Corrected Version, November 1997. Originally submitted per the requirements of Colorado College course General Studies 213: The History of Mathematical Thought, Profs. M. Anderson (Mathematics) and O. Cramer (Classics), et al.
"arithmetic, n. 1. The mathematics of integers under addition,
subtraction, multiplication, division, involution and evolution.
2. Computation or problem solving involving real numbers and the
arithmetic operations....
--The American Heritage Dictionary, Second College Edition,
1982
We've all done arithmetic. In the supermarket when providing for our families. On the highway when comparing our speed with the posted limits. (We may not necessarilty care if we are exceeding the limit, of course, but it's always interesting to know by how much.) And in the restaurant when determining how much of a tip to leave on the table, or whether or not we'll have to do the dishes to pay for the meal. Arithmetic, verily mathematics as a whole, is always around us, from the most mundane tasks to the most embarrasing and profound situations.
With some minor exceptions, we perform our arithmetic operations in base 10. But is base 10 really the best way to do arithmetic? Are calcualtions easier to perform in some other base, say 12 or 16? Let us take a brief look at someof the inherent advantages and disadvantages of the use of alternate number bases in arithmetic, starting with our tried and true friend, base 10.
The primary advantage that base 10 gives us is that we're accustomed to it. The most popular explanation is that we have 10 fingers on our hands. We are able to match our fingers to up to ten of some other object. Counting the number of times we can do this before we run out of whatever we were counting allows us to use numbers greater than ten. Some cultures, for similar reasons, have used number systems of 5 and 20. (Base 20 by calling the toes into play.) [1]
There are some disadvantages that are immediately apparent in bases 5 and 20. Base 5, for instance, is a fairly small base, requiring more units of measurement. (Which is easier to work with, one hand and three fingers, or eight fingers?) Base 20, on the other hand (forgive the pun) disadvantages itself in the fact that not many of us can bend our toes independently of the others. Peoples in colder climes may have to remove shoes of moccasins to do any such counting. For whatever the reason, base 10 seems to have predominated.
This was not always the case, and in any group can be found some oddballs. The Babylonians used a base 60 system of enumeration, although for numbers less than 60 a base 10 notation was used. [2] Some Northern European societies had a quantity known as a "great hundred" made up of 10 dozen or 120, reflecting the rudiments of a base 12 counting system. [3] The Romans, even though they used base 10 for their integer counting, had a system of duodecimal, or base 12, fractions. It is thought they chose this because of easy divisibility in so many different ways. [4]
Again, base 10 has lasted through the years, and is still going strong. However, there is something that is gaining attention for non-decimal enumerating: the computer.
Computers use, at their lowest levels of operation, the concept that an 'on' condition can be assigned a different numerical value than a corresponding 'off' condition. This leads us to a binary, or base 2, system of numeration. This is all well and good for computers, but for humans it can be a bit cumbersome. For example:
842(10) = 1101001010(2)
As you can see, relatively small numbers in base 10 produce some real monster-sized binary numbers. We clever humans, therefore, have developed some convenient shortcuts. These shortcuts are created by converting binary numbers into octal (base 8) or hexadecimal (base 16) numbers. This is actually quite easy. Taking our example from above:
1101001010(2)
we divide the number up three places at a time from the right, thus
1 101 001 010
and convert each group of three into single octal digits:
1 | 101 | 001 | 010 |
---|---|---|---|
1 | 5 | 1 | 2 |
just by finding the decimal values that correspond to each place. Therefore,
842(10) = 1512(8)
Converting binary into hexadecimal is similar; start by dividing the number into four digit groups:
11 0100 1010
and insert the appropriate values:
11 | 0100 | 1010 |
---|---|---|
3 | 4 | 10 |
This leads to a bit of a problem, in that '10' in and of itself is not a single digit. In the computer industry, therefore, it has become customary to represent the values 10 through 15 by the letters A through F, A equalling 10, B equalling 11, etc. Our conversion from above then becomes:
11 | 0100 | 1010 |
---|---|---|
3 | 4 | A |
meaning that
842(10) = 34A(16)
Now that we have seen some of the rationale for the use of some of these number bases, let us try to compare them mathematically. To recapitulate, the bases we are comparing are:
One fine way of comapring number bases is to compare some of their divisibility indicators. (A divisibility indicator would be the fact that, in base 10, all numbers divisible by 5 end in 0 or 5.) Other ways include tests to help identify prime numbers and perfect squares. [5] Let us take a quick look at the divisibility rules first. Only the really "easy" rules will be listed for each base, and we will concentrate only on divisibility by numbers less than the base number itself.
