Is Binary a Fit Numbering System?

[O.J.]

Allright, now we know using the decimal system you can represent any number you want because you use a base 10 system. my question about the binary system is: you can very well represent numbers from 0 to 255 using 8 bits. Now, how can you show that you can represent EVERY number between these 2 extremes?

 

[chroot]

You could write them all out on paper.

- Warren

 

[bob1182006]

No proof here but...it seems that in general if you have base R and you're using N digits.
You can express any number in that base with
$$a*R^N+b*R^{N-1} + ... + c*R^2 + a*R^1 + d*R^0 + e*R^-1 ...$$ a/b/c/d/e being any number in the interval [0,R-1]

to find what numbers you can express you take your base R to the digit's power: $R^N$ and then -1 for 0 so you can express from $[0,R^N-1]$

ex: Base 10, 2 digits gives $10^2-1=100-1=99$ so you can represent any number from 0 to 99 which we know for sure but you can still write out every number between 0 and 99 to be sure.

the same is with base 2/8/16/etc...

 

[O.J.]

what I am saying is: how do the people who came up with this system be sure that you can represent ANY number using ANY base? is there a general proof or intuitive explanation to this?

 

[chroot]

If you accept that you can represent any number in decimal, what makes you think any other base would be any different? Is the number 10 special?

- Warren

 

[Averagesupernova]

Yes Chroot it is special because it is how many fingers I have.

 

[DaveC426913]

what I am saying is: how do the people who came up with this system be sure that you can represent ANY number using ANY base? is there a general proof or intuitive explanation to this?

As chroot points out, you're thinking that base 10 is somehow unique. It's not. You can use 10 symbols to count, or you can use 16 or you can use 2, or you can use any number.

Line up a thouand pebbles on the ground.
Start with the first one, assign a symbol to it (let's make that symbol a "0").
Assign another symbol to the next pebble (we'll make that symbol a "1").
Continue assigning a new symbol to each new pebble until you have a comfortable number of unique symbols (Some people like 10 unique symbols, some people like 16, some people and machines can only manage 2 unique symbols).
Now, rather than creating new symbols for every single subsequent pebble, you can group them. You can keep track of the groups using the symbols you've already created.
So, that first group, the one you've just finished, prefix it to mark it as "group 0", so your first few pebbleas will be labelled 00, 01, 02, 03 etc. The one you're about to start, prefix it with your second symbol "1" (so, 10,11,12,etc.)

Now continue counting pebbles. Everytime you run out of symbols, start a nrew group and prefix it with the next symbol (30,31,32,33, etc.)

(Sidenote: Once you've made as many groups as you have symbols, you can group the groups. pebble 3 of group 5 of supergroup 2 is called 253.)

Note that the above method works regardless of how many unique symbols you start with.


Follow along, here:

[FONT="Courier New"]
Say we have ten unique symbols (Base 10)

row of pebbles:     Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl ...
label them uniquely: 0   1   2   3   4   5   6   7   8   9
label as group 0:   00  01  02  03  04  05  06  07  08  09
start a new group:                                           0   1   2   3   4  5
label as group 1:   					    10  11  12  13  14  15  ...

what do you get?
row of pebbles:     Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl ...
labels:              0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  ...


Now let's try  this with fewer symbols, say 4 (Base 4)


row of pebbles:     Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl ...
label them uniquely: 0   1   2   3
label as group 0:   00  01  02  03
start a new group:                   0   1   2   3
label as group 1:   		    10  11  12  13
start a new group:                                   0   1   2   3 
label as group 2:   		                    20  21  22  23

what do you get?
row of pebbles:     Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl ...
labels:              0   1   2   3  10  11  12  13  20  21  22  23  30  31

Now let's try it with only two symbols (Base 2):

row of pebbles:     Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl ...
label them uniquely: 0   1
label as group 0:   00  01
start a new group:           0   1 
label as group 1:           10  11
start a new group:                   0   1 
You can't label it as group 2, what is your "third" symbol? It's 10.
So label as group 10:               100 101
start a new group:                           0   1
Label as fourth group (11):		    110 111

what do you get?
row of pebbles:     Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl Pbl ...
labels:             000 001 010 011 100 101 110 111 etc.
[/FONT]

Now you can see why every number can be represented in any base.

 

[DaveC426913]

I'll give extra bonus points to anyone who can show, not only how base 1 is represented, but a common place many of us use it when passing the time with some friends.

 

[rbj]

what I am saying is: how do the people who came up with this system be sure that you can represent ANY number using ANY base? is there a general proof or intuitive explanation to this?

you (there are 3 letters in the word) want a constructive proof that every non-negative integer less than B^N can be represented with N digits, each that can take on any integer value from 0 <= digit < B? do you want that constructive proof?

 

[chroot]

"Base one" has only one symbol, usually a vertical line. A group of N vertical lines represents the number N. People often use this number system when keeping score in games.

- Warren

 

[DaveC426913]

10 char