The hierarchy of numbers

The hierarchy of numbers
The numbers used in mathematics form a hierarchy, leading from the simplest numbers to those that are more abstract. Below, we introduce the most important classes of numbers.
•     Natural numbers
•     Integers
•     Rational numbers
•     Real numbers
•     Complex numbers

A. Natural Numbers
These are the numbers 1,2,3,4,…, that is, the numbers that we use every day to count with.
Important information
1. Sometimes zero is also considered a natural number, though most writers consider 1 to be the first natural number.
2. For every pair of natural numbers m and n, the sum m+n and the product m•n are again natural numbers. However, this does not hold in general for differences and quotients.
3. Although these are the simplest numbers, there are many difficult problems and open questions concerning them. There is an entire mathematical subdiscipline devoted to them, called number theory.
4. When mathematicians want to work with the natural numbers in a very precise way, they introduce them axiomatically: the natural numbers are then a set (in fact the smallest such set) that contains the number 1 and has the property and for every element x in the set, its successor x is also in the set; behind the scenes, x is actually the number x+1.  One then postulates the Peano axioms:
- Distinct numbers have distinct successors.
- Every natural number other than 1 is the successor of some natural number. The number 1 is expressly not the successor of any number.
- The induction axiom holds: If the number 1 possesses a certain property and this property holds for every x whenever it holds for x, then all the natural numbers have this property.
This axiomatic system is the basis for everything that is known about numbers.

B. Integers
To form the integers, the natural numbers are extended to include the number 0 and the numbers  –1, –2, –3,…. .Thus the integers comprise the numbers …, –2, –1, 0, 1, 2, 3, …, where the dots indicate that the numbers extend in each direction.
Important information
1. Integers can be added, subtracted, and multiplied: the result is again an integer. Addition and multiplication have many properties that one knows about from one’s schooldays: the commutative, associative, and distributive laws, and that 0 is the identity element for addition, and 1 the identity for multiplication.
The quotient of two integers is not necessarily an integer.
2. Natural numbers are a special type of integer.

C. Rational Numbers
The rational numbers constitute the set of all quotients of the form m/n, where m is an integer and n is a natural number: Thus 4/87 and –345/777 are examples of rational numbers.
Important information
1. Every integer m can be written as m/1, and therefore every integer is also an example of a rational number. Of course, there are rational numbers that are not integers, such as 1/2 and –7/19.
2. In the realm of natural numbers, all four operations are allowed: adding, subtracting, multiplying, or dividing (except by zero) two rational numbers leads invariably to another rational number. At the same time, the rational numbers continue to enjoy the commutative, associative, and distributive properties.

D. Real Numbers
We have now reached a difficult juncture. It took a long time in the history of mathematics, in fact until the middle of the nineteenth century, before mathematicians were able to work with real numbers in a sufficiently precise way.
For our purposes, we are going to simplify our lives a bit and make the following definition: real numbers are those that can be written in a (possibly terminating) decimal expansion: for example 13.1212121212…, or 4.5, or –896626.4142894110….
The real numbers include practically all the numbers that one encounters in school and that are necessary in practical applications. The number ?, the fifth root of 323, the base of the natural logarithm e, all are real numbers.
Important information
1. Every rational number has a (finite or periodic) decimal expansion. Therefore, all rational numbers are special examples of real numbers.
2. It is not easy to see that there are indeed real numbers that are not rational. The most famous example of such a number is surely the number whose square is 2, that is, the square root of 2.
Moreover, this number is one that is encountered in elementary geometry: according to the Pythagorean theorem, the diagonal of the square with side length 1 is precisely equal to the square root of 2.
3. Real numbers that are not rational are called irrational. Thus as just noted, the square root of two is an irrational number.

E. Complex Numbers
By definition the square root of a number a is a number r with the property that the square of r is equal to a. Thus 10 is a square root of 100, and –11 is a square root of 121. Every positive real number has exactly two square roots, one of which is positive and the other negative. Therefore we may agree to call the positive of the two roots the square root. In this sense, 1.414213… is the square root of 2, and 7 is the square root of 49.
Since the square of every real number is positive or 0, no negative real number can have a real square root. However, it is possible to extend the domain of real numbers to the even larger domain of complex numbers, in which all the real numbers do have square roots.
If you imagine the real numbers as points on a line, say the x-axis in a coordinate system, then the complex numbers may be viewed as the set of all the points in the plane. We may then define for complex numbers an addition (which functions like vector addition) and a multiplication (which takes some getting used to) so that all the usual properties of numbers are satisfied: the commutative, associative, and distributive laws. We note in particular that there is a complex number whose square corresponds precisely to the point –1 on the x-axis. The number that is associated with the coordinates (0, 1) has this property. It is called the imaginary unit and is denoted by the letter i.
Thus i is a square root of –1. Furthermore, –i (this number corresponds to the point (0, –1) is also a square root of –1. In contrast to the case of the real numbers, here there is no obvious preference for one of the roots over the other. (Therefore, mathematicians are not entirely happy about saying ‘i is the square root of –1’; just as in everyday speech we don’t say that we ran into the brother of Irene if she in fact has two brothers.)
Complex numbers are a fertile domain for the existence of square roots: the equation x2 – a = 0 always has a solution for any complex number a.
Remarkably, every equation of the form
a0+a1x+a2x2+ ••• +anxn = 0
with arbitrary (real or complex) coefficients can be solved in the domain of complex numbers.
Because of their importance in solving such algebraic equations, these numbers were employed intuitively centuries ago. But for a long time, mathematicians had no secure understanding of how to operate in this domain. One spoke of ‘false roots’ and so on. It was only in the nineteenth century that methods were proposed for eliminating such difficulties.
Today, professional mathematicians and those who use mathematics in applied fields consider complex numbers to be just as valid as the natural numbers 1, 2, 3, ….