The set below, of all positive integers is denoted by ℕ (also known as the natural numbers):

Each positive integer (natural number) $n$ has a successor, $n+1$. The following axioms are known as the Peano Axioms and they define the set of natural numbers:

1. 1 belongs to ℕ.
2. If $n$ belongs to ℕ, then its successor $n+1$ belongs to ℕ.
3. 1 is not the successor of any element in ℕ.
4. If $n$ and $m$ in ℕ have the same successor, then $n=m$.
5. A subset of ℕ which contains 1, and which contains $n+1$ whenever it contains $n$, must equal ℕ.

We expand the set of natural numbers to the set ℤ of all integers, including zero and all negative integers:

When division is introduced, ℤ becomes inadequate. Hence, all fractions must be included as well, leaving us with ℚ, containing all rational numbers which are of the form $\frac{m}{n}$ where $m,n\in \mathbb{Z}$ and $n\ne 0$.

Many algebraic equations (typically polynomials) can be solved for rational solutions, but what about equations like $x^{}=2$? Clearly, there are some gaps in ℚ.

Further, there are numbers like $e$ and $\pi$ that arise naturally in math, yet are not rational numbers.

To begin addressing this problem, we first start by defining the algebraic numbers, all of which satisfy a polynomial equation