The real numbers, as we know them, are all the numbers that can be found on the number line. Mathematically, however, they must be constructed.
The simplest mechanism is to simply directly develop the entire algebra of reals through axioms, as we did the natural numbers. However, for the purpose of relating reals to structures we already know about, namely the integers and rationals, we will build upon them.
Recall that we began with Peano's axioms to define natural numbers, used natural numbers to define integers, and finally defined rational numbers in terms of integers. We will build real numbers by building on this scaffolding. We will define real numbers in terms of rational numbers.
To do so, we will use a method proposed by Richard Dedekind in 1858 and published in 1872.
A Dedekind “cut” is a partitioning of the rational numbers into two non-empty sets:
FIGURE
The term “cut” is meant to illustrate that the precise point of the cut cannot be uniquely identified — it disappears.
Formally, a Dedekind cut is a set with the following properties:
We define real numbers as Dedekind cuts
Consider the following set: \[ C = \{ x | x \in \rationals~AND~x \lt 0~OR~x^2 < 2\} \]
$C$ is the set of all negative rationals and all positive rationals whose squared value is less than 2. On the number line, $C$ is the set of all rationals that lie to the left of the arrow in the figure below:
FIGURE
We will call the above cut $\sqrt(2)$. Note that by this construction, the symbol $\sqrt(2)$ has no significance other than that it is the name of this particular cut.
Here are a few more cuts and their names. \[ \sqrt(3) = \{ x | x \in \rationals~~AND~~x \lt 0~~OR~~x^2 \lt 3\} \\ \sqrt[3](7) = \{ x | x \in \rationals~~AND~~x^3 \lt 7\} \\ \pi = \{ x | x \in \rationals~~AND~~x \lt \text{area of a circle of unit radius}\} \]
Most cuts will not have a nice formula such as the above, as we will see shortly. The key point is is the following:
Definition: Every cut of $\rationals$ is a real number. We will call the set of all such cuts the set of reals and denote it by $\reals$.
The “value” of a “real” number is obtained through its order relation with other reals.
Order: We define an order relaton “$\lt$” as \[ \alpha \lt \beta ~~~\text{ for }~~~ \alpha, \beta \in \reals ~~~\text{ iff }~~~ \alpha \subset \beta \]
A real number $\alpha$ is said to be lesser than another real $\beta$ if $\alpha$ is a strict subset of $\beta$.
Note that the cut we call $\sqrt(2)$ is a subset of the cut we call $\pi$, so by the above definition $\sqrt(2) \lt \pi$. In fact, the above notion of order satisfies all of our intuitive notion of order among real numbers.
Operations are well defined (Addition, multipication)
Density of reals: Between any to reals is a rational. Complex proof
Density of reals: Between any to reals is an IRrational. Simpler proof: x + (y-x)/sqrt(2)
Density of reals corollary: Between any to irrationals is a rational. Actually: Countably infinitely many rationals betweentwo irrationals
Density of reals: Between any to rationals is an irrational proof: y-x/sqrt(2) < y-x. Add to x. Actually: Uncountably infinite irrationals betweentwo rationals
More generally: Any open interval (a,b) is uncountable.
A more formal definition must not draw upon the concept of division, which is not defined for integers, which do not have multiplicative inverses. Instead we will define rationals in terms of pairs of numbers $p$ and $q$, specifying that $q$ cannot be 0.
Rational numbers are instances of constructible numbers. Constructible numbers are numbers that can be built with rulers and compasses.
The ancient greeks, who discovered the rationals, originally assumed that all constructible numbers were rational. They were very shocked to realize that some constructible numbers were not rational.
Rational numbers are defined over ordered pairs $(p,q)$ from the set $\integers \times \integers^*$, where $\integers^* = \integers\setminus\{0\}$, the set of non-zero integers (represented here as the set of integers minus the elements of the set containing only zero).
Formally, a rational number is an equivalence class $[(p,q)]$ of ordered pairs from $\integers \times \integers^*$, defined over the equivalence relation $(p,q) \sim (n,m)$ if $pm = nq$.
Note that we didn't simply define rationals as ordered pairs $(p,q)$, since such a definition would treat every ordered pair of integers as unique, so for instance, $(3,6)$ and $(1,2)$ would be treated as being distinct, whereas we know that they are both the same rational number. Our definition of rationals must also account for this.
In order to do so, we had to first define an equivalence relation for rationals. We know that informally two rational numbers $(p,q)$ and $(n,m)$ are identical if $\frac{p}{q} = \frac{n}{m}$, which is identical to saying that $pm = nq$ (since $q$ and $m$ are not $0$ by definition). We build our definition of rationals on this equivalence.
The set of all rationals is denoted by the symbols $\rationals$.
For formal notation, we will specify any rational as $[(p,q)]$. We will also use the informal notation $\frac{p}{q}$.
More generally, we will simply use a symbol such as $x$, instead of explicitly specifying the elements of the ordered pair, and denote that it is a rational number by saying $x \in \rationals$. We will explicitly use the numerator-denominator representation only if it is needed.
We define a multiplication operation &lquo;$\times$” between two rational numbers $\frac{p}{q}$ and $\frac{n}{m}$ as follows. \[ \frac{p}{q} \times \frac{n}{m} = \frac{pn}{qm} \]
We can verify that this is, in fact, a well-defined operation. For it to be well defined, we must find the following:
This is easily tested. By our definition, $\frac{x}{y} \times \frac{r}{s} = \frac{xr}{ys}$. We must verify that $\frac{xr}{ys} = \frac{pn}{qm}$.
