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.
seeAlso
primaryTopic
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.
has abstract
Aunque la constante matemática ...... da, encontrada por Ivan Niven.
@es
In the 18th century, Johann He ...... l, but transcendental as well.
@en
No século XVIII, Johann Heinri ...... como também é transcendental.
@pt
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。
@ja
Wikipage page ID
12,792,547
Wikipage revision ID
744,655,210
subject
comment
Aunque la constante matemática ...... da, encontrada por Ivan Niven.
@es
In the 18th century, Johann He ...... l, but transcendental as well.
@en
No século XVIII, Johann Heinri ...... é devida a Miklós Laczkovich.
@pt
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。
@ja
label
Demostración de la irracionalidad de π
@es
Proof that π is irrational
@en
Prova da irracionalidade de π
@pt
البرهان على أن π عدد غير جذري
@ar
円周率の無理性の証明
@ja