\begin{equation} \mathbb{N} = \{0, 1,2,3 \dots \} \end{equation}

the set of natural numbers. **start from 0**.

\begin{equation} \mathbb{Z} = \{\dots, -2, -1, 0,1,2, \dots \} \end{equation}

the set of integers. natural language and their negatives

## Key Sequence

- first, we built the ground work of principle of induction in order to construct the WOP
- we defined division, and formalized the algorithm for doing so
- we then defined the greatest common divisor, and the fact that greatest common divisor is a linear combination
- we then constructed the idea of prime numbers, coprimes, and showed that There are infinitely many primes
- Finally, we used yet another lemma from Euler to build the fundamental theorem of arithmetic