## Assuming place values ... d3×b3+ d2×b2+ d1×b1+ d0×b0. notation and similar for the other side of the "decimal point".

Last Up date: 2010 November 21
Recent changes: Added a column for Base 16
Page started: 2007 May 8

The above is implied when we write numbers useing digits 0 - 9 for the dn when the base: b is ten.

Arithmetically, any positive integer b ≥ than two can be used. But, some are superior in different environments. Binary gets more bang for a given number of transistors, or gates, in a computer than any other base. Octal and Hexadecimal are just convenient external representations of binary.

### Since the days of Roman numerals, humans have used base ten. Most likely because we have ten fingers. But is it "best" for humans?

As we shall see: Except for tradition, ten is a poor choice. But tradition is strong. Engineers designing the early computers: Eniac, Univac, Boroughs 205, and IBM 650, 705, 7070, 1401, 1620, RCA 301, 3301 & others all did decimal arithmetic even though it required much more involved circuitry. Other computers using only binary arithmetic such as: CDC machines, DEC machines, SDS machines, IBM 704 and its' descendents through the 7094, offered more "bang for the buck".

Today, the Intel 8086 and its' descendents along with the IBM 360 and its' decedents, all use binary addressing and arithmetic but have a few special instructions to aid in decimal arithmetic. I know of no currently manufactured computer that uses other than binary addressing. Obviously, the pure decimal machines, could not compete and must be considered a failure. I suspect that even the hand held calculators which input and display decimal are doing mostly binary arithmetic inside.

### For human use

It is nice if the base is small enough we can memorize the addition and multiplication tables. This probably limits us to bases 16 or less. Also as many computer people know if the base is too small, like binary or base 4, the numbers we deal with simply have too many digits. Most computer people today prefer Hexadecimal to Octal for this reason. Thus, I will only consider bases 8 through 16.

As most of us know, it is very nice to be able to just look at the last digit of a number and tell if it is even, a multiple of 2 or not. This would not be true for any number base which is odd. We find ourselves frequently looking at the last digit to tell if the number is a multiple of five. But five is not nearly as useful of number as three or four. Its' virtue comes solely from the fact that five is a factor of ten, our number base.

Small integers are used most. Frequence of use decreases as the value increases. Zero and One are most used: Zero answers questions like "Any?". One answers questions like "Do you have? Two is next most useful, and half is the most common divison. Three should and would be next if it were compatible with our number system.

In the following table I have listed the Bases from 8 through 15 along the top row. Listed in the first column are potential divisors. If the last digit can determine it is a divisor, the entry in the table is a "Y" if not it is an "N". If examining the last two digits will determine it is a divisor the table entry is "L2".
 Base → 8 9 10 11 12 13 14 15 16 Y value↓ 2 Y N Y N Y N Y N Y 50 3 N Y N N Y N N Y N 33 4 Y N L2 N Y N L2 N Y 25 5 N N Y N N N N Y N 20 6 N N N N Y N N N N 17 7 N N N N N N Y N N 14 8 Y N L3 N L2 N L3 N Y 12½ 9 N Y N N L2 N N L2 N 11 10 N N Y N N N N N N 10

### Analyzing the table.

The "Y" values are simply 100 divided by the value in the left column. This gives higher weight to the smaller digits. Adding up the "Y" values for any base gives a "figure of merit" for that base.

• Looking only at the "2" row: In any even base one can tell by inspecting the last digit if the number is even or not. By this measure: 8, 10, 12, and 14; all are equally good.
• Looking at the "2" and "3" rows: Inspecting the last digit only in base 12 is it obvious if the number is a multiple of either 2 or 3, or both.
• It is obvious from the top down through two or more rows base 12 has more "Y"s than any other base. Further, base 10 is only slightly better than base 14! (Only because 5 is a smaller number, and hence more used, than 7.)

While I have only spoken of "telling if a number is a multiple" it is also true that where "N"s occur the fractions: 1/v is a repeating value that cannot be expressed exactly as a decimal value. 1/3, 2/3, and 1/6 are examples of simple, useful, fractions that are infinite repeating decimals in base ten. But, not in base 12. Clearly, it is unfortunate that we use base ten, when 12 is so much superior. But, human "mental inertia" will probably get in the way of progress.

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