Proof that π is irrational

<p>In the 1760s, <a href="/facts/Johann_Heinrich_Lambert/CG3IHVB9">Johann Heinrich Lambert</a> was the first to prove that the <a href="/facts/Pi/vC0UBj1q">number π</a> is <a href="/facts/Irrational_number/Psv19TET">irrational</a>, meaning it cannot be expressed as a fraction 
  
    
      
        a
        
          /
        
        b
      
    
    {\displaystyle a/b}
  
, where 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 are both <a href="/facts/Integer/PUwwrawx">integers</a>. In the 19th century, <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a> found a proof that requires no prerequisite knowledge beyond basic <a href="/facts/Calculus/MEmUTYoz">calculus</a>. Three simplifications of Hermite's proof are due to <a href="/facts/Mary_Cartwright/5GPGzsX2">Mary Cartwright</a>, <a href="/facts/Ivan_Niven/SCt1FK9p">Ivan Niven</a>, and <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Nicolas Bourbaki</a>. Another proof, which is a simplification of Lambert's proof, is due to <a href="/facts/Mikl%C3%B3s_Laczkovich/NPJAshyl">Miklós Laczkovich</a>. Many of these are <a href="/facts/Proof_by_contradiction/vHtFzFQ5">proofs by contradiction</a>.
</p><p>In 1882, <a href="/facts/Ferdinand_von_Lindemann/xMlaFGKj">Ferdinand von Lindemann</a> proved that 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 is not just irrational, but <a href="/facts/Lindemann%E2%80%93Weierstrass_theorem/BafvlpWm">transcendental</a> as well.
</p>

Proof that π is irrational open-in-new

Proof that π is irrational