# Base 10?

1. Jun 22, 2010

### manlyman62

Hello! I was just wondering why the most common base is base 10? It seems kind of arbitrary to me, although I may be overlooking something.

2. Jun 22, 2010

### Integral

Staff Emeritus
Counting your thumbs, how many fingers do you have?

3. Jun 22, 2010

### manlyman62

Haha! Duh! Ok, thanks!

4. Jun 22, 2010

### Staff: Mentor

Other bases have been used over time, such as 12, 20, 60, 360.

Base 12 is convenient to use because it can be divided evenly by 2, 3, 4, and 6, so many fractional amounts of 12 things come out as whole numbers. We buy eggs by the dozen and count (in the US) 12 inches to the foot. A gross of things is 144 of them, or 12 dozen.

Base 20 is nearly as obvious as base 10 (counting toes and fingers). I believe that the French word for 60, quatre vingt (4 twenty), might be a remnant of base-20 counting.

Base 60 is still present in the number of seconds in a minute or minutes in an hour or degree. 60 has the advantage of being divisible by 10 or 20 as well as 12.

Base 360 - the number of degrees in a full circle. If my memory is correct, the Babylonians counted numbers using this base.

Other bases -- base 2, base 8, base 16 -- are more modern, and stem from computer technology, with transistors able to attain one of two possible states. A single bit in memory can be either 1 or 0, corresponding in some way to a high voltage or a low voltage. The binary digits are 0 and 1.

If you collect three bits together, you can represent any of eight numbers: 0, 1, 2, 3, 4, 5, 6, or 7, the octal digits. If you collect four bits together, you can represent any of 16 different numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. These are the hexadecimal digits.

5. Jun 22, 2010

### Office_Shredder

Staff Emeritus
I always knew there was something a little funny about the French

6. Jun 22, 2010

### DoctorBinary

This is the prevailing answer, but it always leaves me unsatisfied. Personally, I think base 6 is the better match: the right hand is the 1s place, and the left hand is the 6s place.

7. Jun 22, 2010

### fourier jr

80 is quatrevingt (4 twenties, or 4x20). 60 is soixante. & I think another common base is hexadecimal (base 16) & I guess there's binary also (base 2), but everyone knows about that one

8. Jun 22, 2010

### DaveC426913

I wonder if any societies have ever had a base 21 counting system...

9. Jun 22, 2010

### HallsofIvy

I have read that the Kwakiutle Indians of the Pacific north west counted using the spaces between the fingers. Although their's was not a "place value" numeration system, it was based on "4" rather than "5" or "10".

10. Jun 22, 2010

### Office_Shredder

Staff Emeritus
Apparently the Mayans had a base 20 system

I feel like by the time placeholders were invented, either a base was set already or a base was picked on needs other than the ability to count on your hand

11. Jun 22, 2010

### CRGreathouse

Base 6 is extremely uncommon, though -- base 10, base 20, base 5, base 100, base 12, base 60, and base 4 are all more common in natural languages. Actually, I can't think of a single example.

12. Jun 22, 2010

### Antiphon

When I want to hide a number in plain sight I use base 9 but then add random 9's and simply ignore them.

13. Jun 23, 2010

### DoctorBinary

I don't know of any examples either; it just seems natural to me. Of course, I'm looking at this in hindsight with zero and place value at my disposal.

14. Jun 25, 2010

### dulrich

That's a fantastic idea! Do you do the conversion in your head? With octal you could just use a calculator...

15. Jun 25, 2010

### Martin Rattigan

The real reason is if a number is in any base it's in base 10.

Last edited: Jun 25, 2010
16. Jun 25, 2010

### dulrich

What does this mean? Are you speaking binary?

17. Jun 25, 2010

### Martin Rattigan

Binary, ternary, hex - whatever. If you store the binary number base (two) and print it out in binary it comes out as 10. If you store the hex number base (sixteen) and print it out in hex it comes out as 10. Same for any other base. The base is always 10.

Last edited: Jun 25, 2010
18. Jun 25, 2010

### DaveC426913

Heh. I'd never thought of it that way.

There is, of course, a single exception - and a commonly used one at that (people use it a lot when playing cards): base 1.

i.e.:
||||| |||||
||||| |||

Base 1 is simply: base 1.

19. Jun 25, 2010

### Staff: Mentor

I don't think counting by tickmarks really counts as "base 1." Base 1 doesn't follow the consistent pattern that is present in other bases such as base 2, 3, 8, 10, 16, etc. In base 2, there are 2 binary digits, 0 and 1. In base 3, there are three ternary digits, 0, 1, and 2. If we're counting in base a, the digits go from 0 up through a - 1.

Following the same logic, in base 1, the only digit would be 0, which makes counting pretty difficult.

20. Jun 25, 2010

### DaveC426913

That's not the pattern though. You're looking a the wrong part for significance.

We use 0 and 1 by convention as the first two symbols in our numeral system. In base 1, we can use tickmarks, but it doesn't matter what the symbol is as long as we know what it represents - which is why it can just as easily be 1 - or 0.

The important pattern this:
- we start counting with our list of symbols
- when we use up all our symbols we start a new column, and resume counting

So:

base 3:
0,1,2, (add a new column, keep going) 10,11,12,20,21,22, (new column) 100, etc.
Here it is with different symbols:
a,b,c,ba,bb,bc,ca,cb,cc, baa, etc.

base 2:
0,1 (add a new column, keep going) 10,11, (again) 100,101, etc.
Again, with different symbols:
a,b,ba,bb,baa,bab, etc.

base 1:
1, (add a new column, keep going) 11, (again), 111, etc.
Again, with different symbols:
a, aa, aaa, etc.

Last edited: Jun 25, 2010
21. Jun 25, 2010

### Phrak

I disagree, Dave.

Base 3 digits
0,1,2

Base 2 digits
0,1

Base 1 digits
0

In base b, the contributed numerical value of a digit x, in the nth column is the x n^b. The range of x is (0,1,2,...,b-1).

It would break with established pattern to use one instead of zero for the digit set for base one.

22. Jun 26, 2010

### DaveC426913

You don't have to use the numerical symbols 1,2,3. You can use any symbols, as long as you keep them in order. The problem here, is the confusion between the symbols 1,2,3 and the counting numbers 1,2,3.

23. Jun 26, 2010

### Phrak

No, I don't have that confusion. Using the standard symbols, so we all know what we're talking about, base 1 can represent only zero, and no higher.

24. Jun 26, 2010

### DaveC426913

Zero in all the bases is a way of representing "no unit here". When you want to to start counting, you add your first symbol to that column. Another way of representing "no unit here" is to leave the column blank.

The key to bases is this: how many symbols can you use before you have to add another column to continue counting?

In base 2, you can count exactly 2 things before you need another column; in base 1 you can count exactly one thing before you need another column.

Regardless of where you see a deviation, it's still valid.

An analogy: By your logic, you would be claiming that the first prime number is 3 (the pattern you see is that prime numbers must be odd, therefore 2 is out).

But it isn't; the first prime is 2. 2 is a unique prime in that it breaks some common symmetry of prime numbers (being even), but it does not break the critical rule that truly defines prime: it is divisble only by 1 and itself. 2 fits, no matter how exceptional it is.

Last edited: Jun 26, 2010
25. Jun 26, 2010

### Office_Shredder

Staff Emeritus
In base 2 when you get to the second object you need a new column to represent it, so in base1 you need a new column to represent the first object. In base 2 you need a third column to represent 22 objects, so in base 1 you need a third column to represent 12 objects.

To make a long story short, you need infinite columns to represent the number 1