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commensurable numbers∗
pahio†
2013-03-22 1:17:22
Two positive real numbers a and b are commensurable, iff there exists a
positive real number u such that
a = mu,
b = nu
(1)
with some positive integers m and n. If the positive numbers a and b are not
commensurable, they are incommensurable.
Theorem. The positive numbers a and b are commensurable if and only if
m
their ratio is a rational number
(m, n ∈ Z).
n
Proof. The equations (1) imply the proportion
m
a
=
.
b
n
(2)
Conversely, if (2) is valid with m, n ∈ Z, then we can write
b
a = m· ,
n
b
b = n· ,
n
which means that a and b are multiples of
b
and thus commensurable. Q.E.D.
n
Example. The lengths of the side and the diagonal of square are always
incommensurable.
0.1
Commensurability as relation
• The commensurability is an equivalence relation in the set R+ of the positive reals: the reflexivity and the symmetry are trivial; if a : b = r and
b : c = s, then a : c = (a : b)(b : c) = rs, whence one obtains the transitivity.
∗ hCommensurableNumbersi
created: h2013-03-2i by: hpahioi version: h40760i Privacy
setting: h1i hDefinitioni h12D99i h03E02i
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1
• The equivalence classes of the commensurability are of the form
.
[%] := {r% .. r ∈ Q+ }.
• One of the equivalence classes is the set [1] = Q+ of the positive rationals,
all others consist of positive irrational numbers.
• If one sets [%] · [σ] := [%σ], the equivalence classes form with respect to
this binary operation an Abelian group.
2