6Real numbers
IA Numbers and Sets
6.4 Irrational numbers
Recall Q ⊆ R.
Definition (Irrational number). Numbers in R \ Q are irrational.
Definition (Periodic number). A decimal is periodic if after a finite number `
of digits, it repeats in blocks of k for some k, i.e. d
n+k
= d
n
for n > `.
Proposition. A number is periodic iff it is rational.
Proof.
Clearly a periodic decimal is rational: Say
x
= 0
.
7413157157157
···
.
Then
10
`
x = 10
4
x
= 7413.157157 ···
= 7413 + 157
1
10
3
+
1
10
6
+
1
10
9
+ ···
= 7413 + 157 ·
1
10
3
·
1
1 − 1/10
3
∈ Q
Conversely, let
x ∈ Q
. Then
x
has a periodic decimal. Suppose
x
=
p
2
c
5
d
q
with (
q,
10) = 1. Then 10
max(c,d)
x
=
a
q
=
n
+
b
q
for some
a, b, n ∈ Z
and
0
≤ b < q
. However, since (
q,
10) = 1, by Fermat-Euler, 10
φ(q)
≡
1 (
mod q
), i.e.
10
φ(q)
− 1 = kq for some k. Then
b
q
=
kb
kq
=
kb
999 ···9
= kb
1
10
φ(q)
+
1
10
2φ(q)
+ ···
.
Since
kb < kq <
10
φ(q)
, write
kb
=
d
1
d
2
···d
φ(q)
. So
b
q
= 0
.d
1
d
2
···d
φ(q)
d
1
d
2
···
and x is periodic.
Example. x
= 0
.
01101010001010
···
, where 1s appear in prime positions, is
irrational since the digits don’t repeat.