Since the origin of the study of numurology (known today as "number theory"), which my be rightly accredited to the Greek philosophers of old such as Euclid and Pythagoras, it has been well known and accepted that not all numbers are equal.
Numbers, in keeping with one of the fundamental goals of science itself, may very simply be classified into a number of different "categories." Some of these categories are entirely separate from others, while some very clearly overlap. Because of the nature of the accepted system of numerological classification, perhaps the simplest way to envision how it all works is by way of a simple Ven Diagram, which visibly shows the relationships between the following groups of numbers:
The Counting Numbers
1, 2, 3, 4, 5, 6, etc...
These are the first numbers that most people learn as children. They are the numbers which retain a one-to-one correspondence with real world objects. They are used, as their name implies, for counting things unit by unit.
Not included in this group are negative numbers (how can you count -3 units of something), though zero is sometimes included (as it is possible to count zero of something).
The Integers
...-3, -2, -1, 0, 1, 2, 3...
The set of numbers known as the integers include counting numbers (which can also be referred to as "non-negative integers"), but include the negative end of the spectrum as well.
Integers include only whole numbers, so decimals or fractions are not included in the set of integers. In order to be able to include partial numbers, the view would have to be widened even further to a more inclusive set:
The Real Numbers
Real numbers include every number that can possibly be positioned on a number line. This includes all the integers and everything that lies in between - both rational and irrational. This means that any fraction or decimal is included in the real number category. Whether it is something as simple as 0.5, as endless as 1/3 (0.3333333333...), or as infinitely insequential as π (3.14158...), e (2.17..) and √2.
The difference, just to clarify, between the rational and irrational numbers is that the rationals (which include the integers) are either of finite length, or repeat endlessly. Irrationals, on the other hand, continue forever while following no repeating pattern. They continually possess the power to surprise with the addition of every new digit.
Complex Numbers
Complex numbers are a combination of real numbers and imaginary numbers. The imaginary number is denoted by "i" and simply refers to √(-1). A complex number, then, is any number in the form: A + Bi, where both A and B are real numbers.
By way of simple algebra, this means that any number, really, can be written as a complex number.
"4" is a complex number in the form: 4 + (0)i
"√(-4). " is a complex number in the form: 0 + 2i
In this sense, complex numbers need not explicitly contain an imaginary component.
Further Defining Number Sets
These are just some of the most basic forms of numerical "sets" that can be created, and which overlap each other in various ways. Mathematicians, of course, are a creative bunch, and have certainly found many, many other ways of categorizing numbers - there is the set of odd numbers, the set of even numbers, the set of prime numbers, the set of even prime numbers (which consists of just a single number: "2"), the set of odd primes, the set of perfect squares... the list can go on forever.
Mathematicians have even devised ways of categorizing sets of numbers by the cardinality of their infinite nature, thanks to the nineteenth century realization that not all infinities are equal.
So, beginning with just a basic idea of some of the most essential numerical classifications, one can easily see how this train of thought and reasoning can lead to ever more complicated mathematical realizations.
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