なんで 0.99999… = 1 なの?

昔、疑問に思っていた事を、ふと思い出した

0.99999… = 1

これは本当だろうか?
正しい気もした、間違っているきもした。

  1. 1 / 3 = 0.33333…
  2. 0.33333… x 3 = 0.99999…
  3. 1 / 3 x 3 = 1
  4. よって、0.99999… = 1 は正しい
  1. 0.9 < 1
  2. 0.99 < 1
  3. 0.999 < 1
  4. このように、0.9 の後にどれだけ 9 を付け足しても 1 より小さい
  5. よって、0.99999… < 1
  6. よって、0.99999… = 1 は間違い

どちらの考え方も一見正しいように見えるが、正反対の結論になってしまった。

実は、この議論を始めるために最初に確認しなくてはいけない事がある。
0.99999… とは何だう?

0.9 の後に 9 を無限個、続けた数字?
「無限」って整数だったの?
整数って事は、偶数か奇数のどちらかだ。
「無限」って偶数?奇数?

「無限」について数学的に正しく扱う事は簡単ではない。
だが、ここが上の議論で一番重要な点だ。
0.33333… や 0.99999… を正しく定義できれば、このパラドックスは理解できる。

「無限」という言葉を使うと難しくなるので、その言葉を使わずに 0.99999… を定義してみよう
以下のような数列 an を考える。

a1 = 0.9
a2 = 0.99
a3 = 0.999

ちなみに、数学的に厳密に書くと

a1 = 0.9
ak+1 = ak + (9/10k+1)

となる。
(まあ、表現が難しくなっただけで今回の記事には関係無いが)

さて、この数列 an は以下の 2 個の特徴をもつ。

i) どんな自然数 n においても、an < 10
ii) どんな自然数 n においても、an < an+1

i) と ii) を真面目に証明するとこの記事が長くなりすぎるので、ここは「当たり前」で許してほしい。

では、i) と ii) の条件を満たす数列のグラフはどんな形になるだろう?
(今は an の詳しい事は一回忘れて、「i) と ii) の条件を満たす数列」のグラフを考えよう。)
例えばこんな形になるのではないか?

上図を見ると直観的に分かるかもしれないが、an は「n を大きくするとある数に近づく」という性質を持っている。

なぜなら
ii) どんな自然数 n においても、an < an+1
より、an は、n を大きくすると
「際限なく大きくなる」
「上限となる、ある数に近づく」
のどちらかである。

そして、「際限なく大きくなる」は
i) どんな自然数 n においても、an < 10
と明らかに矛盾する。

この「上限となる、ある数 x」を先ほどの図に加えるとこのようになる。

では、この x について厳密な定義をしてみよう。
(「n を大きくすると近づく数」という表現は数学的には曖昧すぎる)

先ほど an について
i) どんな自然数 n においても、an < 10
と書いた。
当たり前だが、この条件はもっと厳しくする事ができる。
例えば、

i’) どんな自然数 n においても、an < 5
i”) どんな自然数 n においても、an < 3

これはどちらも正しい。

この条件を一番厳しくした時の値で x を定義する。
i), i’), i”) では 10, 5, 3 という数字を使ったが、もっと小さい数を使う事も出来る。
この使用可能な数の中で、最小値を x とするのだ。
正確に書くと以下のようになる。

a)
どんな自然数 n においても、an < k を満たす数 k は 2 個以上存在する。
x はそのような k の中の最小値

別の言い方をすると、こんな風になるかもしれない。

b)
どんな自然数 n においても、an < x。
ただし、y < x を満たす全ての数 y において、 y < an となる n をが 1 個以上存在する

x の定義は上記の a), b) どちらでもよい。

数学的にはいい加減だが少しだけ直観的な表現にすると、

c)
x とは 0.9 の後に 9 を多く続けた数より、少しだけ大きい数
(「多く」とは、100万でも 10億でも良い)

とも言えるだろう。
「少しだけ大きい」という部分がポイントだ。

そして、この数 x を 0.99999… と名付けよう。

すると、「0.99999… = 1」とは、

a) の表現を用いると
0.9 の後に 9 をいくつ続けた数よりも大きい数は 2 個以上存在する。
そのような数の最小値は 1 である

b) の表現を用いると
0.9 の後に 9 をいくつ続けても 1 より小さい。
1 より小さい全ての数 y について、0.9 の後に 9 をいくつか続けると y より大きくなる事がある

c) の表現を用いると
0.9 の後に 9 を多く続けた数より、少しだけ大きい数は 1 である

という事になる。

どうだろう。
「”0.9 の後に 9 を無限個繋げた数” = 1」
という表現とは全然違う印象を受けるのではないか?
そして、この表現ならば直観的にも簡単に納得できるだろう。

では、最初のパラドックスに戻ってみよう。

長くなるので、以下では c) の表現のみを用いてみる

  1. 1 / 3 = 0.33333…
    1 / 3 = “0.3 の後に 3 を多く続けた数より、少しだけ大きい数”
  2. 0.33333… x 3 = 0.99999…
    “0.3 の後に 3 を多く続けた数より少し大きい数” x 3
    = “0.9 の後に 9 を多く続けた数より少し大きい数”
  3. 1 / 3 x 3 = 1
  4. よって、0.99999… = 1 は正しい
    “0.9 の後に 9 を多く続けた数より少し大きい数” = 1 は正しい。

これは全て正しい。

1 / 3 = "0.3 の後に 3 を多く続けた数より、少しだけ大きい数"

ここが直観に反するかもしれないが、実際には正しい。
(数学的に厳密な証明は、今回は控える)

  1. 0.9 < 1
  2. 0.99 < 1
  3. 0.999 < 1
  4. このように、0.9 の後にどれだけ 9 を付け足しても 1 より小さい
  5. よって、0.99999… < 1
    “0.9 の後に 9 を多く続けた数より少し大きい数” < 1)
  6. よって、0.99999… = 1 は間違い
    “0.9 の後に 9 を多く続けた数より少し大きい数” = 1 は間違い

これは間違っている。

0.9 の後にどれだけ 9 を付け足しても 1 より小さい
"0.9 の後に 9 を多く続けた数より少し大きい数" < 1

この間に議論の飛躍がある。
ここで間違えているのだ。

この疑問を始めて感じたのは、小学生の時だったと思う。
その際、周りの大人たちに聞いたが納得のいく解答は得られなかった。
(今にして思うと、大人にも良く分からなかったのかもしれない)

小学生だったころの自分にも分かるような説明を考えてみたが、どうだろうか。

Author: wbcchsyn

未来のスーパーエンジニア とりあえず、新しい物が大好き

192 thoughts on “なんで 0.99999… = 1 なの?”

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