The Secret Pythagorean Society

  • There is geometry in the humming of the strings, there is music in the spacing of the spheres…There is nothing so easy but that it becomes difficult when you do it reluctantly. (Pythagoras)

The secret Pythagorean Society in Greece 400 BC led by Pythagoras was based on the belief that everythingin the World can be represented as relations between the natural numbers 1,2,3,…..But one day somebodydiscovered that \sqrt{2}, the length of the diagonal of a square with side 1, cannot be expressed as a rational number as the quotient of two natural numbers, e.g. as \frac{22}{7} = 3 \frac{1}{7}.

It was thus discovered that \sqrt{2} is not a rational number, that is, that \sqrt{2} is an irrational number. This was first kept as a secret, but like in Climategate a whistleblower revealed the secret public. This was so devastating to the basic belief of the Pythagoreans that their society collapsed, and was replaced by the geometric School of Euclide, which resolve the difficulty of the irrationality of \sqrt{2}, by simply defining \sqrt{2} geometrically as the length of the diagonal of a square with side 1.

The geometric school of Euclide propagated by Aristotle ruled science for almost 2000 years until Descartes in the 17th century replaced geometry by analytic geometry based on numbers thus returning to Pythagoras, and initiating the scientific revolution transforming medieval society into the industrial society of the 19th century leading up to the information society of the late 20th and 21st century, which you are lucky to have been born into. A Pythagorean society based on numbers!

You may recall from school that \sqrt{2}\approx 1.41, but computing 1.41^2=1.9881, we see that \sqrt{2} is not exactly equal to 1.41. A better guess is 1.414, but then we get 1.414^2=1.999386. No matter how many decimals of x we add, x^2 will not become exactly equal 2. For,example, with 415 decimals and x =

1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503 8753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275 579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430 87143214508397626036279952514079896872533965463318088296406206152583523950547457502877599 617298355752203375318570113543746034084988471 60386899970699,

we have that x^2=

1.999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999863810370027  90393547544921481567520719364336722392248627179189098787015809960232640597261312640760405 6912999503092957478318885969500708874056058336501652271573809445593320690045817264222173 935969533242515158760233604272994889141803598971038204956184812333321625160160972831371230 64499497943653479698629776683334066577024031851330600242723212517527304354776748660808998 7807935797774759645877082503170068870585486010 .

No matter how many decimals we take in a guess x of \sqrt{2}, we never get a number which squared gives exactly 2.

Can you prove that? Can you reveal the secret of the Pythagorean society, the knowledge that ended the reign of the Pythagoraens?

Hint: Assume that \sqrt{2}=\frac{p}{q} with all common factors of 2 in the natural numbers p and q. Then consider the equation 2q^2=p^2 obtained by squaring and multiplying by 2q^2. Conclude that p must contain the factor 2 and thus p^2 the factor 4=2\times 2=2^2. Conclude that q must contain a factor 2, which contradicts the assumption that p and q have no common factor 2.

A rational number has a finite or periodic (repeating) decimal expansion, and an irrational number has a neverending non-periodic (non-repeating) decimal expansion. Since it is impossible to compute all decimals of an irrational number, we must acknowledge that an irrational number really is “irrational” in the sense that its exact value cannot be pinned down in the same precise sense as for a natural or rational number. Irrational numbers satisfy the same computational rules as rational numbers, but their exact values are hidden to our inspection: there is always another decimal to be computed/discovered somehow. In particular, given two irrational numbers, it may be impossible to decide if they are exactly equal (all decimals being equal) or not. For example, the statement 0.99999999...=1 is correct only if the dots indicate a (periodic) never-ending squence of the digit 9.

Another  irrational number is \pi =3,14159265..., which has been computed 2 billion digits, but that is not the whole truth…


This was the argument which was kept secret by the Pythagoraen society, and in a form of\sqrt{2}-\emph{gate} led to the collapse of the society, when it leaked. In Climategate the secret revealed is that Anthropogenic Global Warming AGW, is what it says, namely invented man-made warming: Scientific evidence that CO2 causes catastrophical global warming, appears to be missing. The scientific evidence that it does, appears to be fabricated.

True or not true? Can mathematics give an answer?

Watergate was the political scandal in the US in the 1970s caused by the break-in into the Democratic National Committee headquarters at the Watergate office complex in Washington, D.C, which led to the resignation of President Richard Nixon, and indictment and conviction of several Nixon administration officials. Compare with the Lewinsky scandal.

Climategate can be seen as a test of the basic principles of science as well as democracy: open data and free debate.

Pythagoras and Music

Pythagoras belief that the World is based on numbers, was supported by his discovery that the ratio offrequencies of musical scales are simple rational numbers such as \frac{3}{2} for a

fifth (G in a scale of C), \frac{9}{8} for a second (D), \frac{27}{16} for a sixth (A), and \frac{4}{3} for a fourth (F).

Babylonian approximation of \sqrt{2}

A formula for computing \sqrt{2}. From where?

Read More

  • \hyperref[chaptersqrt2]{The Squareroot of Two}.

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