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第10回数学カフェ「数学史」での発表資料です。 タレスから導入し、ユークリッドの『原論』の中身を概観します。
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2016年10/05開催の35時間ぶっ続け数学イベント「mathpower」内の企画「ロマンティック数学ナイト」で数学の自由さについて発表した内容です。
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2016年10/05開催の35時間ぶっ続け数学イベント「mathpower」内の企画「ロマンティック数学ナイト」で数学の自由さについて発表した内容です。
数学が自由すぎて地球がヤバい #ロマンティック数学ナイト
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級数を大改造!! 劇的ビフォーアフター
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数学が他の生物や宇宙人についても普遍的な理由と、そう信じる為に必要な前提条件を書いてみました
何故宇宙人も同じ数学に辿りつくか
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To succeed in your career, you need a strategy for sending out ripples of influence. Here are 12 ways you can start doing just that.
12 Ways to Increase Your Influence at Work
12 Ways to Increase Your Influence at Work
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数学のはじまり
1.
数学のはじまり 日曜数学会 キグロ Twitter :
@kiguro_masanao niconicoID : 1035113 2016/01/31 数学カフェ @dots. in 渋谷
2.
Q.世界初の数学者は誰? ミレトスのタレス (B.C. 650頃 ~
B.C. 546頃) A. 2016/01/31 数学カフェ @dots. in 渋谷
3.
2016/01/31 数学カフェ @dots.
in 渋谷 タレスは、世界で初めて 数学の定理を証明した! 数学には証明が必要不可欠 なぜタレスは世界初の数学者と呼ばれるのか?
4.
タレスが証明した定理 直径に対する円周角は直角である 2016/01/31 数学カフェ
@dots. in 渋谷 この定理の証明には、 「三角形の内角の和は180度」 「二等辺三角形の底角は等しい」 などの定理が使われる。
5.
2016/01/31 数学カフェ @dots.
in 渋谷 疑問
6.
2016/01/31 数学カフェ @dots.
in 渋谷 直径に対する円周角は直角である ↑ 二等辺三角形の底角は等しい ↑ ? ↑ ? ↑ …
7.
2016/01/31 数学カフェ @dots.
in 渋谷 どこまで遡ればよいのか?
8.
2016/01/31 数学カフェ @dots.
in 渋谷 公理までである! アレクサンドリアのユークリッド (B.C. 300頃)
9.
アレクサンドリアのユークリッド 2016/01/31 数学カフェ @dots.
in 渋谷 ・紀元前300年頃の数学者 ・アレクサンドリアのムセイオンという大学で 数学を教えていた。 ・生没年、出生地などは一切不明。 ・『原論』など、12冊の本を書いた。
10.
『原論』(ストイケイア) B.C. 300年頃にユークリッドが書いた初等数学の 入門書。
おそらく教科書だったと考えられている。 登場するすべての定理が、公理から証明されて いる。 2016/01/31 数学カフェ @dots. in 渋谷
11.
公理とは 証明せずとも正しいと認める事柄 例) 2点を通る直線を引ける 互いに重なり合うものは互いに等しい 2016/01/31 数学カフェ @dots.
in 渋谷
12.
直径に対する円周角は直角 ↑ 二等辺三角形の底角は等しい ↑ 2016/01/31 数学カフェ @dots.
in 渋谷 … ↑ 互いに重なり合うものは互いに等しい
13.
『原論』の中身 2016/01/31 数学カフェ @dots.
in 渋谷 ・第1巻 ・第2巻 ・第3巻 ・第4巻 ・第5巻 ・第6巻 ・第7巻 ・第8巻 ・第9巻 ・第10巻 ・第11巻 ・第12巻 ・第13巻 幾何学 幾何学 幾何学 幾何学 幾何学 幾何学 幾何学 幾何学的代数学 比例の理論 数論 数論 数論 無理数論 (平面図形) (円) (内接・外接) (面積比、 相似) (空間図形) (求積法) (正多面体)
14.
第1,3,4巻(平面図形、円、外接・内接) 現代の私たちが中学までで習う内容が大体出てくる。 例)三角形の合同条件、タレスの定理、三平方の定理など 第1~4巻の内容は、ピタゴラス学派や キオスのヒポクラテス(B.C.
