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4.4 The Real Numbers

The rational numbers are still not complete with respect to the basic operations: there does not exist an x in Q with
x * x = 2.
i.e., there is no square root of 2 in Q.

Proof  We show
forall x in Q: x * x != 2.
Take arbitrary x in Q. We assume (1) x * x = 2 and show a contradiction. From the construction of Q, we know x = a/b for some a in Z and b in Z> 0 such that (2) N(a) and N(b) are relatively prime. We have a *Z a/b *Z b = 2 and thus (from now on we operate in Z and drop the corresponding subscripts):
(3) a * a = 2 * b * b.
From (3) we know N(2) | N(a*a) and thus also (4) N(2) | N(a) (a proposition that has to be proved extra). Therefore there exists some c in Z such that
(5) a = 2*c.
From (3) and (5) we have 2*c*2*c = 2*b*b, i.e., 2*c*c = b*b, thus (6) N(2) | N(b*b) and therefore (7) N(2) | N(b). (4) and (7) contradict (2).

We will now introduce the domain R of real numbers which provides a solution for the equation stated above. While there exist also relatively concrete set-theoretic constructions of R (e.g. as a set of infinite decimal fractions), the usual construction is another instance of the kind that will be stated for Z and Q in Chapter Relations.

In this section, we will characterize R in an inconstructive way by a number of axioms.

Axiom 4 (Real Numbers) The domain of  real numbers (reelle Zahlen) is a set R with constants 0 in R and 1 in R, functions +: R x R -> R, -: R -> R, *: R x R -> R, -1: R\{0} -> R and predicate <= subset R x R such that, for all x in R, y in R, and z in R, the following holds:
Field Axioms (Körperaxiome)
x + y = y + x,
x + (y + z) = (x + y) + z,
x + (-y) = 0,
x * y = y * x,
x * (y * z) = (x * y) * z,
x * (x-1) = 1,
x * 1 = x,
x * (y + z) = (x * y) + (x * z);
Order Axioms (Ordnungsaxiome)
x <= x,
x <= y \/ y <= z,
(x <= y /\  y <= z) => x <= z,
x <= y => x + z <= y + z,
(x <= y /\  0 <= z) => x <= y * z.
Completeness Axiom (Vollständigkeitsaxiom)

Every non-empty subset of R that has an upper bound also has an upper limit.

forall S subset R: S != 0 =>
   ((exists B in R: B is upper bound of S) =>
      (exists L in R: L is upper limit of S)).

The predicates used in the completeness axiom are defined below.

Definition 47 (Bounds and Limits) An  upper bound (obere Schranke) of S is as large as every element of S:
B is upper bound of S : <=> forall x in S: x <= B.
An  upper limit (obere Grenze) of S is the smallest upper bound of S:
L is upper limit of S : <=>
   L is upper bound of S /\ 
   (forall M: M is upper bound of S => B <= M).

Intuitively, the completeness axiom makes the real numbers much more "dense" than the rational numbers where above property does not hold.

Example  Take the set S := {si: i in N} where si is the sum of the first i components of the sequence
a: N -> Q
a(i) = 1/i!
and ! denotes the factorial function, i.e.,
a = [1/1, 1/1, 1/1*2, 1/1*2*3, 1/1*2*3*4, ...]
S = {1/1, (1/1+1/1), (1/1+1/1+1/1*2), (1/1+1/1+1/1*2+1/1*2*3), ...}.
S has an upper bound 3/1 but no upper limit in Q. However, it has an upper limit in R traditionally denoted by the constant e (= 2.718281828...).

Then we have the following result which implies that the problem stated at the begin of this section can be solved.

Proposition 52 (Existence of Real Roots) In R every non-negative number has an n-th root, i.e., for R >= 0 := {x in R: x >= 0} the following holds:
forall a in R >= 0, n in N>0: exists x in R: xn = a.

Since we can show that, for every x in R, x * x = x2N, this implies the existence of a square root of 2 in R.

Definition 48 (Real Root Function) 
sqrtn(x) := such y: xn = y
sqrt(x) := sqrt2N(x).

As a consequence of Proposition Existence of Real Roots, we have

forall a in R >= 0, n in N>0: (sqrtn(a))n = a.

Contiguous subsets of the reals are often denoted as shown below.

Definition 49 (Intervals) We define the following  intervals (Intervalle):
[a, b] := {x in R: a <= x <= b};
[a, b[ := {x in R: a <= x < b};
]a, b] := {x in R: a < x <= b};
]a, b[ := {x in R: a < x < b}.
For all a in R and b in R, the intervals [a, b] and [a, b[ are called  left closed (linksseitig abgeschlossen), and the intervals [a, b] and ]a, b] are called  right closed (rechtsseitig abgeschlossen).

Intervals are contiguous in the sense that if two values are in an interval S, then also every intermediate value is in this interval, i.e.,

(forall x in S, y in S: x <= y => (forall z in R: x <= z <= y => z in S)).

The set R "embeds" Q in the sense that will be discussed in Section Embedding Sets. However, unlike the relationship between Z and Q (which are essentially of the "same" size), there are considerably "more" reals than there are rationals. Actually we will see in Section Counting Set Elements that there are so many elements of R that this set cannot be represented in a computer even with an infinite number of memory cells.

Consequently the Logic Evaluator does in file real.txt not implement the reals but fakes real number operations (without square root computation) by rational number operations .

In Chapter Relations we will sketch a "structurally equal" construction that is an instance of the general technique that can be applied for the construction of Z, and Q.

Author: Wolfgang Schreiner
Last Modification: October 4, 1999

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