Binary rules

Jarvis323
Aparently Pythagorus's geometry cult thought that 10 was the perfect complete number because 1+2+3+4=10.

Also, apparently some people on the internet have been convinced that using a different numeric base, like 7, would enable new mathematics that could unlock the mysteries of the universe and give us free energy technology, or allow us to transcend into interdimentional beings.

Personally I'm a little bit skeptical.

cmb
There are ten digits in base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Similarly, in base-2 (binary), there are two digits; in base-3 (ternary or trinary) there are three digits, in base-16 (hex or hexadecimal), there are 16 digits - the ten digits from decimal plus six more from A, B, C, D, E, and F.

In base-X, the digits run from 0 through X - 1.
Yes, of course, but I'm saying I don't see how '0' is a digit that could derive from finger counting.

Count zero on your fingers, then work up. How many do you count?

To get '9' on your last finger you have to fold your first finger and count "zero!". Then hold two fingers down and count "one", three and count "two", etc..

So the Arabic numerals do not match up with fingers. Also, there was no zero in the numeral set in the western world from the time it emerged here (late first millennium, zero appearing mid 2nd millennium).

If '0' was one of the intrinsic digits of the Arabic notation, then '0' would have appeared in Gerbert's original texts where he first described the notation in the west, and it didn't. Apices also wrote on the subject in the West in the 12th century and '0' still hadn't appeared.

I am not at all in a disagreement that our number systems seem to have stemmed from finger counting, but it must have been a multi-stage step. First, develop symbols for number 10 (e.g. 'X' in Roman), then replace it with a positional notation and 'one lot of ten'. So at least a two step process past finger counting.

The Hindu mathematicians had a dot as a place holder from the 7th century which emerged as '0'.

If you count "one" as your first finger bent you get 10 on the last finger. '10' isn't a digit.

I think the only way this could have emerged is that there were representations of numbers-to-fingers and singular symbols for '10' on the last finger (viz. "X" in Roman") so that last finger gets a symbol. After counting "1 to 10" with symbols for a few thousand years, the Indian/Arabic folks figured out that "0 to 9" made more sense.

Fact is, all of the Egyptian, Chinese (rod and later Suzhou), Roman number systems were base 10 and those were also absent '0'. These seem to have done so independently of each other. So the 'because of the number of fingers' argument is pretty solid, difficult to see an alternative explanation.

Last edited by a moderator:
weirdoguy and PeroK
Gold Member
2022 Award
There are ten digits in base-10

In base-X, there are X - 1 digits.
Uh ... ? So base 10 has 10 digits, except that it has 10-1 = 9 digits?

Mentor
Uh ... ? So base 10 has 10 digits, except that it has 10-1 = 9 digits?
Ulp... What I meant but erred in saying is that in base X, the digits run from 0 through X - 1. I'll fix what I wrote.

Bystander
Gold Member
2022 Award
Ulp... What I meant but erred in saying is that in base X, the digits run from 0 through X - 1. I'll fix what I wrote.
Yeah, I figured that's what you meant. I was just ragging you

pinball1970 and Bystander
Gold Member
Of course feel free to pack them together to octs or hex numbers
If this is acceptable, why isn't it acceptable to pack them in decimal numbers?

I'm not saying you are wrong, I just fail to see your point. How does it "make much more sense" to you?

Some have mentioned the alphabet as well. Following your logic, we should also use a binary alphabet over our base-26 system (not counting capital letters and punctuations). That is what our computers do, isn't it? they translate all of these letters in a series of 0 and 1. Imagine reading your favorite masterpiece written in ASCII encoding, where even a space is considered a character (which is logical, isn't it?). One might argue that it is "better", but I'm curious to hear the argumentation explaining how it is better.

cmb
There are ten digits in base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Similarly, in base-2 (binary), there are two digits; in base-3 (ternary or trinary) there are three digits, in base-16 (hex or hexadecimal), there are 16 digits - the ten digits from decimal plus six more from A, B, C, D, E, and F.

(Corrected)In base-X, the digits run from 0 through X - 1.
On the semantic point, I'm unconvinced 0 is a digit.

If I hold up my 10 digits, which one is the 0th?

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members. Is the "10" in "base 10" a cardinal or an ordinal number?

I don't know, I think it is over-doing it to try to make semantic sense of why we do it this way. Just my POV. We all understand it, why try to over-describe it?

Mentor
2022 Award
On the semantic point, I'm unconvinced 0 is a digit.
Well, good that this is irrelevant. 0 is certainly a digit. One may discuss whether it is a natural number, or which I think, a discovery, but it is part of the alphabet and therewith a digit.

If I hold up my 10 digits, which one is the 0th?

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members. Is the "10" in "base 10" a cardinal or an ordinal number?

I don't know, I think it is over-doing it to try to make semantic sense of why we do it this way. Just my POV. We all understand it, why try to over-describe it?

Chomsky shed some light on such discussions. It isn't necessary anymore to babble.

symbolipoint
cmb
Well, good that this is irrelevant. 0 is certainly a digit. One may discuss whether it is a natural number, or which I think, a discovery, but it is part of the alphabet and therewith a digit.
I'd still like someone to address my previous point directly.

"Digit" is latin for "finger".

If you write each of (0 to 9) digits in pen on your fingers, which one do you write "0" on, and which one gets "9"?

I'm sorry but it is pretty bonkers to hold up one's hands, palms up, then bend ones right hand thumb and count 'zero!' then working along to the left hand thumb and say 'nine! all done!'. Just doesn't make much sense to me, but I am content to disagree on that.

