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 everything in the World can be represented as relations between the natural numbers 1,2,3,…..But Hippasus of Metapontum discovered 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}.

Hippasus 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 when Hippasus revealed the secret, he was drowned. But his discovery was so devastating to the basic belief of the Pythagoreans that their society collapsed and was replaced by the geometric School of Euclide, which resolved 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, and not as a number.

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 =

1.414213562373095048801688724209698078569671875376948073176679737990
7324784621070388503875343276415727350138462309122970249248360558507
37212644121497099935831413222665927505592755799950501152782060571470
1095599716059702745345968620147285174186408891986095523292304843087
143214508397626036279952514079896872533965463318088296406206152585
2395054745750287759961729835575220337531857011354374603408498847160
386899970699,

we have that x^2=

1.99999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999
9999999999999998638103700279039354754492148156752071936433672239224
8627179189098787015809960232640597261312640760405691299950309295747
8318885969500708874056058336501652271573809445593320690045817264222
1739359695332425151587602336042729948891418035989710382049561848123
3332162516016097283137123064499497943653479698629776683334066577024
0318513306002427232125175273043547767486608089987807935797774759645
877082503170068870585486010 .

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} without 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…

Squarerootoftwo-Gate

This was Hippasus’ argument which was kept secret by the Pythagoraen society, and in a form of\sqrt{2}-gate led to the collapse of the society, when it leaked.

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.

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 in Chap 180 of MST.

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