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1 Introduction
Definition 1. A rational number is a number which can be expressed in the form a/b where a and b are integers with b > 0_._
Theorem 1. A real number α is a rational number if and only if it can be expressed as a repeating decimal, that is if and only if α = m.d 1 d 2... dkdk+1dk+2... dk+r , where m = [α] if α ≥ 0 and m = −[|α|] if α < 0 , where k and r are non-negative integers with r ≥ 1 , and where the dj are digits.
Proof. If α = m.d 1 d 2... dkdk+1dk+2... dk+r,
then (10k+r^ − 10 k)α ∈ Z and it easily follows that α is rational. If α = a/b with a and b integers and b > 0 , then α = m.d 1 d 2... for some digits dj. If {x} denotes the fractional part of x, then
{ 10 j^ |α|} = 0.dj+1dj+2.... (1)
On the other hand,
{ 10 j^ |α|} = { 10 j^ a/b} = u/b for some u ∈ { 0 , 1 ,... , b − 1 }.
Hence, by the pigeon-hole principle, there exist non-negative integers k and r with r ≥ 1 and
{ 10 k|α|} = { 10 k+r|α|}.
From (1), we deduce that 0 .dk+1dk+2 · · · = 0.dk+r+1dk+r+...
so that α = m.d 1 d 2... dkdk+1dk+2... dk+r,
and the result follows.
Definition 2. A number is irrational if it is not rational.
Theorem 2. A real number α which can be expressed as a non-repeating decimal is irrational.
Proof 1. From the argument above, if α = m.d 1 d 2... and α = a/b is rational, then the digits dj repeat. This implies the desired result.
Proof 2. This proof is based on showing that the decimal representation of a number is essentially unique. Assume α can be expressed as a non-repeating decimal and is rational. By Theorem 1, there are digits dj and d′ j such that
α = m.d 1 d 2... dkdk+1dk+2... dk+r and α = m.d′ 1 d′ 2 d′ 3... ,
where the latter represents a non-repeating decimal. Then there is a minimum positive integer u such that du 6 = d′ u. Observe that there must be a v > u such that |dv − d′ v| 6 = 9; otherwise, we would have that d′ v = 9 − dv for every v > u, contradicting that the d′ v do not repeat. Hence,
0 = |α − α| = |m.d 1 d 2... dkdk+1dk+2... dk+r − m.d′ 1 d′ 2 d′ 3... |
≥
|du − d′ u| 10 u^
∑^ ∞
j=u+
|dj − d′ j | 10 j
10 u^
∑^ ∞
j=u+
10 j^
The last expression is easily evaluated to be 0 (the series is a geometric series). Hence, we obtain a contradiction, which shows that α must be irrational.
We will begin the course by briefly discussing the irrationality of certain numbers, namely
log 10 2 , e, π, log 2 (natural logarithm of 2), and ζ(3) (to be defined). It is nevertheless convenient to define now the main topic of this course.
Definition 3. An algebraic number is a number which is a root of f (x) ∈ Z[x] for some f (x) 6 ≡ 0_. A transcendental number is a number which is not algebraic._
It should be noted that rational numbers correspond to roots of linear polynomials in Z[x]. Examples of transcendental numbers include e, π, and eπ. The number
2 is an easy example of a number which is irrational but not transcendental. There are many open problems concerning the subject. We do not know if the numbers e + π, eπ, or πe^ are transcendental. We know that log 2 and log 3 are transcendental, but we do not know if (log 2)(log 3) is. Euler’s constant is
γ = lim n→∞
( (^) n ∑
k=
k
− log n
we do not even know if it is irrational. The Riemann zeta function is defined as ζ(s) =
n=1 1 /n
s
(for R(s) > 1 ). It is known that ζ(2n) is transcendental whenever n is a positive integer, but the status of ζ(2n + 1) is not very well understood. In 1978, Apery gave the first proof that ζ(3) is irrational, and very recently it was established that ζ(2n + 1) is irrational for infinitely many positive integers n. We now turn to some irrationality examples.
Theorem 3. If the real number α is a root of
f (x) = xn^ + an− 1 xn−^1 + · · · + a 1 x + a 0 ∈ Z[x],
then α is either an integer or an irrational number.
Proof. Prove directly or by using the rational root test. Suppose α = a/b with b > 0 and (a, b) = 1, and show that b = 1 (that is that b has no prime divisors).
