Proof that π is irrational
In the 18th century, Johann Heinrich Lambert proved that the number π (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 π is not just irrational, but transcendental as well.
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Proof that π is irrational
In the 18th century, Johann Heinrich Lambert proved that the number π (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 π is not just irrational, but transcendental as well.
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Aunque la constante matemática ...... da, encontrada por Ivan Niven.
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In the 18th century, Johann He ...... l, but transcendental as well.
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No século XVIII, Johann Heinri ...... como também é transcendental.
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درس العدد π منذ العصور القديمة ...... امبرت على كون π عددا غير جذري.
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円周率の無理性の証明（えんしゅうりつのむりせいのしょうめい） ...... 最も簡単な証明は、初等的な微分積分学のみを用いるものである。
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Aunque la constante matemática ...... da, encontrada por Ivan Niven.
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In the 18th century, Johann He ...... l, but transcendental as well.
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No século XVIII, Johann Heinri ...... é devida a Miklós Laczkovich.
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درس العدد π منذ العصور القديمة ...... امبرت على كون π عددا غير جذري.
@ar
円周率の無理性の証明（えんしゅうりつのむりせいのしょうめい） ...... 最も簡単な証明は、初等的な微分積分学のみを用いるものである。
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Demostración de la irracionalidad de π
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Proof that π is irrational
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Prova da irracionalidade de π
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البرهان على أن π عدد غير جذري
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円周率の無理性の証明
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