You have a universe of one variable and allow one output.
The variable is I and the output is O.

The base of these numbers can be from 2 to n. Start at 2.

O = f(I)

What is f?

(Hint: think of a coin.)

Originally posted by hlreed
You have a universe of one variable and allow one output.
The variable is I and the output is O.

The base of these numbers can be from 2 to n. Start at 2.

O = f(I)

What is f?

(Hint: think of a coin.)
I have no idea what you're asking. You've indicated that a function f relates two quantities in an unspecified way... that's all.

- Warren

Warren
I hate to give the answer so soon, but it is counting which in this case is also integration.

Let I = 1 then O = 1 0 1 0 1 0 ...
That is all you can do with one variable. With the coin you can flip it head to tail, which is the same thing. In other words the only thing you can do with one bit is to flip it.

Harold

Originally posted by hlreed
In other words the only thing you can do with one bit is to flip it.
Or you could leave it alone.

Sorry, but O = f(I) means nothing other than "O is related to I through some unknown mechanism f."

- Warren

Originally posted by hlreed
Let I = 1 then O = 1 0 1 0 1 0 ...

and originally

The base of these numbers can be from 2 to n. Start at 2.

O = f(I)

What is f?

Since when was 1 > 2?

As Warren points out standard nomenclature has f() as any function you want. As in,

f(x) = e^x
f(x) = sin(x)

What you have is f(x) = n where 0 < n < 1 and n an element of the integers.

O = f(I) ; Of course f can be any function. That is what I am asking.

A binary digit is an on - off switch. You can tell it to do sin(I) and it will still count. Actually the output will be 0 0 0 0...

What this means is that number base relates to the discrimination a number can make. A single digit of any base cannot do arithmetic by itself. The only thing you can do with it is count, which makes integration the most primative operator of all.

names value countfrom 0 count from 1
zero 0 0 0 1
one 1 1 0 0
1
0
I doubt if these will line up.
Think of a digit as a wheel with clicks from one value to the next.
Write the number value names on the wheel, starting at 0. Then with this wheel you can click from one name to the next. The number of names is the base of the number. So base is a counting number. What if base is 0 or 1?
base what you can do
0 nothing, no clicks. Always stuck at 0
1 still no clicks, stuck at 1
2 0, 1
3 0, 1, 2
...
256 0 - 255 ; which is eight bits as one digit.

It is also 2^8 which is how you can convert any number into a single digit. I mean any number.

Base is also a number which can grow large but not infinite. There are no infinite numbers just by the fact that it takes wires to represent numbers.
Enough now.
Thanks for the comments.
Harold

Errrmmm, yeh. Right! OK!

Originally posted by hlreed
A binary digit is an on - off switch. You can tell it to do sin(I) and it will still count. Actually the output will be 0 0 0 0...
And you have fulfilled my prophecy that you are in fact a crackpot, are obsessed with the new-found subject of numerical bases, and are self-assured that numerical bases have some deep physical significance.

Unfortunately, you are wrong.

I am anxiously waiting for you to describe the beginning of time and the future of alien civilization with the concepts of arithmetic bases.
A single digit of any base cannot do arithmetic by itself.
A base one counting system only has one symbol, and is a degenerate system. It can only represent itself. The reason is that the place system doesn't work when every place has the same weight, 1.
The only thing you can do with it is count, which makes integration the most primative operator of all.
Integration and counting are two very, VERY different concepts.
Think of a digit as a wheel with clicks from one value to the next.
Write the number value names on the wheel, starting at 0. Then with this wheel you can click from one name to the next. The number of names is the base of the number.
Yes, a numerical base N has N different symbols. WOW.
So base is a counting number. What if base is 0 or 1?
base what you can do
0 nothing, no clicks. Always stuck at 0
1 still no clicks, stuck at 1
2 0, 1
3 0, 1, 2
...
256 0 - 255 ; which is eight bits as one digit.
A base zero system has no symbols at all, which makes it somewhat difficult to represent anything in it. A base one system only has one symbol, and can only represent one thing.
It is also 2^8 which is how you can convert any number into a single digit. I mean any number.
Yes, 2^8 is 256. If you're allowed to pick the base, you can pick a base in which any number can be represented as one symbol. This is not an important consequence. The reason we don't use base gazillion is because us humans don't like having to memorize a gazillion symbols. We'd rather just memorize ten, and then use the place system.
Base is also a number which can grow large but not infinite. There are no infinite numbers just by the fact that it takes wires to represent numbers.
There is one infinite number. We call it infinity for short. It does not seem to depend on any 'wires.'

- Warren

Actually, I take that back... a base 1 system is feasible for counting -- but it cannot represent zero.

- Warren

Surely the simplest function is: f(I) = 0 (zero)

Or maybe: f(I) = I (the do-nothing function)

Base is irrelevant.

If you're talking binary and digital, blah blah blah, it seems to me the simplest mathematical function is in the MIPS architecture:

addi $t0, $zero, 1

It's been a few months, but I think that's it. The first register is always zero. Adding one is a single state change to the output line.

Adam,
I think you got it if I can understand what you are saying.

In this discussion there is only one variable in the universe. Its base is 2 but that is not what matters. The function can have one input only. That is I.
The output O can be written down, so you can have many outputs.
Now O = f(I) ; I say the function is O = i(I) where i is an integral function.

The i function
Start
1 O = 0 ; clear count
Loop
2 Read I ; if I = 1 then we count by 1.
3 O = O + I
4 Write O
5 GOTO Loop ; do this forever
END

I have hardware for this. On the bench I can show you
x = d(i(x)) ; where d is the derivative function.

The i function here puts out a data stream which is a saw tooth wave from 0 to 256 in base 256. It goes 1 255 1 255 1 255 ...
For base 2 the data stream is 1 0 1 0 1 0...

The difference in the integral function and the count function is that the count function does not read a new count each cycle.

When you are counting and run out of symbols you have 3 choices to continue.
1. Just let it roll over
2. add more symbols and make a new base
3. hire another digit that can count by one when you overflow.
3 is what we chose 3000 years ago.
choosing 2 removes infinities.

If the base is large enough you will never notice it.
Symbols here mean names. In base 256 the name 255 is one symbol.

Harold

Yes, exactly -- are you going to explain the beginning of time and the future of alien civilization now for us, as a consequence of your amazingly well-crafted model?

- Warren

My interest is getting the garbage out of language. That includes mathematics, which is a language. The notion of base 0 and base 1 came about because base (b) is a counting number. The obvious question is, what it it were 0. If you put the digit on a wheel, the number of clicks to go around the wheel is also b.
The click count is the number. The symbols or names shown at each click are completely arbitrary. So a base 0 wheel has no clicks and the symbol is 0. The base 1 wheel has no clicks and is alway named one. Makes perfect sense when you are crazy.
The application is: b = 0 is ground and b = 1 is VPP (voltage).
This says that analog voltages are not numbers. (Not that you cannot make numbers from them.)

Harold

You've lost me. I suspect that what you're saying makes no sense, but perhaps I'm wrong.

weeeeee! the wheel goes round and round, having base pi*diameter as the circumference

weeeeee! round and round it goes, when it clicks, no "one" knows!

You betcha. The wheel goes round and round, cycling forever.
Expand the wheel into a drum and you can write all the symbols or names in the world on the drum and still you have the clicks that count them. Let the wheel diameter be 0 and the clicks be infinite and you will raise Pythagoras.

A wheel with two values is a single pole single throw switch.
Divides the universe in half.

Um...yeah....ok.

*backs away carefully*