My title - Departamento de Matemática da Universidade do Minho

4
NÚMEROS E DINÂMICA 21
4 Números e dinâmica
“317 is a prime, not because we think so, or because our minds are shaped in one way
rather than another, but because it is so, because mathematical reality is built that way.”
4.1 Expansão decimal e multiplicação ×10
Ref.
[HW59], chapter IX.
G.H. Hardy, A Mathematician’s Apology, 1940.
Expansão decimal.
When children we learn to represent numbers as decimals, like
3.14159265358979323846264338327950288419716939937510 . . .
Of course, there is nothing especial with the number 10, it is but the number of fingers in our
hands. Any other integer d ≥ 2 would do. Representing a non-negative (for simplicity) real
number x ∈ R + in base 10 means writing x as the sum of a convergent series
x = X m . . . X 2 X 1 X 0 .x 1 x 2 x 3 . . .
= X m · 10 m + · · · + X 2 · 10 2 + X 1 · 10 + X 0 + x 1
10 + x 2
10 2 + x 3
10 3 + . . .
m∑
∞∑
= X n · 10 n + x n · 10 −n
n=0
n=1
where X n , x n ∈ {0, 1, 2, . . . , 9} and m ≥ 0. Some representations terminate, i.e. have x n = 0
starting from some n ≥ N, and some others are recurring, i.e. of the form
X m . . . X 2 X 1 X 0 .x 1 x 2 . . . x k−1 x k . . . x k+n
(and of course a terminating decimal is a recurring one with recurring word 0). The representation is
unique, hence there is a bijection between R and the space of infinite words X m . . . X 2 x 1 X 0 .x 1 x 2 x 3 . . .
as above, if we do not admit recurrent 9’s, i.e. if we substitute . . . x k−1 9 with . . . (x k−1 +1)0 (where
we assume x k−1 ≠ 9).
The finite sum
m∑
[x] = X n · 10 n ∈ Z
n=0
is the integral part of x, the largest of those integers n such that n ≤ x. The possibly infinite sum
{x} = 0.x 1 x 2 x 3 · · · =
∞∑
x n · 10 −n ∈ [0, 1)
is the fractional part of x, the difference {x} = x − [x]. Consequently, [x] + {x} = x.
Division algorithm. The iterative scheme to obtain the decimal representation of a rational
number is the “division algorithm” that we learn when children. Consider a positive rational
number x = p/q with p, q ∈ N:
p
q = x 0.x 1 x 2 x 3 . . . .
The integer [x] = x 0 is “the number of times q is contained in p”, i.e. the unique integer such that
n=1
p = x 0 · q + r 0
for some rest r 0 which is an integer 0 ≤ r 0 < q. Hence, p/q = x 0 + r 0 /q and 0 ≤ r 0 /q < 1. The
“geometric” meaning of x 1 is that the point r 0 /q lies between 0.x 1 and 0.x 1 + 0.1. Multiplying by
10 and then by q this means that
x 1 · q ≤ 10 · r 0 < x 1 · q + q