# Mathematical tools that help Bitcoin ensure system consistency discovered

More than 14 years ago, Satoshi Nakamoto announced the Bitcoin network to the world, creating the first three-term accounting system known to man. With a current market cap of \$558 billion, this technological marvel has cleverly integrated cryptography and mathematical formulas to underpin its security. In this exploration, we delve into two mathematical choices behind Bitcoin’s complex architecture, determining block rewards, transaction inputs and outputs, and limiting mining difficulty while adjusting the discovery rate of new blocks.

## Consider Bitcoin’s use of integers

Bitcoins are created using various coding processes and mathematical formulas, each with a specific purpose. A design element built into Bitcoin is the use of integers or integers and their symmetric negative integers.

The Bitcoin network uses integer math to prevent potential disagreements that could arise when using decimals or fractions. Their symmetrical use of positive and negative integers ensures that all computing devices can synchronize specific network changes more efficiently and uniformly. Using integers to maintain Bitcoin’s rule set includes block rewards and halvings that occur at specific block heights divisible by 210,000. Bitcoin mining difficulty also uses integers to adjust the difficulty every 2,016 blocks. Integers are a type of numeric data commonly used in computing software and are also used for inputs and outputs in Bitcoin transactions.

Also, integer operations are generally faster and less error-prone than floating-point numbers. If Bitcoin used floating-point real numbers, rounding errors could result, leading to inconsistencies and disagreements between different nodes on the network.

Since Bitcoin uses integers, block rewards from future halvings will eventually be truncated or rounded down to the nearest integer using shift operators or bit manipulation operations. (bitwise operations). Because the smallest unit of Bitcoin is Satoshi, it cannot be divided equally. Therefore, the much-disputed Bitcoin supply cap will actually be lower than 21 million.

## Adjust block time using Poisson distribution

In addition to integers, Bitcoin also uses mathematical formulas such as the Poisson distribution to adjust the consistency of block times. The Poisson distribution model was proposed in 1837 by French mathematician Simeon Denis Poisson. Using this model, Bitcoin is designed to ensure that blocks are discovered every 10 minutes or so. In a May 2022 paper titled “Interpreting Bitcoin Mining as a Poisson Distribution,” author Suhail Saqan stated, “The orange line in the graph represents the average wait time (approximately 10 minutes). Between Bitcoin blocks The simulation of the waiting time approximates a Poisson process.”

Due to the probabilistic nature of the mining process, the actual time it takes to mine a block may vary, but typically it takes 8 to 12 minutes to find a block. Satoshi set a difficulty setting every 2,016 blocks, using a formula to maintain a rough average of 10-minute block times.

Integer math and the Poisson distribution are both essential mathematical tools in Bitcoin, providing a consistent framework for performing calculations and modeling various aspects of the system.

Bitcoin uses many other mathematical mechanisms and coding schemes to ensure the accuracy, consistency and efficiency of the entire system. These include concepts and formulas such as PoW, Merkle trees, elliptic curve cryptography, cryptographic hash functions, and finite fields.

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