How many decimals -> to surpass 20,999,999.99999999

[remotemass]

Right now BTC has 8 decimals and can reach a maximum of 20,999,999.97690000.
What is the minimum number of decimals that BTC need to be able to surpass 20,999,999.99999999.
That is, for how many decimals would BTC have to be extended to so that the cumulative value could reach 20,999,999.99999999?

[pawel7777]

? You mean if the halving doesn't end on 1 satoshi, but goes further (0.5 sat; 0.25 sat)?

[2double0]

? You mean if the halving doesn't end on 1 satoshi, but goes further (0.5 sat; 0.25 sat)?

Block rewards won't reach 1 satoshi.
Someone correct me if im wrong.

[franky1]

why do people care about the year 2140.. YOU WONT BE ALIVE

but if you wish to write a book for your grandkids to read when they are adults.. heres some stuff

in the last 4 years before rewards per block stops.. the circulation will be
20,999,999.99790000

during that last 4 years each block would only give 1 satoshi per block. (maths brings the total at the end of 4 years to be the 21mill)..

at the end of the 4th year (2140) no one will notice or care that block rewards just stop, as their mindset has moved over to concentrate on tx fee's many years before

[pawel7777]

? You mean if the halving doesn't end on 1 satoshi, but goes further (0.5 sat; 0.25 sat)?

Block rewards won't reach 1 satoshi.
Someone correct me if im wrong.

It says here it will (Projected Bitcoins Long Term):

https://en.bitcoin.it/wiki/Controlled_supply

[remotemass]

Yes, the rewards will always keep halving but the 8 decimals limitation truncates the total limit.
What I was wondering is how many more decimals we would need to surpass 20,999,999.99999999 BTC was reached.
Does anyone have an answer?

[Beliathon]

...no one will notice or care that block rewards just stop, as their mindset has moved over to concentrate on tx fee's many years before

Listen to this man, he knows what he's talking about.

[cr1776]

Yes, the rewards will always keep halving but the 8 decimals limitation truncates the total limit.
What I was wondering is how many more decimals we would need to surpass 20,999,999.99999999 BTC was reached.
Does anyone have an answer?

This may give you a hint on calculating the answer:
http://en.m.wikipedia.org/wiki/Geometric_progression. Or geometric series.

Perhaps no one will be alive, but it is an interesting exercise for the mathematically inclined. 

Does it converge or diverge?

This link has a bit more:
http://www.quora.com/Why-is-Bitcoins-cap-set-at-circa-21-million-coins-and-not-more-or-less


(Btw Alcor and others might contest the assertion that for sure no one will be alive.)

[Jamacn]

As long as needed, regardless of how much can be created

[Dabs]

Between the years 2104 and 2140 only 1 bitcoin is added.
That's about 36 years to generate the last whole bitcoin.

Per the rules of the protocol, and geometric progression, no amount of decimal places will result in accumulating more than 21 million, unless there is rounding (up), and even then it will take another thousand years.

[iwillwin]

And why can BTC only reach a maximum of this value ? Any possible explanation ? I mean why this specific figure only ? That too with decimals ........

[jonald_fyookball]

What I was wondering is how many more decimals we would need to surpass 20,999,999.99999999 BTC was reached.
Does anyone have an answer?

Actually, you will NEVER get there.

With unlimited decimals,
the limit will be 20,999,999.979.
 
Check out the chart on

https://en.bitcoin.it/wiki/Controlled_supply

look at line 34, in the column that
says "BTC added" -- you will see
we add .0021 BTC in the last era
of 210,000 blocks where this is
a reward of 1 satoshi.  (which
also makes sense since there's
a hundred million satoshis in 1 BTC).

Now...in a geometric progression where
you keep halving something, the
limit will be double of what you
started with.

So that means the next era would be
half of .0021, etc... with all the remaining
eras combined approaching  .0021.

So if you add .0021 to the current limit
of 20999999.97690000    you get
20,999,999.979.

This (and 21 million) may seem like a
funny, abritrary number.  It is
simply a consequence of the 210,000 block
halving number chosen by Satoshi.

