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.