Can't We Get Rid of Recurring Fractions ?

When I was writing about Dozenal Number System (or Duodecimal Number System), this thought struck me.

In Decimal Number System, just looking at the last digit (least significant digit or unit’s digit) we can say whether the whole number is divisible by 2, 5 and 10. Looking at the last two digits we can say its divisibility by 2, 4, 5, 10, 20, 25, 50 and 100. Looking at the last three digits the divisibility list grows little more and so on. It also means that if there is a fraction of the form k/2, k/5 or k/10 (where ‘k’ is an integer and the fractions are in their minimal form), in real number format, there will be one digit after the fractional point. Similarly, fractions of the form k/4, k/20, k/25, k/50 and k/100 will have two fractional digits. We know that some fractions like 1/3 and 1/7 will never end becoming recurring fractions. It means, to know whether the divisibility by 3 or 7, we need to see the whole number, not just a constant number of digits. What makes some denominators to end after constant number of digits (after the fractional point) and some repeat forever?

To understand the behavior of recurring fractions, let’s dissect the base number. Prime factorization of the base number 10 is 10=2*5. This should explain the divisibility list for the last digit {2, 5, 2*5}. When two digits are considered, the prime factorization comes down to 10*10=2*2*5*5, which gives us the divisibility list for the last two digits {2, 2*2, 5, 2*5, 2*2*5, 5*5, 2*5*5, 2*2*5*5}. Consider any number of digits; they are made up of only 2s and 5s. That’s why if the denominator of a fraction (in minimal form) is made up of only 2s and 5s (and no other prime numbers), only then it ends in constant number of fractional digits. For example, 1/2000 where 2000=2^4 * 5^3 (four 2s and three 5s) would surely end with four fractional digits. And, for example, 1/6 where 6 has a 2 and a 3 will end up as recurring fraction because it has a prime factor other than 2s and 5s.

Recurring fractions are not just limited to Decimal NS, they occur in any n-base Number System (where ‘n’ is any finite integer >= 2). If the denominator of a fraction (in minimal form) has a prime number which is not in the base number, then the fraction ends up as a recurring fraction. For example, in Dozenal NS, the base 12 is made up of 12=2*2*3. A fraction in Dozenal NS, 1/5 for example, would be a recurring fraction. We can see that in any n-base NS, reciprocal of a prime number which is not in the base number, would be a recurring fraction. So, isn’t there any way of having a number system without the fuss of recurring fractions?

I think there is a way!

Regards,
Channa Bankapur

Dozenal Number System

During my IISc days (2003-05), I had this raw thought. I began to think that Dozenal (base twelve or duodecimal) number system should have been the de facto number system in the world instead of decimal (base 10). Couple of days back, when I set out to pen down my thoughts, I got this page on Wikipedia - Dozenal/Duodecimal Number System http://en.wikipedia.org/wiki/Duodecimal. Déjà Vu! My raw thoughts explained! Not just that, there is a whole bunch of people who feel the same. To my surprise, a book has been written on this topic long back in 1935 by Emerson Andrews; New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. And, there are groups formed in the interest of Dozenal system; Dozenal Society of America (DSA) and Dozenal Society of Great Britain (DSGB).

The wiki page I mentioned above describes what I wanted to describe. However, I will write whatever I had set out to, in as far as layman’s words. I have picked majority of the terminology from the wiki page including the word Dozenal in place of Duodecimal.

What would be one reason which made humans to think of using Decimal number system? Even Roman number system basically follows Base Ten. The main reason being, people could count using fingers of both hands up to ten. That made them to invent ten symbols to start with, and then for multiples of ten and powers of ten. For all of us, Decimal system is deeply rooted in our mind. So, any new system won’t look better than that. If you are being introduced to a number system for the first time and the whole world follows the same, then which one would be great.

First, let’s see what’s wrong with Decimal system. Of the five most elementary fractions (1/2, 1/3, 2/3, 1/4, 3/4), two of them end with recurring decimals, the other two end at second decimal point, and only 1/2 ends at first decimal point. When you had to divide something for three people, haven’t you felt weird to say each one gets 33.33 percent? And, adding up 33.33 of three people will get 99.99 instead of 100. When you are given a large number, to find whether it can be divisible by small numbers, it is easy to see whether it’s divisible by 2, 5 and 10 just by looking at the last digit. But, finding divisibility by as small number as 3 needs more effort. This matters in known situations like finding modulus. Modulus is simply the reminder function. For example, if I am asked to share 127 bucks among 5 people, I immediately yell out – how do I share the last 2 bucks. Because just by looking at the last digit, we can say modulus 5 of 127 is 2. If I am asked to share among 3 people, I had to put more effort to see whether it can evenly divide for 3. We see that knowing a number whether it’s even or odd almost instantly comes very handy. In a similar way, if we could find whether dividing the number by 3 yields a reminder 0, 1 or 2, it would have been widely used by common people.

