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Generalized Cantor Expansions Rose-Hulman Undergraduate Mathematics Journal Volume 5 Issue 1 Article 4 Generalized Cantor Expansions Joseph Galante University of Rochester, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Galante, Joseph (2004) "Generalized Cantor Expansions," Rose-Hulman Undergraduate Mathematics Journal: Vol. 5 : Iss. 1 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol5/iss1/4 Generalized Cantor Expansions Joseph Galante University of Rochester 1 Introduction In this paper, we will examine the various types of representations for the real and natural numbers. The simplest and most familiar is base 10, which is used in everyday life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is unique for every natural number. However, if one views factorials as special type of product, then it becomes natural to ask what happens if one uses other types of products as bases. It can be shown that there are an uncountable number of representations for the natural numbers. Additionally this paper will show that it is possible to extend the concept of mixed radix base systems to the real numbers. A striking conclusion is that when the proper base is used, all rational numbers in that base have terminating expansions. In such base systems it is then possible to tell whether a number is rational or irrational just by looking at its digits. Such a method could prove useful in providing easy irrationality proofs of mathematical constants. 2 Motivation 2.1 Definition The Cantor expansion of the natural number n is n = ak * k!+a(k − 1) *(k −1)!+.... + a2 * 2!+a1 *1! where all the ai (digits) satisfy 0 ≤ ai ≤ i This definition is standard and found in several sources [see 8.2, 8.5]. The identity n−1 n! = 1+ ∑i(i!) (Equation 2.1.1) i=0 is crucial for the Cantor Expansion since it allows “carries” to occur. When adding numbers, Equation 2.2.1 provides a meaning to advance to the next term in the base. 2.2 Example 23 = 3*3!+2*2!+1*1! 24 = 23 + 1 = 3*3!+2*2!+1*1! + 1 = 4! 1 It is reasonably easy to show via induction that all natural numbers have a unique Cantor expansion. Rather than prove uniqueness at this time, we will offer a generalization of this concept, and then show that the regular Cantor expansion is a special case. 3 Generalization Knowing that identity 2.1.1 is the key, it is conceivable that a more general identity will yield a more general result. Realizing that the original expansion relies on factorials and their properties, a generalization should therefore rely on defining a new and more general product. Rather than multiplying together the sequence of numbers 1,2,3,4…, we will consider functions which multiply positive integers in a given list. 3.1 Lemma Let S = {1, x1, x2,…| where xi is a natural number strictly greater than one}. (Note that the index for the first element will be n=0.) If p(n) is the nth element of S (note n p(0)=1) and P(n) = ∏ p(i) , then our generalized identity becomes i=0 n P(n +1) = 1+ ∑( p(i +1) −1)P(i) (Equation 3.1.1) i=0 An inductive proof of Lemma 3.1 is presented in section 7.1. Using the generalized identity we can extend the concept of a Cantor expansion. 3.2 Definition The generalized Cantor expansion (GCE) of the natural number n with respect to the ordered sequence S is n = ak * P(k) + a(k − 1) * P(k −1) + .... + a1 * P(1) + a0 * P(0) (Equation 3.2.1) 0 ≤ ai < p(i + 1), 0 ≤ i ≤ k where S, p, and P are as defined above in Lemma 3.1. The sequence S is referred to as the base set or base sequence. It is important to note that the order of the elements in S matters, and the ak‘s are called the digits of n in its GCE representation. Notation varies, but in this paper we will use the format (ak …a0)S where S is the base set or sometimes just (ak …a0) when the base set is implied. Other notations use matrix format to combine the digits and base sets [see 8.13]. Lastly, as an example, we often measure time itself using a limited form of a mixed radix system, which has a base set corresponding to S={1,60,60,12,...