# B Most Efficient Method for Counting

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1. Mar 13, 2016

### ProfuselyQuarky

I'm talking about different bases here. I've heard a bunch of people say that base 12 is the best way to go, since it would make basic math easier. After all, 10 from the decimal system has only 4 factors (1,2,5,10), whereas 12 from the duodecimal system has 6 (1,2,3,4,6,12). But, looking from a math-based perspective, would base 12 really improve anything? When I think of theoretical math and all of the higher divisions of math that people study, numbers are hardly even used, so base 12 wouldn't make much of an impact. Or am I just mistaken?

Last edited: Mar 13, 2016
2. Mar 13, 2016

### Staff: Mentor

It would be relevant for everday life - it is much easier to divide numbers. It can also be useful in science, if numbers pop up. In mathematics it does not matter at all.

3. Mar 13, 2016

### Staff: Mentor

Base-12 and related counting systems were in use thousands of years ago by the Babylonians, and are still present in how we measure angles and time (60 seconds in a minute, and 60 minutes in a degree or hour). Because 60 = 5 * 12 it is evenly divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The main advantage in having a base that has many factors is that it makes fractions easier to work with. Other than for angle time measure, base 12 isn't used very much, but it hasn't gone away entirely. The English words "dozen" and "gross" (meaning 144 of some item) are remnants from an earlier time.

Base 10 is the predomimant counting base, due to the fact that we humans generally have 10 fingers and 10 toes. However, in computer science, binary (base 2) and hexadecimal (base 16) are arguably as important as the decimal system. Each transister in a memory cell can have one of two possible values. By grouping eight, sixteen, thirty-two, etc. of these transistors we can represent a large range of values.

4. Mar 13, 2016

### ProfuselyQuarky

Yes, duodecimal definitely makes dividing a whole lot easier, no question about that. But dividing in base 10 really isn't that bad. Some individuals are just so adamant that base 12 is the most perfect way to count. Aside from simpler mental math, I can't see anything better about it.
People could, possibly, count using the lines on their fingers (not including thumb). Four fingers, with two lines from the joints on each, makes 12 "sections" to count.
I'm familiar with binary (I'm trying to learn how to write in Binary to ascii text), but I don't know much about hexadecimal.

5. Mar 13, 2016

### Staff: Mentor

Each group of four binary digits (bits) makes one hexadecimal number. The hex digits are 0, 1, ..., 9, A, B, C, D, E, and F.
$\begin{matrix} \text{Decimal} & \text{Binary} & \text{Hex} \\ 0 & 0000 & 0 \\ 1 & 0001 & 1 \\ 2 & 0010 & 2 \\ 3 & 0011 & 3 \\ 4 & 0100 & 4 \\ 5 & 0101 & 5 \\ 6 & 0110 & 6 \\ 7 & 0111 & 7 \\ 8 & 1000 & 8 \\ 9 & 1001 & 9 \\ 10 & 1010 & \text{A} \\ 11 & 1011 & \text{B} \\ 12 & 1100 & \text{C} \\ 13 & 1101 & \text{D} \\ 14 & 1110 & \text{E} \\ 15 & 1111 & \text{F} \\ 16 & 1~0000 & 10 \\ 32 & 10~0000 & 20 \\ 33 & 10~0001 & 21 \end{matrix}$

Hex numbers often include some mark to signify that they are hex, such as 0x2F in C and related languages, or 2Bh in some assembly languages.

6. Mar 13, 2016

### Staff: Mentor

And you can use the thumb to point to the joint. Easy 2 digit-counting with the fingers. But you could also include fingertips for base 16. One byte (8 bit) if you combine both hands.

7. Mar 13, 2016

### ProfuselyQuarky

Thanks a lot, I’ll try to learn this, too . . . not sure what need I have for it, but I always do, anyway It’s curious how letters find their way into base 16.
That’s true, I guess I’ll use my fingers when learning hexadecimal. Thanks guys!

Last edited: Mar 13, 2016
8. Mar 13, 2016

### Staff: Mentor

Not really, if you think about it. In base n, where n is a positive integer greater than 1, there have to be n digits. In base 10, the digits are 0, 1, 2, ... , 8, and 9. In base 2, the digits are 0 and 1. In base 16 we have the digits 0 through 9, but we need 6 more, so whoever came up with this numbering system made the decision to use the letters A through F for the remaining six digits.

9. Mar 13, 2016

### ProfuselyQuarky

Ah, okay, that makes sense.

10. Mar 13, 2016

### Staff: Mentor

There is also base36 using all 10 digits and 26 letters to represent numbers in a shorter format that is still easy to read and compatible with all systems that process text. It needs about two digits where base 10 needs 3.

11. Mar 14, 2016

### pwsnafu

Base 36 is pretty bad because O (or o) and 0, and 1 and I look similar. Better would be Base 32 or Base 58.
Of course, at this level memorizing multiplication tables become tedious.

12. Mar 14, 2016

### Svein

A long time ago (when hex was starting to replace octal) I heard a rant from a language professor: "Hexadecimal! Hex is greek, decimal is latin - what a linguistic mongrel! Either stick to greek, making it "hexadekadik" or latin, making it "sedecimal"!"

13. Mar 14, 2016

### ProfuselyQuarky

I looked into base36 after reading Mark44's post. It's really neat. What characters are used in even higher bases, once digits and letters run out?
Ha! That language professor should know that the words in the English language are often a big salad mixed with bits (see what I did there?) and pieces from multiple languages. I'd second the use of hexadekadik over hexadecimal, however.

14. Mar 14, 2016

### jbriggs444

It is irrelevant to the mathematics. Essentially nobody writes down numbers in such bases. The digits can be thought of as abstract entities. One obvious notation would be:

(10)(31)(0)(55).(71)(1)

You can see some actual notational conventions here: https://en.wikipedia.org/wiki/Sexagesimal

15. Mar 14, 2016

### ProfuselyQuarky

Thanks, I guess that higher numbers in higher bases become a bit unnecessary in practical math.

16. Mar 14, 2016

### micromass

Clearly the best base is $e$.

17. Mar 14, 2016

### ProfuselyQuarky

18. Mar 14, 2016

### micromass

19. Mar 14, 2016

### ProfuselyQuarky

So base e is sort of like the most efficient base (lowest radix economy)? Not too short like binary, but not too long like the decimal or duodecimal? I'm not sure if I completely understand how irrational bases can even exist and how radix economy is calculated . . .

20. Mar 14, 2016

### Staff: Mentor

It has some advantages, but 1+1+1 = 10.02001120... is just ugly.

21. Mar 14, 2016

### MrAnchovy

Base 64 is used for encoding binary files in emails etc., google it :)