Download A Quick Change of Base Algorithm for Fractions

We now return to our original question concerning an array with the entries
1, 2, . . . , n 2 . The key to showing that an n-by-n matrix A is a linear combination
of R2 , . . . , Cn is to show that for each value of k we have aik − ai1 = a j k − a j 1 for all
i and j. In particular, if we let A be the n-by-n matrix formed by listing the numbers
from 1 to n 2 consecutively, then ai j = (i − 1)n + j. This means that aik − ai1 = k − 1
for every i. Thus we can do the trick described at the start of this note not just for the
first 25 numbers, but for the first n 2 numbers for any n, and the predicted sum will
always be the sum of the diagonal entries, 12 (n 3 + n).
◦
A Quick Change of Base Algorithm for Fractions
Juan B. Gil ([email protected]) and Michael D. Weiner ([email protected]), Penn State
Altoona, Altoona, PA 16601
In this note we discuss the digital (floating-point) representation in various arithmetic bases of a fraction m1 with m ∈ N. With a base b and a denominator m, we
associate a corresponding key, defined as the string of digits consisting of the residues
of b modulo m, and will use it to speed up some computations. For example, if m is
a prime for which m1 has a maximum period expansion of m − 1 digits, and if b is a
primitive root mod m, then the associated key can be used to obtain quickly the digital
representation of mi for i = 2, 3, . . . , m − 1, from the representation of m1 . Recall that
if m is prime, b is a primitive root mod m if b j ≡ 1 (mod m) for 1 ≤ j < m − 1.
On the other hand, for arbitrary integers b and m greater than 1, we will use the key
to give a surprisingly simple algorithm to change the representation of m1 in base b to
its representation in base b + mt for any t ∈ N. Our arguments rely on basic modular
arithmetic and well-known results that can be found in any textbook on elementary
number theory; see, for example, [1] or [2].
Fractions with cyclic periods. Let us start with the simple and commonly used
example of m = 7. The number 17 has a couple of fascinating properties that delight
even those who are familiar with the mysteries of math. In the decimal system 17 =
0.142857. Let 132645 be the key associated with 10 and 7 by means of the residues
100 ≡ 1
(mod 7),
101 ≡ 3
(mod 7),
102 ≡ 2
(mod 7),
103 ≡ 6
(mod 7),
104 ≡ 4
(mod 7),
105 ≡ 5
(mod 7),
and let k(i) be the digit in the period 142857 of
key
1
7
that corresponds to the digit i in the
1 3 2 6 4 5
↓ ↓ ↓ ↓ ↓ ↓ ,
1 4 2 8 5 7
that is, k(1) = 1, k(2) = 2, k(3) = 4, and so on. Then 7i = 0.k(i) · · ·, where the missing 5 digits in the period are placed as to get a rotation of the period of 17 . That is,
2
7
56
= 0.285714,
3
7
= 0.428571,
4
7
= 0.571428,
5
7
= 0.714285,
6
7
= 0.857142.
c THE MATHEMATICAL ASSOCIATION OF AMERICA
6 6
In other words, each number i × 10 7−1 is a rotation of 10 7−1 = 142857. For this
reason, 142857 is called a cyclic number. This property is preserved in some other
arithmetic bases. For instance, in base 3, the key is the same, and the map k from the
key to the period 010212 of 17 is given by
1 3 2 6 4 5
↓ ↓ ↓ ↓ ↓ ↓ .
0 1 0 2 1 2
Thus
1
7
= (0.010212)3 ,
2
7
= (0.021201)3 ,
3
7
= (0.102120)3 ,
4
7
= (0.120102)3 ,
5
7
= (0.201021)3 ,
6
7
= (0.212010)3 .
Therefore, we say that 7 generates the 3-cyclic number (010212)3 . Similarly, in base
17, we get
1
7
= (0.274e9c)17 ,
2
7
= (0.4e9c27)17 ,
3
7
= (0.74e9c2)17 ,
4
7
= (0.9c274e)17 ,
5
7
= (0.c274e9)17 ,
6
7
= (0.e9c274)17 ,
where c = 12 and e = 14. Thus 7 also generates a 17-cyclic number. However, in the
binary system, we have
1
7
= (0.001)2 ,
2
7
= (0.010)2 ,
3
7
= (0.011)2 ,
4
7
= (0.100)2 ,
5
7
= (0.101)2 ,
6
7
= (0.110)2 ,
so 7 does not generate a 2-cyclic number. Note that in this case the key is 124, which
only provides information about the behavior of 17 , 27 , and 47 .
If we are flexible about the base b chosen to represent a number, then for every
odd prime p we can always choose b so that 1p has a maximum period in that base.
In other words, every prime generates a b-cyclic number for some base b. How many
such bases are there? Are there b-cyclic numbers for every b? The answer to these
questions is given by the following fact:
An odd prime p generates a b-cyclic number if and only if b is a primitive root
mod p.
A discussion of this statement as well as other interesting facts about cyclic numbers
can be found in [2] and [3].