So, there are benefits of having more divisors for a base number. Ten has only 4 divisors; 1, 2, 5 and 10. We can find bigger numbers, which has lot more divisors, say 60 which has 12 divisors. But choosing bigger base number means more symbols. Using 60 symbols for a digit is not a great idea. On the other hand using very small base like 2 or 4 would mean the whole numbers will have lot more digits even for not so big numbers. For example, a three digit number in base 10 such as 800 or 900 would require 9 digits in base 2. So, we need to find a base number around 10 and has good number of divisors. I keep calling this property of having number of divisors as Composite Potency. The higher the potency, the better it is! There is a known property by the name Highly Composite Number (HCN). A positive integer is an HCN if it has more divisors than any smaller positive integers. So, HCNs are good candidates for choosing a good base number. Here are first few HCNs (in parenthesis, number of divisors).
1 (1), 2 (2), 4 (3), 6 (4), 12 (6), 24 (8), 36 (9), 48 (10), 60 (12), 120 (16), 180 (18), 240 (20), 360 (24).
Ten is not an HCN! Ten has just 4 divisors, but a smaller number six has the same potency. Why is number 10 left behind? Look at the divisors of an HCN. They get good number of lower prime numbers. 10 has got 2 and 5. 6 has got 2 and 3. Both of these get two prime numbers, but number 6 has grabbed the lower one (number 3) and hence emerges as a winner. That’s why 6 is better than 10.

Highly Composite Number (HCN) concept was presented by Srinivasa Ramanujan in 1915. He also mentioned of Superior Highly Composite Numbers, which is a subset of HCN. First few Superior HCNs are 2, 6, 12, 60, 120, 360, and 2520. Now, these are superior candidates for a great base number! Because 2 and 6 are too small and 60 (and above) are too big for a base, it boils down to 12.

We see many of these numbers (HCNs) are used in non-metric measurement systems mostly due to their ease of use in calculations. For example, there are 12 inches in a foot, 36 (=12*3) inches in a yard, 12 ounces in a pound, 24 (=12*2) hours in a day, 60 minutes in an hour, 60 seconds in a minute, 12 months in a year, 12 old British pence in a shilling, 12 items in a dozen, 360 degrees in a cycle, etc. Metric system has been introduced to align these measurements with Decimal number system.

With Dozenal number system, the all five most elementary fractions (1/2, 1/3, 2/3 , 1/4 , 3/4) ends at first decimal point, which simplifies a great deal of calculations. Look at the factors of 12; 1, 2, 3, 4, 6, 12. It covers five of first six natural numbers. However, handling with number 5 becomes tougher when compared base 10. If we try to include 5 also in the base, the base number would jump to 60 (=12*5), which is too big a base number. It’s interesting to note that base 60 (Sexagesimal) number system has been used in some ancient cultures like Babylonian.

To adopt Dozenal number syste, we need twelve symbols to represent a digit. We could use 0 to 9 similar to decimal and two different for decimal equivalents of ten and eleven. 10 of Dozenal would be 12 of Decimal. We could simply use A and B for decimal ten and eleven for now. However, there have been varied suggestions for the two new symbols. Emerson Andrews suggested symbols which mostly looked like “x” and “e”, which were inspired by “X” of Roman 10 and “E” of word “Eleven”. Dozenal Society of America and Dozenal Society of Great Britain promote the widespread use of Dozenal number system. In Dozenal system, “Dozen” is the word for 10 of Dozenal (12 in Decimal), “Gross” for 100 (144=12*12 in Decimal), and “Great Gross” for 1000 (1728=12*12*12 in Decimal).

I personally feel that if we could replace Decimal by Dozenal Number System, the coming generations would grasp Number System fast and will hate Mathematics little less. And, even for higher end Mathematics, things would become easier and faster. Obviously, overnight replacement is impossible and doing it in couple of years is also foolishness. It should be over generations and phase out Decimal system gracefully. Next generation should be introduced to both systems and they will treat them as two languages of Mathematics in which Dozenal is for the newer world and Decimal is legacy. Wherever needed, conversions across systems should be simple. Does it makes sense to spawn “Dozenal Society of India”? What do you say?

Regards,
Channa Bankapur