Natural number: the story on HearLore

Natural number

The Ishango bone, discovered in the Democratic Republic of the Congo and dating back approximately 20,000 years, bears notched marks that suggest early humans were already performing arithmetic with natural numbers long before written language existed. This artifact, now housed in the Royal Belgian Institute of Natural Sciences, serves as the earliest known physical evidence of natural number arithmetic, proving that the concept of counting is as old as human consciousness itself. Before the invention of numerals, our ancestors used their fingers to represent quantities, a method so intuitive it remains the foundation of modern counting. They would put down a tally mark for each object, allowing them to test for equality, excess, or shortage by striking out a mark and removing an object from a set. This primitive abstraction was the first step in transforming the physical world into a system of abstract relationships, turning the tangible act of holding up fingers into the conceptual understanding of quantity.

The Birth of Zero

For millennia, the number zero was a ghost in the machine of mathematics, existing as a concept of absence but lacking a symbol of its own. 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. It was not until the 7th century CE that the Indian mathematician Brahmagupta formally introduced the numeral 0, allowing it to function as a distinct number with its own arithmetic properties. Even earlier, the Olmec and Maya civilizations had developed 0 as a separate number as early as the 4th century CE, but their usage remained confined to Mesoamerica and never spread to the rest of the world. In medieval Europe, the calculation of the date of Easter, known as the computus, utilized 0 as a number beginning with Dionysius Exiguus in 525 CE, though it was never denoted by a numeral. Standard Roman numerals, which dominated Western thought for centuries, had no symbol for 0, instead employing the Latin word nulla, meaning none, to denote a zero value. The struggle to define zero as a number rather than a placeholder was a pivotal moment in the history of mathematics, transforming it from a mere absence into a fundamental building block of the natural number system.

The Philosophers of Number

The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes, who began to treat numbers not merely as tools for counting but as objects of philosophical inquiry. 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. Euclid, for example, 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, as any two units from indefinitely many units is a 2. However, in his definition of a perfect number which comes shortly afterward, Euclid treats 1 as a number like any other. In definition VII.3, a part was defined as a number, but here 1 is considered to be a part, so that for example 6 is a perfect number. This tension between the unit and the multitude persisted for centuries, influencing how mathematicians approached the very nature of existence and quantity. The debate over whether 1 was a number or the building block of numbers shaped the trajectory of mathematical thought, leading to the complex definitions that would emerge in the 19th century.

Common questions

What is the earliest known physical evidence of natural number arithmetic?

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.

When was the numeral 0 formally introduced as a distinct number?

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.

Who defined the natural numbers as a multitude of units in ancient Greece?

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.

What are the five Peano axioms established in 1889?

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.

How is the natural number 0 defined in set theory?

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.

Why are natural numbers not closed under subtraction?

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.