22 I The Expanding Universe of Numbers
This completes the first stage of Dedekind’s construction. In the second stage we
pass from cuts to real numbers. Intuitively, a real number is the difference of two cuts.
We will deal with the second stage rather briefly since, as has been said, it is completely
analogous to the passage from the natural numbers to the integers.
On the setP×Pof all ordered pairs of cuts an equivalence relation is defined by
(A,B)∼(A′,B′) ifA+B′=A′+B.We d e fi n e areal numberto be an equivalence class of ordered pairs of cuts and, as is
now customary, we denote the set of all real numbers byR.
Addition and multiplication are unambiguously defined by
(A,B)+(C,D)=(A+C,B+D),
(A,B)·(C,D)=(AC+BD,AD+BC).They obey the laws(A2)–(A5),(M2)–(M5)and(AM1)–(AM2).
A real number represented by (A,B)issaidtobepositiveifB<A. If we denote
byP′the set of all positive real numbers, then(P1)–(P3)continue to hold withP′in
place ofP. An order relation, satisfying(O1)–(O3), is induced onRby writinga<b
ifb−a∈P′. Moreover, anya∈Rmay be written in the forma=b−c,where
b,c∈P′. It is easily seen thatPis isomorphic withP′. By identifyingPwithP′,
we may regard bothPandQas subsets ofR. An element ofR\Qis said to be an
irrationalreal number.
Upper and lower bounds, and suprema and infima, may be defined for subsets of
Rin the same way as for subsets ofP. Moreover, the least upper bound property(P4)
continues to hold inR. By applying(P4)to the subset−S={−a:a∈S}we see
that if a nonempty subsetSofRhas a lower bound, then it has a greatest lower bound.
The least upper bound property implies the so-calledArchimedean property:
Proposition 19For any positive real numbers a,b, there exists a positive integer n
such that na>b.
Proof Assume, on the contrary, thatna ≤ bfor everyn ∈ N.Thenbis an
upper bound for the set{na:n∈N}.Letcbe a least upper bound for this set. From
na≤cfor everyn∈Nwe obtain(n+ 1 )a≤cfor everyn∈N. But this implies
na≤c−afor everyn∈N.Sincec−a<candcis a least upper bound, we have a
contradiction.
Proposition 20For any real numbers a,b with a<b, there exists a rational number
c such that a<c<b.
Proof Suppose first thata≥0. By Proposition 19 there exists a positive integern
such thatn(b−a)>1. Thenb>a+n−^1. There exists also a positive integermsuch
thatmn−^1 >a.Ifmis the least such positive integer, then (m− 1 )n−^1 ≤aand hence
mn−^1 ≤a+n−^1 <b. Thus we can takec=mn−^1.
Ifa<0andb>0 we can takec=0. Ifa<0andb≤0, then−b<d<−afor
some rationaldand we can takec=−d.
Proposition 21For any positive real number a, there exists a unique positive real
number b such that b^2 =a.