} and has the common format of “hh:mm:ss.” 3.3 Theorem Given a base set S, any natural numbers can be written in generalized Cantor expansion. Proof The proof is by induction. It is easy to see that Equation 3.1.1 holds for n=0 and 1. That is, 0=0*P(0) and 1=1*P(0) since P(0)=1 and p(1)>1. 2 Now assume the first n natural numbers can be written in GCE form. We need to show that (n+1) can be written this way as well. So we know then that: n = ak * P(k) + a(k − 1) * P(k −1) + .... + a1 * P(1) + a0 * P(0) for 0 ≤ ai < p(i + 1), 0 ≤ i ≤ k where ak is the first nonzero digit of n, so that n ≥ P(k) . (If the first term was zero, we could consider a smaller number of terms and re-label subscripts accordingly.) We break the inductive step up into two cases. See 3.5 for concrete examples of the cases. th Case I: There exists a place i strictly less than k, such that the i digit ai is strictly less than p(i+1)-1. This case will cover the addition of one to a number n without any arithmetic carries into the kth place. We want the number n to have some digit before the kth place which when one is added to it, will not produce a carry. Using this idea we see that the digits of n satisfy for some i, n > ai * P(i) + .... + a0 * P(0) = y The number y will always be strictly less than n since n will always have an additional nonzero term which y does not, namely the term ak*P(k). When adding one, the strict inequality, becomes only the inequality n ≥ ai * P(i) + .... + a0 * P(0) +1 By the inductive hypothesis, we see that there exists a valid generalized Cantor expansion for y since n≥ y. Thus we can rewrite y+1 as y +1 = ai * P(i) + .... + a0 * P(0) + 1 = (ai' ) * P(i) + .... + (a0' ) * P(0) Our initial assumption about ai will tell us that it will not grow larger than p(i+1)-1 from a carry and so the rewrite will not require using another place. It now follows that n +1 = ak * P(k) + .... + (ai + 1)P(i + 1) + (ai * P(i) + .... + a0 * P(0) + 1) n + 1 = ak * P(k) + .... + (ai + 1)P(i + 1) + (ai' ) * P(i) + .... + (a0' ) * P(0) which is a valid generalized Cantor expansion. Case II: Digits in all places except the kth place are equal to p(i+1)-1. This case covers a series of carries that terminates at the greatest digit place of the number, or possibly advances to the next place. In this case n = ak * P(k) + ( p(k) −1) * P(k −1) + .... + ( p(1) −1) * P(0) 3 Therefore, n + 1 = ak * P(k) + ( p(k) −1) * P(k −1) + .... + ( p(1) −1) * P(0) + 1 = ak * P(k) + P(k) from Lemma 3.1 = (ak +1) * P(k) + 0* P(k −1) + ... + 0* P(0) which is a valid generalized Cantor expansion if ak+1< p(k+1) Otherwise, if ak+1=p(k+1) then n = ( p(k + 1) −1) * P(k) + ( p(k) −1) * P(k −1) + .... + ( p(1) −1) * P(0) and n + 1 = ( p(k + 1) −1) * P(k) + ( p(k) −1) * P(k −1) + .... + ( p(1) −1) * P(0) + 1 n + 1 = 1* P(k + 1) + 0 * P(k) + .... + 0 * P(0) from Lemma 3.1 which is a valid generalized Cantor expansion. Therefore by the principle of induction, we can conclude that all natural numbers can be ▄ expressed in a generalized Cantor expansion. Now that we know every natural number has a generalized Cantor expansion, the question of uniqueness arises. 3.4 Theorem The generalized Cantor expansion of a natural number is unique. The proof of this theorem uses induction and is complicated slightly by needing several different cases. The proof is found in appendix section 7.3 for the curious reader. 3.5 Examples 3.5.1 – Base 10, S={1,10,10,10…} Example of Case 1 Let n=32378 in regular base 10, and then n+1 = 32378 + 1 = 32379. The addition of the number one does not produce a carry which changes the digit in the leftmost place. Thus we can think of n as 30000+2378, and n+1 as 30000+2378+1. In the proof, we used the inductive hypothesis to argue that 2378 is strictly less than n, and so 2378+1 is less than or equal to n, and so has a valid expansion, in this case 2379. Then 30000+2379=32379 = n+1 is also valid as an expansion.
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