Therefore, there are infinitely many bases for which a given prime p gives rise to a
cyclic number. On the other hand, whether a given b (not a square) is a primitive root
for infinitely many primes is still an open problem (known as Artin’s conjecture). In
any case, in the decimal system, the numbers 7, 17, 19, 23, 29, among many others, are
primes that generate cyclic numbers. However, in the widely used hexadecimal system
(b = 16) for example, there are no cyclic numbers at all because a square is never a
primitive root.
Thus the reason why 7 generates cyclic numbers in the bases 3, 10, and 17 is that
these numbers are primitive roots mod 7 (while 2 is not). In general, if p is an odd
prime and b is a primitive root mod p, then the corresponding key consists of p − 1
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57
distinct digits, and the representation of
period as described above.
i
p
can be obtained from
1
p
by rotating the
Change of base algorithm. We continue to examine the fraction 17 . Observe that
its digital representations in the bases 3, 10, and 17, are related as follows. The period
in base 10 can be obtained from that in base 3 by adding the key as follows:
010212 ← base 3
+ 132645 ← key
142857 ← base 10;
and in the same way,
142857 ← base 10
+ 132645 ← key
274e9c ← base 17.
More generally, the period of 17 in base 3 + 7t consists of the 6 digits obtained by
adding t times 132645 to the period 010212. In particular, since 2012 = 3 + 7 × 287,
1
7
= (0.[287][862][574][1724][1149][1437])2012 ,
where [287] denotes the 287th nonzero digit in base 2012, and so on. This is a consequence of the following results.
Lemma. Let b and m be integers greater than 1. Then in base b the fraction
the digital representation m1 = (0.a1 a2 · · · ai · · · )b with
ai =
1
m
has
b i−1
1
(b (mod m)) − (bi (mod m)).
m
m
Proof. For any i, we may write
a1 a2 · · · ai . ai+1 · · · =
Note that 0 ≤
Therefore,
1
(bi
m
1
bi
1
= (bi (mod m)) + (bi − (bi (mod m))).
m
m
m
(mod m)) < 1, so the last term must be the integer part of
a1 a2 · · · ai−1 ai =
bi
.
m
1 i
(b − (bi (mod m))),
m
and so
a1 a2 · · · ai−1 =
1 i−1
(b − (bi−1 (mod m))).
m
a1 a2 · · · ai−1 0 =
b i−1
(b − (bi−1 (mod m))).
m
This implies that
The formula for ai follows by taking the appropriate difference.
58
c THE MATHEMATICAL ASSOCIATION OF AMERICA
Theorem. Let b and m be integers greater than 1. If m1 = (0. a1 a2 · · · ai · · · )b , then
for any t ∈ N, in base (b + mt), the fraction m1 has the digital representation
1
m
= (0. a1 a2 · · · ai · · · )b+mt ,
where ai = ai + tki with ki = (bi−1 (mod m)).
Proof. By the lemma,
b i−1
1
(b (mod m)) − (bi (mod m)), and
m
m
b
+
mt
1
((b + mt)i−1 (mod m)) − ((b + mt)i (mod m)).
ai =
m
m
ai =
On the other hand, (b + mt)i−1 ≡ bi−1 (mod m) and (b + mt)i ≡ bi (mod m). Thus
we get ai − ai = t (bi−1 (mod m)) = tki , as claimed.
Earlier we discussed 17 in bases 3, 10, and 17. As a second example, consider the
fraction 14 . According to the theorem, if we find the representation of 14 in the bases
2, 3, 4, and 5, together with the corresponding keys, then we can easily get the digital
representation of 14 in any base. Recall that the key k1 · · · k associated with m1 in base
b is defined by ki = (bi−1 (mod m)), where is either the length of the fundamental
period of m1 or the length of its nontrivial fractional part. Thus
1
4
= (0.01)2 → 12,
1
4
= (0.02)3 → 13,
1
4
= (0.1)4 → 1,
1
4
= (0.1)5 → 1.
Hence, 14 = (0.13)6 since 01 + 12 = 13, and
larly, using 1 as key, one gets for instance
1
4
= (0.2)8 ,
1
4
= (0.4)16 ,
In particular, in base 2009, we have
1
4
1
4
= 0.25 because 13 + 12 = 25. Simiand
1
4
= (0.3)13 .
= (0.[502])2009 .
References
1. J. Conway and R. Guy, The Book of Numbers, Copernicus, 1996.
2. L. E. Dickson, History of the Theory of Numbers, Vol. I: Divisibility and primality, Chelsea, 1966.
3. S. Guttman, On cyclic numbers, Amer. Math. Monthly 41 (1934) 159–166.
◦
A Waiting-Time Surprise
Richard Parris ([email protected]), Phillips Exeter Academy, Exeter, NH 03833
Let x1 , x2 , x3 , . . . be a sequence of numbers chosen randomly (and uniformly) from
the unit interval 0 < x < 1. For each real number t ≥ 0, the first n for which
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59