For this to be true, we require that $[(xr,ys)] \sim [(pn,qm)]$ according to the equivalence relation for rationals. This means that we require $ xrqm = yspn$.
This is easy to show. Consider $xrqm$. We have $\frac{x}{y} = \frac{p}{q} \Rightarrow yp = xq$. We have $xrqm = xqrm$. Replacing $xq$ by $yp$, $xrqm = yprm$. Similarly from $\frac{r}{s} = \frac{n}{m}$ we get $sn = rm$. Using this equality, $xrqm = yprm = ypsn = yspn$.
Note that this definition of multiplication was defined such that it had this “well-defined” property.
We define the addition operation “+” in the following manner. For any two rational numbers $\frac{p}{q}$ and $\frac{r}{s}$, the addition of the two is defined by \[ \frac{p}{q} + \frac{r}{s} = \frac{ps + rq}{qs} \]
It is easy to verify that the above definition is well defined.
Consider any rational number $\frac{x}{y}$ that is equivalent to $\frac{p}{q}$ (i.e. $qx = py$). Similarly consider any rational number $\frac{n}{m}$ that is equivalent to $\frac{r}{s}$ (i.e. $ns = rm$). By the above definition, their sum is given by \[ \frac{x}{y} + \frac{n}{m} = \frac{xm + ny}{ym} \]
To show that our definition is well defined, we only need to show that $ \frac{xm + ny}{ym} = \frac{ps + rq}{qs}$. To show this we must show that $(xm +ny)qs = (ps + rq)ym$.
Expanding the right hand side, $RHS = (ps + rq)ym = psym + rqym$.
The left hand side becomes $(xm + ny)qs = xmqs + nyqs$. Using the equivalences $qx = py$ and $ns = rm$, we can rewrite the LHS as \[ LHS = xmqs + nyqs = xqms + nsyq = pyms + rmyq = psym + rqym = RHS. \]
If one were to define addition analogously to multiplication, we would define it as \[ \frac{p}{q} + \frac{n}{m} = \frac{p+q}{n+m}, \] but a quick inspection shows this to be not well defined.
For it to be well defined, we should be able to add any rational number that is equivalent to $\frac{p}{q}$ and add it to any rational number that is equivalent to $\frac{n}{m}$, and the result must be equivalent to $\frac{p+q}{n+m}$. We can show that this isn't always true with a simple example.
Consider $\frac{1}{2} + \frac{3}{4}$. According to this dubious definition, we would get the sum to be $\frac{1+3}{2+4} = \frac{4}{6}$.
Now consider $\frac{2}{4} + \frac{3}{4}$. The above definition of addition would give the sum as $\frac{5}{9}$. But $\frac{2}{4} = \frac{1}{2}$. If the definition of addition were well defined, we should have $\frac{5}{9} = \frac{4}{6}$ But $5\times 6 \neq 4\times 9$, so $\frac{5}{9} \neq \frac{4}{6}$. The operation is not well defined.
Arithmetic operations over the rationals have most of the properties of the natural numbers.
The identity properties of rationals must be more carefully defined than for integers. Moreover rationals include at least one identity operation that is not permitted in the integers.
Addition:
Multiplication:
The rationals are closed under exponentiation to integer powers. For any non-negative $n \in \integers$, $n \geq 0$ \[ \left(\frac{x}{y}\right)^n = \frac{x^n}{y^n} \]
For $n \lt 0$, the following holds only if $x \neq 0$ \[ \left(\frac{x}{y}\right)^{-n} = \frac{y^n}{x^n} \]
Positive Numbers: A rational number $\frac{m}{n}$ is defined as being positive if $m\times n \gt 0$. Note that here we are ussing multiplication and “$\gt$” as defined for integers, not rationals. As a matter of fact, we are yet to define order relations for rationals.
Negative Numbers: Any rational number $\frac{m}{n}$ is negative if $m\times n \lt 0$.
We define an order relationship $\lt$ between two rationals analogously to integers.
For any two rational numbers $x, y \in \rationals$, we $x \lt y$ if $x - y$ is a positive rational number. The order relation “$\lt$” is generally read as “ less than ”.
Notation: We will write $x \gt y$ (where $\gt$ is read as “greater than”) if $y \lt x$. We will say $x \leq y$ if x may either be less than or equal to $y$. Similarly $x \geq y$ means that $x$ may either be greater than or equal to $y$.
The order naturally permits us to place the rationals on a line. FIGURE.
The set of rationals $\rationals$ extend the set of integers, $\integers$. Alternately stated, $\integers$ is embedded in $\rationals$, i.e. there is a subset $\integers_\rationals$ of $\rationals$ that has a one-to-one correspndence with $\integers$, and the order, equality and other relationships within this subset $\integers_\rationals$ are identical to those of $\integers$.
The set $\integers_\rationals$ is easy to define: for each integer $n$ in $\integers$ the corresponding rational number in $\integers_\rationals$ is $\frac{n}{1}$.
We can now verify that the addition and multiplication operations on the rational equivalents of the integers are consistent with the corresponding operations on the integers.