, Page 11

FIGURE FOIBLES

Peculiar Habits of Some Numbers

(Written For "The Listener" By

R. W.

C.

N the Puzzle Page of The Listener some time ago, a correspondent, J. A. Reid, of Glenorchy, calls attention to the peculiarity of the number 142857, and says "I knew that this was the only sixfigure number that would repeat its figures in the same manner if multiplied by 2, 3, 4, 5 or 6." I hope that he, as well as, perhaps, other readers, will be interested in the following comments. Recurring decimals, or groups of decimals, are shown within parentheses. First, here is the reason for this number’s seemingly fantastic habits. 142857 are the six figures which recur in the decimal form of the fraction 1/7. Most school-children are familiar with the fact that all the "sevenths" fractions when converted consist of recurring decimals using these same six figures in the same cyclic order but beginning in turn with 1, 2, 4, 5, 7 and 8. Now the decimal form of 1/7 is found by dividing 1.0 by 7, and in this simple division each step leaves us with a certain remainder which must obviously be less than the divisor 7. For instance, 7 into 10 gives 1 with a remainder of 3; 7 into 30 gives 4 with a remainder 2, and so on. There are, counting the original 1, six possible remainders before any need be _ repeated, namely the figures 1 to 6 inclusive. This means that we get a six-figure decimal before the recurrence. The multiples of the fraction -2/7, 3/7, etc. -are of course found by multiplying this decimal by 2, 3, etc. (If we multiply the isolated six-figure group by 7 to find 7/7, or unity, we certainly do get .(999999), but as this is recurring it is equal to 1.) . Fractions and Decimals The correspondent mentions dividing a row of 9’s by prime numbers, and gives certain results. Actually this is again a question of the decimal forms of certain fractions. Most of us remember from our school days that simple fractions with denominators 5, powers of 5, 2, powers of 2, multiples of 2 and 5, or of powers of 2 and powers of 5, "come out evenly" in decimal form. For example 1/25=~.04, 1/16=-.0625 and so on. We also know that denominators of 3 and its powers and multiples, give some sort of recurring decimal. For example 1/6=-.1(6), 1/9=.(1) and so on. Decimal forms of all other fractions we usually consider to be like the brook and to go on for ever. This is not so. Sweet 17 Take the number 17, which Mr. Reid dismissed as "only. middling." If we find the decimal form of 1/17 by dividing 1.0 by 17, we have sixteen possible remainders in this division before repetition must occur, namely the figures 1 to 16 inclusive. This gives us the sixteen-figure recurring decimal .(0588235294117647). Everyone of the other "seventeenths" decimals from

2/17 to 16/17 is a sixteen-figure group consisting of these figures in the same cyclic order, beginning with the eleventh, the twelfth, the fifth, the eighth, the sixth, the tenth, the fifteenth, the seventh ... etc. figures in turn. (There is no apparent rhyme or reason for the order of these commencing figures. ) Anyone sufficiently interested to do the calculations will find the same thing with 19, 23, 29 and so on; in fact with all prime numbers greater than 13. In each case there will be a recurring decimal consisting of one fewer figures than the number in the denominator, and every fraction with each particular denominator will give a decimal group consisting of the same figures, in the same cyclic order, but commencing with @ different figure each time. Consider 11 and 13. To return to the cases of 11 and 13, which" the correspondent said are "no good." They certainly do not give the above type of permutations of the same groups of figures, but watch! 1/11==.(09), 2/11==.(18), 3/11=-= -(27), 4/11=.(36), etc. Thirteenths are even more peculiar. 1/13--.(076923), 2/13== .(153846). No connection at all! But we find that 3/13, 4/13, 9/13, 10/13 and 12/13 consist of the first group of figures, while 5/13, 6/13, 7/13, 8/13 and 11/13 consist of the second. Instead of a twelve-figure group (13-1) we have two six-figure groups. The Figure 3 The figure 3 and its powers give some strange results. 1/13==.(3), 1/9 (or 1/3?) ==" (1), 1/27 (or 1/3%)=.(037), 1/81 (or 1/3*) ==.(012345679) (notice the unaccountable omission of 8). 1/243 (or 1/35) is even more peculiar — .(004115226337448559670 781893). Upon examination this formidable decimal is found to consist of nine sets of triplets, 004, 115, 226, 337, etc. (a complicated sort of Arithmetic Progession), and is consistent right to the end as 670 is the same as 66(10), 781 as 77(11) and 893 as 88(12) with the extra 1 as a kind of carryover to the next recurring group. : Magic in 9 . The figure 9 has always been regarded as a "magic" number, and if space permitted many amusing and surprising calculations could be given involving this figure. As a matter of fact, the whole question of strange and apparently inexplicable properties of certain numbers, and of the relations between numbers of certain forms, is intensely interesting, and has excited the attention. of mathematicians for centuries-probably since some prehistoric man noticed that he had the same number of toes as he had fingers, and used them to count his herd of tame brontosauri. ~