Section 1.1 The various types of numbers
Subsection 1.1.1 \(\mathbb N, \mathbb Z\) and \(\mathbb Q\)
We assume knowledge of the natural numbers:
\begin{equation*}
{\mathbb N}=\{1,2,3,\ldots\},
\end{equation*}
the integers:
\begin{equation*}
{\mathbb Z}=\{0, \pm 1, \pm 2, \ldots\},
\end{equation*}
and the rational numbers \(\mathbb Q\text{,}\) consisiting of all fractions that have an integer in the numerator, and a natural number in the denominator, such as \(2/3\) or \(-1/7\text{.}\) So we can write the definition of natural numbers in set notation as
\begin{equation*}
{\mathbb Q} =\left\{\frac{n}{m}: n\in {\mathbb Z}, m\in {\mathbb N}\right\}.
\end{equation*}
Note that in the definition of rational numbers, the denominator could be 1, and since \(n/1=n\text{,}\) all the integers are automatically also rational numbers. So every natural number is an integer, and every integer is a rational number. In the discussion following Definition 1.2.4, we will see how to write this statement using symbols.
The natural numbers (sometimes also called positive integers) can be added and multiplied. The number 1 is rather special because it has the property that \(x\cdot 1=x=1\cdot x\) for all natural numbers \(x\text{.}\) The natural numbers cannot always be subtracted, for example \(3\) and \(5\) are natural numbers, but \(3-5\) is not.
The integers can be added, multiplied and subtracted. Beside 1, they have another special element, \(0\text{,}\) that has the property that \(x+0=x=0+x\) for all integers \(x\text{.}\) They cannot always be divided. For example, \(2\) and \(3\) are integers, but \(2/3\) is not.
The rational numbers can be added, multiplied, subtracted, and divided (with the only exception that we cannot divide by \(0\)). So we can do all the usual operations of arithmetic and algebra with the rational numbers. We say that they form a field, a concept that we will describe more precisely in the next chapter. In this course, the natural number, integers and rational numbers will be considered as already defined. 1
Subsection 1.1.2 Order
Another important concept that applies to all three types of numbers is that of order. Given any two numbers \(x\) and \(y\) (whether they are natural, integers, or rational), unless the two numbers are the same, one must be smaller than the other. In symbols, if \(x\neq y\text{,}\) then either \(x\lt y\) or \(y\lt x\text{.}\) So we say that the rational numbers form an ordered field, and this will be the main object of study of the next chapter.
Subsection 1.1.3 Incompleteness of \(\mathbb Q\)
We already said that we can do all four operations with the rational numbers. But there is still something missing. Suppose we want to build a fence for a field (as in a corn or strawberry field, not the math concept of field mentioned earlier!) so that it will be a square of area \(A\) square feet. Because the area of a square of side \(x\) is \(A=x^2\text{,}\) this means that if we want to know how long the side of the square is, we need to solve the equation \(x^2=A\text{,}\) and the answer is \(x=\sqrt{A}\text{.}\)
So being able to find the square root of a positive number is of practical use. But already the ancient Greeks, in the 5th century BC, discovered that, for example, there can be no rational number whose square is \(2\text{.}\) In other words, \(\sqrt{2}\) cannot be a rational number (we will soon see a proof of this fact). This means that we need to go beyond the rational numbers, and consider a larger set of numbers, called the Real numbers. These are the numbers that give the name “Real Analysis” to this course.
This is not the case in more advanced treatments, where the natural numbers are defined purely in terms of sets (see this lecture for example), then the integers and rational numbers as derived concepts.
