The Bitcoin system has been designed so that the reward for mining a block is halved after every 210000 blocks. When block 0 (also known as the genesis block) was mined in early 2009 it was 50 BTC (five billion satoshi)
https://blockchain.info/block-index/14849/000000000019d6689c085ae165831e934ff763ae46a2a6c172b3f1b60a8ce26fThe first halving took place on reaching block 210000 towards the end of 2012, after which the reward for mining a block became 25BTC (2.5 billion satoshi)
https://blockchain.info/block-index/270670/000000000000048b95347e83192f69cf0366076336c639f9b7228e9ba171342e It remains at this value today and will continue untill reaching block 419999. From block 420000 the reward will be 12.5 BTC.
It is generally mentioned that the total bitcoins that will ever be mined is approximately 21 million BTC. The math involved here uses geometric progressions (GP) . This is an ordered sequence of numbers where each number (or term) is obtained by multiplying the previous number by a fixed amount. e.g. consider the sequence 1,2,4,8,... here the fixed amount is 2. This is known as the common ratio of the geometric progression and is referred to by the letter 'r'. The letter 'a' is used to refer to the first term of a GP. So for the above GP a = 1 and r = 2.
If the common ratio (r) is a fractional value less than 1, this results in a GP where the terms decrease. e.g. 100000,50000,25000,12500,6250,... is a GP where a = 100000 and r = 1/2. GPs that have such common ratios are said to be convergent. It is a property of all convergent GPs, that the total obtained by adding any number of terms in such a GP (no matter how many terms) will always be less than a fixed value reffered to as the 'sum to infinity' of the GP. We shall refer to this as 'S'. It is obtained according to the simple formula given below
S = a/(1-r)
In other words, to obtain the 'sum to infinity' of a given convergent GP, subtract the common ratio from one, and then divide the first term by the answer obtained. Accordingly, for the GP 100000,50000,25000,12500,6250,... where a = 100000 and r = 1/2, S = 100000/(1-1/2) which gives 200000. What this means in practice is, that no matter how far we continue the above GP, the total obtained by adding together all the terms will always remain below 200000.
Now, let us see how this applies to Bitcoin. Here we will form a GP where the terms will be the total mining reward for a group of 210000 blocks.
for block 000000 to block 209999 total reward = 50 x 210000 = 10500000 BTC
for block 210000 to block 419999 total reward = 25 x 210000 = 5250000 BTC
for block 420000 to block 629999 total reward = 12.5 x 210000 = 2625000 BTC
for block 630000 to block 839999 total reward = 6.25 x 210000 = 1312500 BTC
and so on
It can be seen that we get a GP where a = 10500000 and r = 0.5 (or 1/2)
from this we can see that S = 10500000/(1-0.5) = 21000000
this means that the total reward obtained by mining all blocks that exist at any point in time (whether in the past or the future) will be less than 21000000 BTC. Since it is the bitcoins produced through mining that enter and circulate in the bitcoin economy, this means the total BTC in circulation will always be less than 21 million BTC. It should be noted that this also means that exactly 21 million BTC will never come into existence.
The table given below shows the mining reward in satoshis per block as the block chain grows
( 1 BTC = 100000000 satoshi )
start finish reward per block
block block (satoshi)
000000 209999 5000000000
210000 419999 2500000000
420000 629999 1250000000
630000 839999 625000000
840000 1049999 312500000
1050000 1259999 156250000
1260000 1469999 78125000
1470000 1679999 39062500
1680000 1889999 19531250
1890000 2099999 9765625
2100000 2309999 4882812
2310000 2519999 2441406
2520000 2729999 1220703
2730000 2939999 610351
2940000 3149999 305175
3150000 3359999 152587
3360000 3569999 76293
3570000 3779999 38146
3780000 3989999 19073
3990000 4199999 9536
4200000 4409999 4768
4410000 4619999 2384
4620000 4829999 1192
4830000 5039999 596
5040000 5249999 298
5250000 5459999 149
5460000 5669999 74
5670000 5879999 37
5880000 6089999 18
6090000 6299999 9
6300000 6509999 4
6510000 6719999 2
6720000 6929999 1
Since 1 satoshi is the smallest unit of transaction in the Bitcoin system, when halving results in fractions of a satoshi, the fractional part is discarded. This will first happen on reaching block 2100000 and many times thereafter until ultimately the block reward comes down to 1 satoshi on reaching block 6720000. This means that from block 6930000 there will be no mining reward, and hence no new bitcoins produced through mining. At such a time miners will only get the tranaction fees, but it is expected that by such time the value of 1 BTC will be high enough for transaction fees alone to be sufficient compensation for miners.
If we use the above figures and calculate the total quantity of BTC produced, the figure obtained is 20999999.97480000 BTC, which is 2520000 satoshi less than 21 million BTC, and confirms the fact that the total BTC mined will be less than 21 million BTC as mentioned above.
The Bitcoin system has been designed so that new blocks are produced through mining at an approximate rate of 1 block per 10 minutes, which comes to around 144 blocks per day. So the time taken to reach block 6930000 should be 6930000/144 = 48125 days, which is a little less than 132 years. We know that the genesis block was mined, most probably by Satoshi Nakamoto himself, on 03-01-2009 as indicated on the blockchain at
https://blockchain.info/block-index/14849/000000000019d6689c085ae165831e934ff763ae46a2a6c172b3f1b60a8ce26f. This means that by the end of 2140, block 6930000 would be reached. This is why it is said that the last satoshi would be mined in 2140, and that no new bitcoins would ever be produced thereafter.