Simply, a digit "1" in the second column represents the number ten. The "0" in the first column represents nothing at all, it is just a place holder to show where the second column isn't that column.

Mentor
On the semantic point, I'm unconvinced 0 is a digit.
I agree with @fresh_42 that 0 is definitely a digit.
If I hold up my 10 digits, which one is the 0th?
Here you are confusing two meanings of the word "digit," one of which refers to numbers while the other has an anatomical meaning.

From the Merriam-Webster dictionary (https://www.merriam-webster.com/dictionary/digit) with emphasis added by me
1a: any of the Arabic numerals 1 to 9 and usually the symbol 0

3: any of the divisions in which the limbs of most vertebrates terminate, which are typically five in number but may be reduced (as in the horse), and which typically have a series of phalanges bearing a nail, claw, or hoof at the tip
I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members.
Yes, but so what? The number of digits in any base is always one more than the largest digit; e.g. in binary the number of digits is 2, but the largest digit is 1; in octal, the number of digits is 8, but he largest octal digit is 7, and so on.

Is the "10" in "base 10" a cardinal or an ordinal number?
Cardinal

Last edited:
phinds and symbolipoint
Mentor
2022 Award
I'd still like someone to address my previous point directly.

"Digit" is latin for "finger".

Not very convincing, if it dates back to only 1640.

digit (n.)
late 14c., "numeral below 10," from Latin digitus "finger or toe" (also with secondary meanings relating to counting and numerals), considered to be related to dicere "to say, speak" (from PIE root *deik- "to show," also "pronounce solemnly"). The numerical sense is because numerals under 10 were counted on fingers. The "finger or toe" sense in English is attested from 1640s.
https://www.etymonline.com/search?q=digit

Nevertheless, it doesn't matter where it stems from. Its current usage is relevant, especially if binary is the subject. As such it is an element of the alphabet we use to express calculations, ergo a letter. Digit is just another word for a certain kind of letter.

symbolipoint
Homework Helper
On the semantic point, I'm unconvinced 0 is a digit.

If I hold up my 10 digits, which one is the 0th?
This seems to go to the distinction between cardinal numbers (how many) and ordinal numbers (which one).

As much as anything, it is an accident of language that we a use the same set of numbers for each and that we denote the first cardinal by zero (no objects) and the first ordinal by one (the first object).

It is not always so. In many computer languages, the first element of an array is at index zero. An array with n positions (cardinal number n) has a last position denoted by n-1 (ordinal number n-1).

Note that if you have a set of ten objects and if want to have a numbering system to be able to express the cardinality of any possible subset then you need at least eleven cardinalities.

I don't think the size of the set of "base 10 digits" can be directly compared with the names of that set's members. Is the "10" in "base 10" a cardinal or an ordinal number?
It is a cardinal. The cardinality of the set of digits.

I don't know, I think it is over-doing it to try to make semantic sense of why we do it this way. Just my POV. We all understand it, why try to over-describe it?
Wait, I thought you were the one trying to over-describe it.

Last edited:
pinball1970
cmb
Wait, I thought you were the one trying to over-describe it.
I was just looking at the different possibilities to see if there is one that offers the simplest explanation that is more self-evident than others, but I think there isn't and we're looking at naming conventions, so no clear answer possible from my POV.

jbriggs444
Gold Member
I'm loving this nine or ten digits discussion! Let me add fuel to that fire!

So in a base-1 numeral system, following the same logic of other numeral systems, do we consider counting this way:

0, 00, 000, 0000, 00000, ...

But if 0 is nothing, isn't a series of 0 still nothing?

Or this way:

1, 11, 111, 1111, 11111, ...

Still the same problem from a different point of view: Can 1 represent nothing? Or is the absence of a 1 enough to represent 0?

Or this way:

0, 1, 11, 111, 1111, 11111, ...

But then we would be using 2 digits in a base-1 system? Or is it a 1-digit system + a zero?

And still with 2 digits, there is unary code :

0, 10, 110, 1110, 11110, 111110, ...

alternative unary code :

1, 01, 001, 0001, 00001, 000001, ...

So in a unary numeral system, do we consider:
• no zeros;
• a single zero;
• only zeros;
• a mix of both for every number?

BillTre and symbolipoint
Gold Member
A compromise. Since quantum computers are the future, let‘s just used a counting system based on a superposition all of the proposed bases. I’ll leave the derivation as an exercise to the OP.

BillTre and phinds
Staff Emeritus
Also, apparently some people on the internet have been convinced that using a different numeric base, like 7, would enable new mathematics that could unlock the mysteries of the universe and give us free energy technology, or allow us to transcend into interdimentional beings.

Personally I'm a little bit skeptical.
They wouldn't let it on the internet if it wasn't true.

phinds, Jarvis323 and jack action
cmb
They wouldn't let it on the internet if it wasn't true.
Nor if it was true... You do know Big-Oil-Gov steals free energy machines from their inventors and supresses the discoveries?

Oh, hang on, sorry, let me go find the citations for that first ...

Jarvis323
Nor if it was true... You do know Big-Oil-Gov steals free energy machines from their inventors and supresses the discoveries?

Oh, hang on, sorry, let me go find the citations for that first ...
This is obvious, it was just surprising to me that they were able to supress free energy by making us use base-10 . Turns out free energy doesn't work in that base.

Mentor
2022 Award
We seemingly reached a point of conversation where the anyway problematic topic isn't in the focus any longer.