Franky, not sure where you got your number
but it is incorrect...

in the last 4 years before rewards per block stops.. the circulation will be
20,999,999.99790000

Check the chart in the link
https://en.bitcoin.it/wiki/Controlled_supply


EDIT:

After thinking further about it, I realize the answer
above is true only if we extend the decimal
precision at that point in the future when
we are at the last block.  If we wind the
clock back to 2009 when Bitcoin got started,
and introduced unlimited digits there, the theoretical
limit WOULD be 21 million. (210,000 blocks x 50 BTC,
and then doubled)...The bitwise rotation
rounding is what causes it to go lower.

Calculating the exact number of initial decimal
places needed to eventually reach .99999999
if we started Bitcoin over from the beginning
is non-trivial because you would
need to figure out the cumulative effects on
rounding over the geometric progression, which
I don't have knowledge of or time to research...
but this was an interesting problem!

If someone is really curious, you can always
write a simple program to do it, but you
will need an unlimited decimal library
otherwise you'll run out of room.

[odolvlobo]

And why can BTC only reach a maximum of this value ? Any possible explanation ? I mean why this specific figure only ? That too with decimals ........

The number of bitcoins created in each block is already known. Beginning with block 6930000 around the year 2140, the number will forever be 0. Thus you can add up the numbers for all the blocks and you will compute a total of 20999999.97690000 BTC.

[remotemass]

Per the rules of the protocol, and geometric progression, no amount of decimal places will result in accumulating more than 21 million, unless there is rounding (up), and even then it will take another thousand years.

I'm saying more than 20,999,999.99999999 not more than 21 million.

I experimented with a script but one of my variables gets out of range...

Code:

<!DOCTYPE HTML>
<html>
<body>
<script>
var total=0;
var block = 1;
var reward = 50;
var year = 2009;
while(total<20999999.99999999){
	for(block=1; block<=210000; block++){
		total += reward;
	}; 
	reward /= 2;
        year += 4; 
	document.write('reward: '+reward+'<br/>');
	document.write('total: '+total+'<br/>');
	document.write('year: '+year+'<br/>');
};
</script>
</body>
</html>

But I got to the conclusion we would need hundreds, if not thousands, of extra decimals and years of rewards.

[jonald_fyookball]

you need something like php binary calculator.
http://php.net/manual/en/intro.bc.php

Your script is not doing the bit shift rounding
btw....but think of it this way, any rounding
down is going to be a permanent loss of precision,
so you need enough decimal places to avoid any
rounding for all the way out to a single satoshi.

not sure why you think it would be thousands of decimal places though...

it seems that 100 or points will be sufficient
to avoid rounding errors out to eight points don't you think?

[Dabs]

But I got to the conclusion we would need hundreds, if not thousands, of extra decimals and years of rewards.

Yah, I meant the same thing. It will approach it, but not reach it, when the sun is about to implode (or something heat death of the universe).

[Q7]

Can the system intepret and round up mathematically that the smallest number which is 0.00000001 if divided by 2 will become zero. So here it can never reach the full 21mil. Ok most probably I'm wrong here. Can somebody provide a more accurate answer and explanation. Getting myself confused

[jonald_fyookball]

Can the system intepret and round up mathematically that the smallest number which is 0.00000001 if divided by 2 will become zero. So here it can never reach the full 21mil. Ok most probably I'm wrong here. Can somebody provide a more accurate answer and explanation. Getting myself confused

if you round up you would exceed 21M.....we round DOWN, that's how it stays below 21M.

[remotemass]

Can the system intepret and round up mathematically that the smallest number which is 0.00000001 if divided by 2 will become zero. So here it can never reach the full 21mil. Ok most probably I'm wrong here. Can somebody provide a more accurate answer and explanation. Getting myself confused

if you round up you would exceed 21M.....we round DOWN, that's how it stays below 21M.

It's not rounded. It's truncated. It's different.

[dserrano5]

at the end of the 4th year (2140) no one will notice or care that block rewards just stop, as their mindset has moved over to concentrate on tx fee's many yearsdecades before

Fixed that. In fact, some of us will be alive to see a time when the block subsidy is already irrelevant, e.g. 1.5 BTC per block around 2028.