Base 2: A number is even if it ends in 0, odd if it ends in 1
Base 5: 2: Any number whose digits add to a multiple of 2 4: Any number whose digits add to a multiple of 4
Base 8: 2: Any number ending in an even digit 4: Any number ending in 0 or 4 7: Any number whose digits add to a multiple of 7
Base 10: 2: Any number ending in an even digit 3: Any number whose digits add to a multiple of 3 5: Any number ending in 0 or 5 6: Any even number whose digits add to a multiple of 3 9: Any number whose digits add to a multiple of 9
Base 12: 2: Any number ending in an even units place 3: Any number ending in 0, 3, 6, 9 4: Any number ending in 0, 4, 8 6: Any number ending in 0, 6 11: Any number whose digits add to a multiple of 11
Base 16: 2: Any number ending in an even units place 3: Any number whose places add to a multiple of 3 4: Any number ending in 0, 4, 8, 12 5: Any number whose places add to a multiple of 5 6: Any even number whose places add to a multiple of 6 8: Any number ending in 0 or 8 10: Any even number whose places add to a multiple of 5 15: Any number whose digits add to a multiple of 15
Base 20: 2: Any number ending in an even units place 4: Any number ending in 0, 4, 8, 12, 16 5: Any number ending in 0, 5, 10, 15 10: Any number ending in 0, 10 19: Any number whose digits add to a multiple of 19
Base 60: 2: Any number ending in an even units place 3: Any number whose units place is a multiple of 3 4: Any number whose units place is a multiple of 4 5: Any number whose units place is a multiple of 5 6: Any number whose units place is a multiple of 6 10: Any number ending in 0, 10, 20, 30, 40, 50 12: Any number ending in 0, 12, 24, 36, 48 15: Any number ending in 0, 15, 30, 45 20: Any number ending in 0, 20, 40 30: Any number ending in 0, 30 59: Any number whose digits add to a multiple of 59
In some bases, identifying prime numbers and perfect squares is trivial, in others more difficult. The best way is to see how many different digits each of these numbers end in, compared to the size of the base. A table showing which digits apply to each type of number follow.
Base | Prime Number End Digits | % | Perfect Square End Digits | % |
---|---|---|---|---|
2 | 1 | 100 | 0, 1 | 100 |
5 | 0, 1, 2, 3, 4 | 200 | 0, 1, 4 | 60 |
8 | 1, 3, 5, 7 | 100 | 0, 1, 4 | 38 |
10 | 1, 3, 7, 9 | 80 | 0, 1, 4, 5, 6, 9 | 60 |
12 | 1, 5, 7, 11 | 66 | 0, 1, 4, 9 | 33 |
16 | 1, 3, 7, 9, 11, 13, 15 | 80 | 0, 1, 4, 9, | 25 |
20 | 1, 3, 7, 9, 11, 13, 17, 19 | 80 | 0, 1, 4, 5, 9, 16 | 30 |
60 | 1, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59 | 57 | 0, 1, 4, 9, 16, 21, 24, 25, 36, 40, 45, 49 | 20 |
Notes on the table: the columns marked % refer to the percentage of digits which appear against the given base. The percentages given after the prime digit column refer to the number of odd digits which appear. Base 5 reads 200% in this column as numbers ending in even digits can also be prime.
Another index that can be used is called the regularity index. This index gives the percentage of regular numbers from 2 to one less than the base that are in a given base. (A regular number is one whose reciprocal, when expressed as a decimal fraction, terminates in the given base.) A table of regular numbers appears below, along with the regularity index for each base.
Base | Regular Numbers | Regularity Index (%) |
---|---|---|
2 | [none] | 0 |
5 | [none] | 0 |
8 | 2, 4 | 33 |
10 | 2, 4, 5, 8 | 50 |
12 | 2, 3, 4, 6, 8, 9 | 60 |
16 | 2, 4, 8 | 21 |
20 | 2, 4, 5, 8, 10, 15 | 33 |
60 | 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 54 | 41 |
Note on the table: The regular numbers listed reflect only those between 2 and base-1.
Now that we have all of this data, how do we compare the relative usefulness of the various number bases? We need to create some sort of index that will quantify this usefulness. A positive consideration needs to be given to the regularity index, and a smaller positive consideration to the fact that a larger base yields a more compact notation. Negative consideration should be given for the number of different digits that ca end primes and squares, as well as the negative influence brought on by the size of the multiplication table. A possible formula could be:
I = (R * ln(b)) / (P * S * b)
where:
b | is the base in question |
---|---|
R | is the regularity index |
P | is the percentage of odd digits found at the ends of prime numbers |
S | is the percentage of all digits found at the ends of perfect squares |
This yields the following table:
Base | Index |
---|---|
2 | .000 |
5 | .000 |
8 | .231 |
10 | .240 |
12 | .559 |
16 | .146 |
20 | .206 |
60 | .245 |
This table indicates that base 12, by far, is a much more logical base to do arithmetic in. Bases 8, 10 and 60 seem to have fared about equally well (even with base 60's enormous multiplication table). It would seem that base 12 is so much higher because it combines good divisibility patterns (noted by the regularity index) with a fairly small set of operation tables.
Contrariwise, note bases 2 and 5 bringing up the rear. For base 5, the reason is that there are no terminating decimal fractions and the fact that, as an odd-numbered base, we have more difficulty finding odd and even numbers. This is reflected in the fact that a prime number may end in any digit. For example: 31(5) equals 3 * 5 + 1 or 16(10), an even number, has an odd last digit. By contrast, 21(5) = 11(10), an odd number. Base 2 fails mainly because it is so cumbersome to work with, and that the nature of a binary number is not intuitively obvious (given by high values of P and S). The regularity index of base 2, zero, may not be entirely accurate, however. There are simply no integers between 2 and 1. We might set the regularity index to 50%; this would give a final index value of .173, still quite low, but more appropriate.
Should we convert to base 12? Probably, but given the inertia inherent in the present number system, conversion would be far more trouble than it's worth.
The paper was returned with the following marks from Prof. Cramer:
A very nice presentation
A
For myself, looking at this ten years later, I wonder just what Prof. Cramer and I were smoking. It isn't that I don't believe in my own thesis. It's just that the urge to rewrite the paper completely was at times overpowering. I did notice some errors in the original and these have been corrected. Other than that, I've left it alone. I wrote it. I get to live with it. CJO