430頃)の業績を 引用したと考えられている。 2016/01/31 数学カフェ @dots. in 渋谷 サモスのピタゴラス
15.
第5巻(比例の理論) 比の定義に始まり、25個の比の定理を証明している。 クニドスのエウドクソス(B.C.
400頃)の比例の理論に完全な 証明を与え、まとめたものと考えられている。 この巻の内容を受け、第6巻(相似、面積比)が展開される。 2016/01/31 数学カフェ @dots. in 渋谷
16.
第7,8,9巻(数論) 有名な「ユークリッドの互除法」や、「素数が無限に存在する ことの証明」が登場する。 等比数列の和や、完全数についての定理も登場する。
第7巻の最初で、「数」が定義されている。 2016/01/31 数学カフェ @dots. in 渋谷
17.
『原論』第7巻における数の定義 a. 単位とは存在するもののおのおのがそれによって1とよばれるも のである。(単位の定義) b. 数とは単位から成る多である。(数の定義) 2016/01/31
数学カフェ @dots. in 渋谷 (超訳) a. 「1」は単位である。 b. 数とは単位が集まったものである。 つまり「1」は数ではない!(当然ゼロや負の数の概念もない)
18.
第2巻(幾何学的代数学) 現在の代数学に相当する内容で、展開や因数分解の公式、 二次方程式の解法が書かれている。 「幾何学的代数学」と呼ばれる理由は、当時は文字式がなく、 すべて幾何学的に描かれているから。 (数が線分の長さで表されている) 2016/01/31
数学カフェ @dots. in 渋谷
19.
第2巻(幾何学的代数学) 2016/01/31 数学カフェ @dots.
in 渋谷 𝑎 + 𝑏 2 = 𝑎2 + 𝑏2 + 2𝑎𝑏 A C B 𝑎 = 𝐴𝐶 𝑏 = 𝐵𝐶
20.
第2巻(幾何学的代数学) 2016/01/31 数学カフェ @dots.
in 渋谷 𝑥2 = 𝑎 𝑎 − 𝑥2次方程式 の解 A C B 𝑎 = 𝐴𝐵 𝑥 = 𝐴𝐶
21.
実は、 第5巻(比例の理論)や第7~9巻(数論)にも、 わざわざ線分を使った挿絵を入れている。 2016/01/31 数学カフェ @dots.
in 渋谷
22.
2016/01/31 数学カフェ @dots.
in 渋谷 どうして幾何学を 使ったのか?
23.
それは、 無理数を扱うため! 2016/01/31 数学カフェ @dots.
in 渋谷
24.
無理数の発見 2016/01/31 数学カフェ @dots.
in 渋谷 タレスの弟子のピタゴラスの弟子のヒッパソスがB.C. 500頃に 「整数同士の比で表せない数(無理数)が存在すること」 を証明した。 無理数の発見は、当時の数学界を震撼させる大事件だった。
25.
無理数の発見 代数的に数を作る方法として、加減乗除しか知らなかった 当時の人々にとって、無理数を表現する方法がなかった。 一方、幾何学的な方法(作図)なら、無理数も簡単に作れた。
そのため、当時の人々は幾何学的代数学を扱った。 2016/01/31 数学カフェ @dots. in 渋谷
26.
第10巻(無理数論) 1巻まるまる無理数の解説に充てている。 全13巻中最も分厚く、登場する定理も最多の115個。
このことからも、当時の人々がいかに無理数を難物扱いし ていたかわかる。 2016/01/31 数学カフェ @dots. in 渋谷
27.
第11,12,13巻 (空間図形、求積法、正多面体) 立体の体積を求める方法などが解説されている。 『原論』の最後の定理は、 「正多面体は正四面体、正六面体、正八面体、正十二面体、 正二十面体の5種類しか存在しない」 2016/01/31
数学カフェ @dots. in 渋谷
28.
『原論』がのちの数学に与えた影響 2016/01/31 数学カフェ @dots.
in 渋谷 『改訂版 中学校 数学3』 数研出版
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