# Why is $\pi$ irrational?

I've a few questions in mind:

• Why is $\pi$ irrational?
• If it is, then how can 2 rational quantities (circumference, diameter) can produce irrational number?
• How are we able to determine digits of $\pi$ accurately?
• The diameter of the unit circle is indeed rational. Now, what makes you think the circumference is also rational? – guest Sep 12 '15 at 19:00
• @MehulMohan: If you agree that there do exist irrational numbers, then obviously you can have lines whose length is an irrational number. – Zev Chonoles Sep 12 '15 at 19:04
• There you go : en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational – krirkrirk Sep 12 '15 at 19:04
• – Mauro ALLEGRANZA Sep 12 '15 at 19:05
• The diagonal of the unit square is already a straight line, no stretching required, and has irrational length by the Pythagorean theorem. – guest Sep 12 '15 at 19:06

The name "irrational number" has an ancient source...

It was an (implicit) assumption common to all "archaic" Greek mathematics that given two magnitudes, e.g. to segments of lenght $a$ and $b$, it is always possible to find a segment of "unit lenght" $u$ such that it measure both, i.e. such that [using modern algebraic formulae which are totally foreign to Greek math] :

$a = n \times u$ and $b = m \times u$, for $n,m \in \mathbb N$.

From the above instance of the assumption, it follows that :

$a/b = (n \times u) / (m \times u) = n/m$.

The assumption amounts to saying that the ratio between two magnitudes is always a ratio between numbers (i.e. in modern terms : a rational number; but note that for Greek math the only numbers are the natural ones and they are distinguished from magnitudes : a segment, a square, ... which are "measured" by numbers).

The discovery of the existence of irrational magnitudes, through the proof that the case where $a=1$ is the side of the unit square and $b=\sqrt 2$ is the diagonal is not expressible as a ratio between (natural) numbers, leads Greek math to the withdrawal of the "commensurability assumption" and to the axiomatization of geometry.

Those couples of magnitudes was called "incommensurable" (i.e. without common measure).

For the same reason, $\sqrt 2$ is an irrational number, exactly because the ratio "diagonal/side" is not expressible as a ratio between natural numbers.

The irrationality of $\pi$ was proved by Johann Heinrich Lambert in 1761.