Short answers, pulled from the story.
The Ishango bone, discovered in the Democratic Republic of the Congo and dating back approximately 20,000 years, serves as the earliest known physical evidence of natural number arithmetic. This artifact, now housed in the Royal Belgian Institute of Natural Sciences, bears notched marks that suggest early humans were already performing arithmetic with natural numbers long before written language existed.
The Indian mathematician Brahmagupta formally introduced the numeral 0 in the 7th century CE, allowing it to function as a distinct number with its own arithmetic properties. The Babylonians used a placeholder digit for zero as early as 700 BCE, yet they omitted it when it would have been the last symbol in a number, treating it as a positional convenience rather than a true number.
Euclid defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all, creating a hierarchy where the unit was distinct from the multitude.
Giuseppe Peano published a simplified version of Richard Dedekind's axioms in 1889, establishing that 0 is a natural number, every natural number has a successor which is also a natural number, 0 is not the successor of any natural number, if the successor of one number equals the successor of another, then the numbers are equal, and the axiom of induction which ensures that if a statement is true of 0 and implies its truth for the successor, it is true for every natural number.
In the realm of set theory, the natural number 0 is defined as the empty set, a collection containing no elements, while the number 1 is defined as the set containing the empty set, and 2 is the set containing 0 and 1. This construction, proposed by John von Neumann, defines each natural number n as a set containing n elements in a way that creates an iterative definition satisfying the Peano axioms.
Subtracting a larger natural number from a smaller one results in a negative number, and the lack of additive inverses means that the natural numbers form a commutative semiring rather than a ring. The procedure of division with remainder, or Euclidean division, is available as a substitute, ensuring that for any two natural numbers a and b with b not equal to 0, there exist natural numbers q and r such that a equals bq plus r.