The Natural numbers, N, are extremely important in the world of mathematics. They are often referred to as counting numbers. An easy way to think about the Natural numbers is as positive whole numbers or Integers. Some mathematicians consider 0 to be an element of the Natural numbers, but others such as myself do not consider them apart. For example, 1, 2, 3, are all elements of the Natural numbers.
There are two main purposes of the Natural numbers: counting and ordering. For example, there are 22 people in Ms. Jones class or there are four tables at the dinner table. Those are examples of counting. Here are some examples of ordering. Jim is the tallest boy in the class or Stacy finished second in the race.
The Natural numbers are often studied in number theory; a branch of mathematics related to the property of numbers and in particular the Integers. Since the Natural numbers are a subset of the Integers, it follows that the Natural numbers are a part of number theory.
One area of number theory that the Natural numbers are important is with divisibility. A divisor or factor of an Integer n, is an Integer which evenly divides n without leaving a remainder. For example, 2 is a divisor of 4 since 2 evenly divides 4 and does not leave a remainder. On the other hand, 3 is not a divisor of 5 because 3 does not evenly divide 5 and there is a remainder other than 0.
One of the most important topics of number theory that deals with the Natural numbers comes from the Greek mathematician, Euclid, The Father of Geometry. It is known as Euclid’s first theorem or Euclid’s Lemma. It states: If p is prime and p | ab, then either p | a or p | b. This is read as: if p is prime and p divides a times b, then either p divides a or p divides b. For example, if p is 5 and a*b equals 3*20, then p clearly divides 3*20. 5 does not divide 3, but 5 does divide 20.
This is extremely important because it is fundamental in The Fundamental Theorem Of Arithmetic, which states that any integer can be written as a unique product of primes. For example, 6 is equal to 2*3 and that is the only way to represent 6 as a product of primes.
The Natural numbers are constantly being studied and are always turning up new and important results.