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In the 18th century, Johann Heinrich Lambert proved that the number (pi) is irrational. That is, it cannot be expressed as a fraction ''a''/''b'', where ''a'' is an integer and ''b'' is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven and Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. In 1882, Ferdinand von Lindemann proved that is not just irrational, but transcendental as well. == Lambert's proof == In 1761, Lambert proved that is irrational by first showing that this continued fraction expansion holds: : Then Lambert proved that if ''x'' is non-zero and rational then this expression must be irrational. Since tan(/4) = 1, it follows that /4 is irrational and therefore that is irrational. A simplification of Lambert's proof is given below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「proof that π is irrational」の詳細全文を読む スポンサード リンク
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