Fractions continues

a0+1
a1+1
a2+1
a3+
a0, a1, . . . a0
p
qp q
p q
p
q
77708431
2640858 ≈29,425
206
7≈29,428

a0+1
a1+1
a2+1
a3+1
+1
ak
a0, . . . , akk≥1a1, . . . , ak
[a0, . . . , ak]
n∈Zk= 0
n=a0k= 0 a0
1
2k= 1 a0= 0 a1= 2
4
3
4
3= 1 + 1 3k= 1 a0= 1 a1= 3
3
4
3
4=1
4
3
=1
1 + 1
3
,
k= 2 a0= 0 a1= 1 a2= 3
α
a b α =a
b
•k= 0
•k= 1 a0+1
a1=a0a1+1 a1
•k= 2
a0+1
a1+1
a2
=a0+a2
a1a2+ 1 =a0(a1a2+ 1) + a2 a1a2+ 1 .

[a0, . . . , ak]
a0, . . . , ak
α=24
17
24 17
24 = 1 ×17 + 7
17 = 2 ×7+3
7=2×3+1
α
α=24
17 =1×17 + 7 17 = 1 + 7 17
7
17 =1
17
7
=1
2×7+3
7
=1
2 + 3
7
,
α= 1 + 1
2 + 3
7
.
3
7=1
7
3
=1
2×3+1
3
=1
2 + 1
3
,
α= 1 + 1
2 + 1
2 + 1
3
.
1
a
p
qa0, . . . , ak−1
p q b0, . . . , bk
p=a0q+b0
q=a1b0+b1
b0=a2b1+b2
bk−2=ak−1bk−1+bk
p q
bk= 1

p
q=a0+b0 q
=a0+1
a1+b1
b0
=a0+1
a1+1
a2+b2
b1
ai+bi
bi−1
bi6= 1
bi
bi−1
=1
bi−1
bi
bi−1bi
k ak−1+1
bk−1bk= 1
ak=bk−1
p
q= [a0, a1, . . . , ak].
a0+1
a1+1
a2+1
a3+
a0a1, a2, . . .
[a0, a1, a1, . . .]
n n
rn= [a0, . . . , an] = a0+1
a1+1
a2+1
a3+1
+1
an
α
α
√2π5 √7+2

x x x
[x]x x −[x]{x}
0≤ {x}<1.
n[n] = n{n}= 0 [2,3] = 2 {2,3}= 0,3
[−1,9] = −2{−1,9}=−1,9−(−2) = 0,13
2= [1,5] = 1 3 2={1,5}= 0,5
π≈3,14 [π] = 3
α α
[α]< α {α} 6= 0 a0= [α]α1=1
{α}>1α
α=a0+1
α1
.
α1α1=a1+1
α2a1
α2>1α2αi>1αi
α α αi
ai= [αi]αi+1 =1
{αi}>1
αiα
a0, a1, a2, . . . ai≥1i≥1
α
π
•π≈3,141592 a0= 3 α1≈1
0,141592 ≈7,062513
•a1= 7 α2≈1
0,062513 ≈14,996594.
•a2= 14 α3≈1
0,996594 ≈1,003417
•a3= 1 α4≈1
0,003417 ≈292,634591
•a4= 292 α5≈1
0,634591 ≈1,575818
•a5= 1
π
[3,7,14,1,292,1].
3 + 1
7=22 7π
α= [a0, a1, a2, . . .]
(pn) (qn)
p0=a0, q0= 1 p1=a0a1+ 1 q1=a1,
n≥1
pn+1 =pnan+1 +pn−1
qn+1 =qnan+1 +qn−1
n≥0rnn α pn
qn.