Why is the most common base 10?

[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.

 

[Integral]

Counting your thumbs, how many fingers do you have?

 

[manlyman62]

Haha! Duh! Ok, thanks!

 

[Mark44]

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 80, 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.

 

[Office_Shredder]

French word for 80, quatre vingt (4 twenty)

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

 

[DoctorBinary]

Counting your thumbs, how many fingers do you have?

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.

 

[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

 

[DaveC426913]

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.

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

 

[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".

 

[Office_Shredder]

Apparently the Mayans had a base 20 system

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.

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

 

[CRGreathouse]

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.

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.

 

[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.

 

[DoctorBinary]

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.

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.

 

[dulrich]

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

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

 

[Martin Rattigan]

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.

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

 

[dulrich]

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

What does this mean? Are you speaking binary?

 

[Martin Rattigan]

What does this mean? Are you speaking binary?

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.

 

[DaveC426913]

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) in binary and print it out in hex it comes out as 10. Same for any other base. The base is always 10.

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

Same for any other base. The base is always 10.

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.

 

[Mark44]

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.

 

[DaveC426913]

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.

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.

 

[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.

 

[DaveC426913]

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.

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.

 

[Phrak]

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.

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.

 

[DaveC426913]

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.

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.

 

[Office_Shredder]

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.

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

 

[Phrak]

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

OK. We use your notion that a base N can count N things in the first column. So the first digit in base 1 can represent one thing. There would be ten symbols in base 10 representing counts from one to ten.

Continuing this pattern, the first symbol in base 2 represents one thing or two things. How would you respresent numbers of higher basis in this modified system?

 

[DaveC426913]

OK. We use your notion that a base N can count N things in the first column. So the first digit in base 1 can represent one thing. There would be ten symbols in base 10 representing counts from one to ten.

Continuing this pattern, the first symbol in base 2 represents one thing or two things. How would you respresent numbers of higher basis in this modified system?

It can represent one (1) thing or no (0) thing.

Note that this occurs with zeroes in embedded columns (such as 101), not leading zeroes. Base 1 has no embedded zeroes.

And like all other bases, base 1 does not use symbols for leading zeros, it just uses blank.

 

[DaveC426913]

I did start the discusion by stating that base 1 is exceptional, so there's no contention there, but exceptional doesn't mean something is disqualified (see prime numbers, above).

You're protesting details, but I think you're missing the forest for the trees.

Any number that can be counted using 2 or 6 or 10 symbols, you can just as easily count using only one symbol. You simply follow the exact same procudure:
- Count using all the symbols available to you until you run out of symbols.
- Add a new column to the left, containing your lowest symbol, then iterate through all your symbols again.
- Repeat as needed.

The only way base 1 is exceptional is that it does not happen to have zero as one of its symbols. It doesn't need it - it starts with blank in its leading column (like all other bases do).

 

[Phrak]

I did start the discusion by stating that base 1 is exceptional, so there's no contention there, but exceptional doesn't mean something is disqualified (see prime numbers, above).

You're protesting details, but I think you're missing the forest for the trees.

Any number that can be counted using 2 or 6 or 10 symbols, you can just as easily count using only one symbol. You simply follow the exact same procudure:
- Count using all the symbols available to you until you run out of symbols.
- Add a new column to the left, containing your lowest symbol, then iterate through all your symbols again.
- Repeat as needed.

The only way base 1 is exceptional is that it does not happen to have zero as one of its symbols. It doesn't need it - it starts with blank in its leading column (like all other bases do).

This seems to be a good enough system to respresent elements of an ordered set set, as counting numbers do, but in the system of the basis respresentations of numbers it can only represent one number without having a symbol for zero and at least one other, or three, if we count blankspace as delimiters.

 

[Max™]

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.

Hmm, interesting thought, why are arabic numerals automatically the standard symbols?

Which ones did you mean?

These: 0123456789?

Perhaps these: ٠١٢٣٤٥٦٧٨٩?

 

[Phrak]

Hmm, interesting thought, why are arabic numerals automatically the standard symbols?

Which ones did you mean?

These: 0123456789?

Perhaps these: ٠١٢٣٤٥٦٧٨٩?

Was I confusing? You can use any symbols you want. It's simply a good idea to have a common language for purposes, you know, communication.

 

[Max™]

No, I was asking about what specific benefit arabic numerals grant, do they represent the numbers well? Are they easier to write than any alternative? Do they avoid issues such as dyslexia type reversal errors?

 

[Phrak]

Oh, I see. If we were all fairly unimpressed with left-right distinctions, all symbols would be left-right symmetrical or equally well be written either way. But it's an historical fact of life that westernized Arabic numerals are the de facto standard today. Trying to change it is like trying to stop a train with your bare hands.

People also get peculiar notions about how spoken and written language should be improved. I once attended a lecture presented by some guy at UCLA who had his own ideas of a common global language. His peccadillo was order. He wanted a language with a high degree of order.

For instance, the words:

---we, they, I, you, he, and she---

would be improved.

I don't recall the actual spelling, but his idea of improvement was something like:

---tad, tid, ted, tyd, tod, tud---

This is not improvement; this is the destruction of redundant audio information that helps us tell similar sentences apart.

I have a special concern about dyslexia. How would you recommend improving the written language--or getting over it?

 

[DaveC426913]

I don't recall the actual spelling, but his idea of improvement was something like:

---tad, tid, ted, tyd, tod, tud---

This is not improvement; this is the destruction of redundant audio information that helps us tell similar sentences apart.

good point.

This guy clearly has not read - but desparateley needs to read - 1984. 'The destruction of words' was an awesome subplot.

 

[Phrak]

good point.

This guy clearly has not read - but desparateley needs to read - 1984. 'The destruction of words' was an awesome subplot.

Doubleplusgood of you to say so. I read 1984 when I was 15. I had to look it up to recall your meaning.