長さ√aを作図せよ「笑わない数学」(11/8、NHK) |
|
|
L⋍√(a^2+a) |
√(a+1) |
|
|
|
|
|
|
|
x⋍√a |
|
|
|
|
|
|
K点 a |
1 |
|
|
|
|
|
|
1:x⋍x:a |
(相似) |
|
|
|
|
|
|
x^2⋍a |
∴x⋍√a |
|
|
|
斜辺 |
|
|
s(零点) |
1-s⋍a⋍2k+1 |
|
x⋍√a |
|
a(a+1) |
ΔL |
Δb |
-2 |
3 |
⋍√ |
3 |
⋍√ |
12 |
|
|
-4 |
5 |
⋍√ |
5 |
⋍√ |
30 |
18 |
|
-6 |
7 |
⋍√ |
7 |
⋍√ |
56 |
26 |
8 |
-8 |
9 |
⋍√ |
9 |
⋍√ |
90 |
34 |
8 |
-10 |
11 |
⋍√ |
11 |
⋍√ |
132 |
42 |
8 |
-12 |
13 |
⋍√ |
13 |
⋍√ |
182 |
50 |
8 |
-14 |
15 |
⋍√ |
15 |
⋍√ |
240 |
58 |
8 |
-16 |
17 |
⋍√ |
17 |
⋍√ |
306 |
66 |
8 |
-18 |
19 |
⋍√ |
19 |
⋍√ |
380 |
74 |
8 |
-20 |
21 |
⋍√ |
21 |
⋍√ |
462 |
82 |
8 |
α⋍1/2+δ⒤、 |
β⋍1/2-δ⒤とおく |
|
|
|
|
(x-α)(x-β)⋍ |
x^2-(α+β)x+αβ |
|
|
|
|
∵α+β⋍1、αβ⋍1/4+δ^2 |
|
|
|
|
|
|
⋍x^2-x+1/4+δ^2 |
|
|
|
|
|
|
⋍(x-1/2)^2+δ^2 |
|
|
|
|
|
|
(s)ー2-4-6ー8ー10… |
|
|
|
|
|
|
(1-s)3,5,7,9.11,… |
|
|
|
|
|
|
sと1-sを入れ替えても成立する,s=1+ℓとおくと |
|
|
|
∴s(1-s)=ーℓ(ℓ+1)=ーα(α+1)ルジャンンドル |
|
|
|
|
x(x+1)+121[(2k+1.25)-δ]⋍0 |
|
|
|
|
|
|
x^2+x+3.25-δ⋍0 ∴x⋍(-0.5)±[11√3-δ]⒤ |
|
|
|
p⋍(1/2)±11⒤[√(1-s)-(q/11)]⋍(1/2)±11⒤[L+S] |
|
|
|