The Natural Numbers

The natural numbers are the counting numbers {1, 2, 3,…}, sometimes with 0 included.  There is no universal definition for the natural numbers; sometimes the natural numbers are defined to be the set of non-negative integers {0, 1, 2, 3,...}, and sometimes they are defined to be the set of positive integers {1, 2, 3,...}.  Natural numbers have two primary uses—counting and ordering.  They are the simplest set of numbers in many ways, but they have some amazing properties, and are still being studied today.  The set of natural numbers is often represented by a fancy N.

The exclusion or inclusion of zero depends on your particular mathematical discipline.  Number theorists usually define the natural numbers to the set of non-negative integers.  Number theory is one of the oldest mathematical disciplines, and the Greeks, Indians, and Islamic mathematicians of antiquity studied it extensively.  Number theorists enjoy a body of theorems stretching back further than any other mathematical discipline, with the possible exception of geometry, so you might say the study of the natural numbers is steeped in tradition.  Given the comparatively recent discovery of zero, the number theorists’ exclusion of zero from the natural numbers is understandable, but to avoid confusion, most mathematicians, when referring to the set of natural numbers will name the set the non-negative integers or the positive integers, depending on whether zero is included or not.

The natural numbers have several important algebraic properties.  They are closed under addition and multiplication.  That is, for all natural numbers a and b, both a+b and a*b are also natural numbers.  Addition and multiplication on the natural numbers are both associative and commutative, and there is both an additive and a multiplicative identity (0 and 1, respectively).  The natural numbers are also ordered:  for any two natural numbers a and b, either a<b, a=b, or a>b. 

The systematic study of the natural numbers as abstract entities most likely begins with the Greeks, and is usually credited to the Greek philosopher-mathematicians Pythagoras and Archimedes. 

Some of the special types of numbers from Pythagorean number theory.

Pythagoras was an interesting character.  He is best known for the Pythagorean Theorem, though he almost certainly did not discover this theorem, as it was known and used by the Babylonians.  Pythagoras was born around 570 BC on the island of Samos in the eastern Aegean Sea, and is thought to have traveled widely, being exposed to Babylonian and Egyptian mathematics.  He was the founder of a mystical religious sect known as the Pythagoreans, heavily influenced by mathematics, though as Pythagoras himself wrote nothing down, the exact belief system of the Pythagoreans is unknown.  So much of what has been written about Pythagoras is myth and legend—for example, according to Aristotle, some ancients believed that Pythagoras had the ability to travel through space and time and could communicate with animals—that the reality is hard to know.  His sect’s fascination with numbers is evident in early music theory.  By examining the vibrations of a single string (called a monochord) they discovered that harmonious tones occurred only when the string was fixed at points along its length that were ratios of natural numbers. For instance when a string is fixed 1/2 way along its length and plucked, a tone is produced that is one octave higher and in harmony with the original. Harmonious tones are produced when the string is fixed at distances such as 1/3, 1/4, 1/5, 2/3 and 3/4 of the way along its length. Fixing the string at points along its length that were not a simple fraction produces a note that is not in harmony with the other tones.  These ratios still form the foundation of modern music theory.  The Pythagoreans carried things a bit further, to the harmony of the spheres, in which the planets and stars moved according to mathematical equations which corresponded to musical notes.

The Pythagoreans did some solid mathematical work, but they also went a little bit off the deep end.  In Pythagorean numerology numbers had mystical attributes. Odd numbers were regarded as male and even numbers as female. 

1.   The number of reason (the generator of all numbers)‏
2.   The number of opinion (The first female number)‏

3.   The number of harmony (the first proper male number)‏
4.   The number of justice or retribution, indicating the squaring of accounts (Fair and square)‏
5.   The number of marriage (the union of the first male and female numbers)‏
6.   The number of creation (male + female + 1)‏
10.  The number of the Universe.

The tetractys was an important symbol to the Pythagoreans, representing for them all possible geometric dimensions. The rows were:  1 point (0 dimensions), 2 points (a line, 1 dimension), 3 points (a plane, 2 dimensions), and 4 points (a surface, 3 dimensions) ‏.  These added up to 10, the number of the Universe.

The Tetractys, a sacred symbol for the Pythagoreans.

Modern mathematics doesn’t view natural numbers in quite the same mystical, religious light as the Pythagoreans, but there are still fascinating properties about the naturals that make them vitally important.  The factorization of a natural number is one example.  If n, m, and q are natural numbers, and n=mq, then m and q are the factors (or divisors) of n.  The process of finding factors is factoring.  The product mq is called the factorization of n.  There are many factorizations of a natural number, but there is only one prime factorization, i.e. when all the factors are prime numbers.  A positive natural number whose only factors are 1 and itself is a prime number.  Prime numbers are critically important in number theory, and have an application central to modern society:  cryptography.

There are hints that the ancient Egyptians had some knowledge of primes.  Egyptian arithmetic was recorded in the form of recipes.  The forms for the expansion of fractions were very different for prime numbers versus composite numbers.  However, the earliest records of the explicit study of primes come from Ancient Greece.  The best known example is Euclid's Elements, written around 300 BC, which contained several important theorems about prime numbers.  One of these is The Fundamental Theorem of Arithmetic, which states that every positive integer (excluding 1) can be expressed as a product of primes and this factorization is unique.  Euclid’s proof of this result and the proof of the infinitude of primes remain to this day beautiful examples of the use of reductio ad absurdum, or proof by contradiction. 

A sketch of Euclid’s proof of the Fundamental Theorem of Arithmetic (FTA) is as follows.  Assume N is the smallest number that is neither prime nor can it be expressed as a product of primes.  Since N is composite (otherwise it would be prime), N = p * q, both less than N.  If p and q are primes, then the assumption is wrong and N can be written as a product of prime factors.  If p and q are composite, then since they are smaller than N they a product of primes, as again the assumption leads to a contradiction.  Thus there is no smallest N that cannot be expressed as a product of prime numbers; hence any number can be expressed as a product of primes. QED.  (This is Latin for quod erat demonstrandum, or “what was to be demonstrated”).

Euclid’s proof of the infinitude of prime numbers is equally ingenious.  We again assume the contrary and consider the finite set of primes: p₁, p₂, …. p_n-1, p_n.  Let S = p₁* p₂* …* p_n-1* p_n.  Consider the number T=S+1, which is either prime or composite.  Since T = p₁* p₂* …* p_n-1* p_n+1.  If T is prime then we have a prime that is not on our list and is larger than any prime on our list, which is a contradiction.  If T is composite, then it can be written as the product of prime numbers by the FTA.  But clearly T is not divisible by any of the primes on our list, as that would leave a remainder of 1, so there must be a prime p>p_n, that divides T, and again we have a contradiction.  Hence there is no largest prime number, and the number of primes is infinite.  QED. 

Euclid’s work on prime numbers doesn’t stop there.  He also showed how to construct a perfect number,  a number that is the sum of its prime factors, from a Mersenne prime, which are prime numbers with the form 2^p-1, with p  prime.  The Greeks also gave us the Sieve of Eratosthenes, a simple method for finding prime numbers.

This barely scratches the surface of number theory, which, while not the focus area for every mathematician remains interesting to all of us, perhaps because the natural numbers are so fundamental.  The German mathematician Leopold Kronecker, said it best:  “God created the integers, all the rest is the work of man.”