Bukti bahwa e irasional

Templat:Konstanta matematika Bilangan e diperkenalkan oleh Jacob Bernoulli pada tahun 1683. Setengah abad kemudian, Euler (yang merupakan siswa adik Jacob, Johann) berhasil membuktikan bahwa e adalah bilangan irasional, atau dalam kata lain tidak dapat dinyatakan sebagai hasil bagi dua bilangan bulat.

Bukti Euler

Euler menulis bukti pertama irasionalitas e pada tahun 1737 (tetapi tulisannya baru diterbitkan tujuh tahun kemudian).^[1]^[2]^[3] Ia menghitung e sebagai pecahan berlanjut, yaitu:

e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,\ldots ,2n,1,1,\ldots ].

Bukti Fourier

Bukti yang paling dikenal adalah reductio ad absurdum Joseph Fourier,^[4] yang didasarkan pada pernyataan berikut:

e=\sum _{n=0}^{\infty }{\frac {1}{n!}}\cdot

Pada awalnya e diasumsikan sebagai bilangan rasional dengan bentuk ^a⁄_b. Perlu dicatat bahwa b tidak mungkin sama dengan satu karena e bukan bilangan bulat. Dari pernyataan di atas dapat ditunjukkan bahwa e hanya berada di antara 2 dan 3.

2=1+{\tfrac {1}{1!}}\ \ <\ \ e=1+{\tfrac {1}{1!}}+{\tfrac {1}{2!}}+{\tfrac {1}{3!}}+\cdots \ \ <\ \ 1+(1+{\tfrac {1}{2}}+{\tfrac {1}{2^{2}}}+{\tfrac {1}{2^{3}}}+\cdots )\ \ =\ \ 3

Bukti lain

Bukti lain^[5] dapat diperoleh dengan mencatat bahwa

(b+1)x=1+{\frac {1}{b+2}}+{\frac {1}{(b+2)(b+3)}}+\cdots <1+{\frac {1}{b+1}}+{\frac {1}{(b+1)(b+2)}}+\cdots =1+x,

Pernyataan berikut juga dapat dijadikan sebagai bukti:^[6]

{\frac {1}{e}}=e^{-1}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\cdot

Lihat pula

Teorema Lindemann–Weierstrass

Catatan kaki

^ Euler, Leonhard (1744). "De fractionibus continuis dissertatio" (PDF). Commentarii academiae scientiarum Petropolitanae. 9: 98–137.
^ Euler, Leonhard (1985). "An essay on continued fractions". Mathematical Systems Theory. 18: 295–398. doi:10.1007/bf01699475.
^ Sandifer, C. Edward (2007). "Chapter 32: Who proved e is irrational?". How Euler did it. Mathematical Association of America. hlm. 185–190. ISBN 978-0-88385-563-8. LCCN 2007927658.
^ de Stainville, Janot (1815). Mélanges d'Analyse Algébrique et de Géométrie. Veuve Courcier. hlm. 340–341.
^ MacDivitt, A. R. G.; Yanagisawa, Yukio (1987), "An elementary proof that e is irrational", The Mathematical Gazette, London: Mathematical Association, 71 (457): 217, doi:10.2307/3616765, JSTOR 3616765
^ Penesi, L. L. (1953). "Elementary proof that e is irrational". American Mathematical Monthly. Mathematical Association of America. 60 (7): 474. doi:10.2307/2308411. JSTOR 2308411.

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